A logicincorporated gene regulatory network deciphers principles in cell fate decisions
Abstract
Organisms utilize gene regulatory networks (GRN) to make fate decisions, but the regulatory mechanisms of transcription factors (TF) in GRNs are exceedingly intricate. A longstanding question in this field is how these tangled interactions synergistically contribute to decisionmaking procedures. To comprehensively understand the role of regulatory logic in cell fate decisions, we constructed a logicincorporated GRN model and examined its behavior under two distinct driving forces (noisedriven and signaldriven). Under the noisedriven mode, we distilled the relationship among fate bias, regulatory logic, and noise profile. Under the signaldriven mode, we bridged regulatory logic and progressionaccuracy tradeoff, and uncovered distinctive trajectories of reprogramming influenced by logic motifs. In differentiation, we characterized a special logicdependent priming stage by the solution landscape. Finally, we applied our findings to decipher three biological instances: hematopoiesis, embryogenesis, and transdifferentiation. Orthogonal to the classical analysis of expression profile, we harnessed noise patterns to construct the GRN corresponding to fate transition. Our work presents a generalizable framework for topdown fatedecision studies and a practical approach to the taxonomy of cell fate decisions.
eLife assessment
The study presented in this manuscript makes important contributions to our understanding of cell fate decisions and the role of noise in gene regulatory networks. Through computational and theoretical analysis, the authors provide solid support for distinguishing distinct driving forces behind fate decisions based on noise profiles and reprogramming trajectories. While acknowledging the potential limitations of small gene regulatory networks in capturing the richness of wholetranscriptome sequencing datasets, this study offers a creative approach for formulating hypotheses about gene regulation during stem cell differentiation using singlecell sequencing data.
https://doi.org/10.7554/eLife.88742.3.sa0Introduction
Waddington’s epigenetic landscape is a fundamental and profound conceptualization of cell fate decisions (Waddington, 1957). Over decades, this insightful metaphor has facilitated researchers to distill a myriad of models regarding cell fate decisions (MacArthur, 2022; Schiebinger et al., 2019; Olsson et al., 2016; Bhattacharya et al., 2011; Fisher and Merkenschlager, 2010; Hota et al., 2022; Huang, 2012; Li et al., 2021). While introducing various quantitative models and dissecting diverse fatedecision processes, researchers have further elaborated the Waddington landscape (Shakiba et al., 2022; Loeffler and Schroeder, 2019; Coomer et al., 2022; Shi et al., 2022; Stanoev and Koseska, 2022; Glauche and Marr, 2021). An outstanding question is whether the landscape is static or not, i.e., whether cell fate decisions are driven by noise or signal (Stanoev and Koseska, 2022; Simon et al., 2018; ZernickaGoetz and Huang, 2010). On one hand, some perspectives hold that cells reside in a stationary landscape, where decisions are made by switching through discrete valleys (Lord et al., 2019; Desai et al., 2021), as a result of gene expression noise (Chang et al., 2008; Guillemin and Stumpf, 2021) (termed as ‘noisedriven’; Figure 1A). Meanwhile, some researchers argued that the epigenetic landscape is dynamic during fate decisions. That is, the distortion of the landscape orchestrates fate transitions (Hota et al., 2022; Zhu et al., 2022; Li et al., 2020) and is driven by extrinsic signals (termed as ‘signaldriven’; Figure 1B).
Under the noisedriven mode, the bias of cell fate decisions largely depends on the spontaneous heterogeneity of gene expressions in the cell population (Wheat et al., 2020; Kovary et al., 2018). Consequently, the initial cellular state predominantly impacts the direction of the fate decision. Chang et al., 2008 uncovered that hematopoietic stem cell (HSC) population possesses intrinsic and robust heterogeneity of Scal1 expression (also known as Ly6a [van de Rijn et al., 1989]). Notably, populations with discrete expression levels of Scal1 confer different propensities for downstream lineage commitment. Considering the signaldriven mode, cell fates are tightly steered by extrinsic signals (e.g., cytokines, chemical molecules, mechanical strength, and genetic operations) that reshape the landscape (Figure 1B). In this circumstance, the impact of the initial state on fate decisions is relevantly inconsequential. Additionally, due to the accessibility of signal manipulation, the signaldriven mode has been widely utilized for cell fate engineering (Xu et al., 2015; Del Vecchio et al., 2017), leading to invitro induction systems centered on induced pluripotent stem cells (iPSC) for obtaining desired cell types (Ng et al., 2021). Recently, researchers reported a ‘fatedecision abduction’ of erythroidtomyeloid transdifferentiation induced by various types of cancer, which facilitates tumor escape from the individual’s immune system (Long et al., 2022). Collectively, driving forces couple the foundational and crucial features of fate decisions, serving as an essential basis for further decoding fate decisions and interpreting the development of organisms (Simon et al., 2018). By examining the two driving modes, we can gain a better understanding and characterization of cell fate decisions, including invivo cell differentiation, oncogenesis, and invitro reprogramming systems.
Nevertheless, the driving forces that underlie fate decisions remain largely elusive. The intricate nature of GRNs presents a challenge in deciphering driving modes. It has been generally acknowledged that corresponding core GRNs orchestrate cell fate decisions (Qiu et al., 2022; Graf and Enver, 2009), where the lineagespecifying TFs interact to implement fatedecision procedures. Furthermore, researchers transferred specific TFs into donor cells to reconfigure the intracellular GRNs for acquiring cell types of interest (Hammelman et al., 2022; Mazid et al., 2022). Although some studies suggested that perturbation of a single TF is sufficient to transform certain cell fates (Ng et al., 2021), a large number of TFs are inevitably involved in most differentiation/reprogramming processes (Hersbach et al., 2022). In particular to orchestrate decisions among multiple cell fates, it is necessary for TFs to regulate target genes cooperatively (Trojanowski and Rippe, 2022). As crucial determinants of cell fates, TFs function via binding to cisregulatory elements (CRE, e.g., promoter and enhancer). CREs of a single gene in metazoans can simultaneously accommodate numerous TFs (Lambert et al., 2018; Reiter et al., 2017). While experimental protocols have been developed to assess TF binding and onetoone up or downregulatory relationships, it is more challenging to quantify these combinatorial regulations. For instance, given two factors activate and inhibit the same target gene, respectively, does the target gene turn on or off when both factors are present in its CREs?
Computational approaches in systems biology can be utilized to tackle complex networks (Macarthur et al., 2009; Krumsiek et al., 2011; Collombet et al., 2017; Moignard et al., 2015; Kirouac et al., 2009). A concise GRN model typically entails the following two elements. The element 1 is the topology. Much research efforts have been devoted to investigating network topologies on cellular behaviors, e.g., toggle switch (Graf and Enver, 2009; Davies et al., 2021), and feedforward loop (Kittisopikul and Süel, 2010; Oliver Metzig et al., 2020; Sciammas et al., 2011). In particular, the CrossInhibition with Selfactivation (CIS) network is one of the most studied twonode GRNs in cell fate decisions (Orkin and Zon, 2008; Kato and Igarashi, 2019), with examples found in Gata1PU.1 and FLI1KLF1 in hematopoiesis (Hoppe et al., 2016; Palii et al., 2019), NanogGata6 and Oct4Cdx2 in gastrulation (Simon et al., 2018; Zhou and Huang, 2011), and Sir2 and HAP in yeast aging (Li et al., 2020). In this topology, two lineagespecifying factors inhibit each other while active themselves. For example, in the wellknown Gata1PU.1 circuit, Gata1 directs fate of megakaryocyteerythroid progenitor (MEP), and PU.1 (also known as SPI1 in humans) specifies the fate of granulocytemonocyte progenitor (GMP) (Orkin and Zon, 2008). Namely, the antagonism of two TFs implicates two cell fates in competition with each other.
The another element is the logic for regulatory functions (Ocone et al., 2015; Matsuura et al., 2018). Exemplified by the CIS network, each node (e.g., X and Y) receives the activation by itself and inhibition by the counterpart. Hence there is naturally the logic function between these two inputs. Given the logic function is AND, in the context of biological mechanism of regulation by TFs, X gene expresses only when X itself is present in X’s CREs but Y is not. Researchers observed in the E. coli lac operon system that changes in one single base can shift the regulatory logic significantly, suggesting that logic functions of GRNs can be adapted on the demand of specific functions in organisms (Collombet et al., 2017; Mayo et al., 2006; Buchler et al., 2003). Additionally, considering that the combination and cooperativity of TFs are of great significance in development (Zaret, 2020; Spitz and Furlong, 2012; Balsalobre and Drouin, 2022), theoretical investigation of the logic underlying GRNs should be concerned in cell fate decisions. However, despite the existence of large number of mathematical models on fate decisions, the role played by the regulatory logic in cell fate decisions is still obscure. Some theoretical studies put emphasis on specific biological instances, adopting logic functions that best fit the observations derived from experiments (Hota et al., 2022; Krumsiek et al., 2011; Huang et al., 2007). As a result, the models incorporated different regulatory logic received limited attention. Other research assigned logic to largescale multinode GRNs, confining the interpretation of the role of logic (Moignard et al., 2015; Wu et al., 2008). Collectively, the bridge between the logic of nodes in GRNs and cell fate decisions has not yet been elucidated systematically and adequately. Current research already encompassed a wealth of cell fate decisions: embryogenesis (Xu et al., 2022; Yu et al., 2022; Simunovic et al., 2022), lineage commitment (Palii et al., 2019; Pei et al., 2020; Psaila et al., 2020), oncogenesis (AguadéGorgorió et al., 2021; Yang et al., 2022; Uthamacumaran, 2021), invitro reprogramming (Tang et al., 2022; Liang et al., 2019; Yu et al., 2021), and largescale perturbations (Ng et al., 2021; Dixit et al., 2016; Joung et al., 2023; Replogle et al., 2022). Analogous to the effort on taxonomy of cell types and tumors (Domcke and Shendure, 2023), how cell fate decisions can be classified and distilled to the common properties is a challenge for further exploring systematically and application on fate engineering (MacArthur, 2023; Casey et al., 2020).
In this work, we integrated the fatedecision modes (noisedriven/signaldriven) and the classical logic operations (AND/OR) underlying GRNs in a continuous model. Based on our model, we investigated the impact of distinct logic operations on the nature of fate decisions with driving modes in consideration. Additionally, we extracted the difference in properties between the two driving modes. We unearthed that in the noisedriven stem cells, regulatory logic results in the opposite bias of fate decisions. We further distilled the relationship among noise profiles, logic motifs, and fatedecision bias, showing that knowledge of two of these allows inference of the third heuristically. Under the signaldriven mode, we identified two basic patterns of cell fate decisions: progression and accuracy. Moreover, based on our findings in silico, we characterized cell fate decisions in hematopoiesis and embryogenesis and unveiled their decision modes and logic motifs underlying GRNs. Ultimately, we applied our framework to a reprogramming system. We deciphered the driving force of this transdifferentiation, and utilized noise patterns for nominating key regulators. We underscored that clustering of gene noise patterns is an informative approach to investigate highthroughput datasets. Together, we underlined regulatory logic is of the significance in cell fate decisions. Our work presents a generalizable framework for classifying cell fate decisions and a blueprint for circuit design in synthetic biology.
Results
Section 1: Mathematical model of the CIS network with logic motifs
Binary treelike cell fate decisions are prevalent in biological systems (Zhou and Huang, 2011; Domcke and Shendure, 2023; Stadler et al., 2021; Macnair et al., 2019), orchestrated by a series of the CIS networks. Accordingly, we developed our ordinary differential equations (ODE) model based on this paradigmatic and representative topology (Equations 1 and 2; see ‘Materials and methods’ for details).
X and Y are TFs in the CIS network. n_{1} and n_{2} are the coefficients of molecular cooperation. k_{1}k_{3} in Equation 1 and k_{4}k_{6} in Equation 2 represent the relative probabilities for possible configurations of binding of TFs and CREs. (Figure 2A). d_{1} and d_{2} are degradation rates of X and Y, respectively. Here, we considered a total of four CRE’s configurations as shown in Figure 2A (i.e., TFs bind to the corresponding CREs or not, 2^{2}=4). Accordingly, depending on the transcription rates (i.e., r_{0}^{x}, r_{1}, r_{2}, r_{3} in Equation 1, similarly in Equation 2) of each configuration, we can model the dynamics of TFs in the SheaAckers formalism (Shea and Ackers, 1985; Olariu and Peterson, 2019).
Thus, the distinct logic operations (AND/OR) of two inputs (e.g., activation by X itself and inhibition by Y) can be further implemented by assigning the corresponding profile of transcription rates in four configurations (Figure 2A). From the perspective of molecular biology, the regulatory logics embody the complicated nature of TF regulation that TFs function in a contextdependent manner. Considering the CIS network, when X and Y bind respective CREs concurrently, whether the expression of target gene is turned on or off depends on the different regulatory logics (specifically, off in the AND logic and on in the OR logic; Figure 2A). Notably, instead of exploring the different logics of one certain gene (Kittisopikul and Süel, 2010), we focus on different combinations of regulatory logics due to dynamics in cell fate decisions is generally orchestrated by GRNs with multiple TFs.
Benchmarking the Boolean models with different logic motifs (Figure 2B; see ‘Materials and methods’), we reproduced the geometry of the attractor basin in the continuous models resembling those represented by corresponding Boolean models (Figure 2C; see ‘Materials and methods’). Under double AND and double OR motifs (termed as ANDAND motif and OROR motif, respectively), there are typically three stable steady states (SSS) in the state spaces (Figure 2C): two attractors near the axes representing the fate of lineage X (denoted as LX, X^{high}Y^{low}) and the fate of lineage Y (denoted as LY, X^{low}Y^{high}), and the attractor in the center of the state space representing stem cell fate (denoted as S).
Evidently, the stem cell states exhibit different expression patterns between the two logic motifs. Stem cells in the ANDAND motif do not express X nor Y (Figure 2B top panel; expressed in low level in Figure 2C top panel), while in the OROR motif, stem cells express both lineagespecifying TFs (Figure 2B bottom panel; expressed in high level in Figure 2C bottom panel). The difference in the status of S attractors relates to the coexpression level of lineagespecifying TFs in stem cells in real biological systems (Loeffler and Schroeder, 2019; Palii et al., 2019). Intuitively, from the view of the Boolean model, stem cell state in the ANDAND motif ([0,0] state) needs to switch on lineagespecifying TFs to transit to downstream fates (Figure 2B top panel). Whereas in the OROR motif, fate transitions are subject to the switchoff of TF expression (Figure 2B bottom panel). Furthermore, we introduced the solution landscape method. Solution landscape is a pathway map consisting of all stationary points and their connections, which can describe different cell states and transfer paths of them (Yin et al., 2020; Yin et al., 2021). From the perspective of the solution landscape, two logic motifs possess akin geometric topologies in their steadystate adjacencies (Figure 2D): when there are three fates coexisting in the state space, S attractor resides in the middle of LX and LY as the possible pivot for fate transitions (Figure 2D). To investigate noise, we developed models with stochastic forms (see ‘Materials and methods’). Simulations display the primary distribution of cell populations, corresponding to SSSs in deterministic models (Figure 2E and F).
Section 2: Two logic motifs exhibit opposite bias of fate decisions under the noisedriven mode
We first investigated the difference between the ANDAND and OROR motifs under the noisedriven mode. Here, we assigned the stem cell state as the starting point in simulation. In biological systems, it is unlikely that the noise level of different genes is kept perfectly the same. Asymmetry of the noise levels was thus introduced. First, we set the noise level of TF X higher than that of Y (σ_{x}=0.18, σ_{y}=0.12). Under this asymmetric noise, we observed that stem cells shifted toward LX in the ANDAND motif (Figure 3A and B), but toward LY in the OROR motif (Figure 3C and D). From the perspective of the state space, such properties intuitively originate from the distinctive status of stem cell attractors in two logic motifs (Figure 2B and C). In the ANDAND motif, the stem cell state resides at the origin of coordinates. Thus, with increasing X’s noise level, the stem cell population crossed the boundary between S and LX basins with a rising probability (Figure 2C top panel). Consequently, the fate decision of the stem cell population manifests a bias toward LX. Likewise, in the OROR motif, the stem cell population has a higher probability of entering LY basin following an increase in X’s noise (Figure 2C bottom panel). Next, we simulated multiple sets of noise levels for X and Y. We quantified the distribution of cell types, which was determined by the basin in which the final state of each round of stochastic simulation ended up. We observed that stem cell population displays almost opposite differentiation preference under identical noise levels but distinct logic motifs (Figure 3E). Conversely, when two distinct logic motifs exhibiting the same fatedecision bias, cell populations need to employ opposite noise patterns (Figure 3E and F). Collectively, if two of the three (noise profile, logic motif, fatedecision bias) are accessible, the last is inferential.
Next, we wondered whether noise could act as a driving force for reprogramming (e.g., from LY to S). We assigned LY state as the starting cell type in simulation. Apparently, in the ANDAND motif, transition of LY to S can be realized by increasing the noise level of TF Y (Figure 3—figure supplement 1A). Meanwhile, in the OROR motif, it is the increased noise level of X that can drive the transition from LY to S (Figure 3—figure supplement 1B), which is also intuitive by viewing the basin geometry of the state space (Figure 2C). These observations suggested that under the noisedriven mode, experimental reprogramming strategy need to take consideration of the regulatory logic (e.g., in reprogramming of LY to S, perturb the high expression TF of LY in the ANDAND motif, while in the OROR motif, perturb the low expression TF).
Section 3: Two logic motifs decide oppositely between differentiation and maintenance under the signaldriven mode
In addition to noise, cell fate decisions can also be driven by signals, e.g., GMCSF in hematopoiesis (Mojtahedi et al., 2016), CHIR99021 in chemically induced reprogramming (Zhao, 2019). The change conducted by signals corresponds to the distortions of the cell fate landscape. To simulate the signaldriven mode, we focused on the effect of parameters in the mathematical models on the system’s dynamical properties. To simulate models feasibly and orthogonally, we added parameters u (u_{x} in Equation 3, u_{y} in Equation 4) to Equations 1; 2:
The increase of u represents an elevation in the basal expression level of lineagespecifying TFs, reflecting an induction signal from the extracellular environment. From an experimental standpoint, this signal can be the induction of small molecules or overexpression by gene manipulations, such as the transfection of cells with expression vectors containing specific genes.
We first explored the impact on the system when the two induction parameters are changed symmetrically (u=u_{x}=u_{y}). As the increase of u, the number of SSS in the ANDAND system decreases from three to two, where S attractor evaporates after a subcritical pitchfork bifurcation (Figure 4A and Figure 4—figure supplement 1A). Whereas in the OROR motif, after the increase of u, LX and LY attractors disappear with saddlenode bifurcations, respectively. Only the SSS representing stem cell fate is retained in the state space (Figure 4B). We then portrayed all the topology of the steadystate adjacency that accompanied the increase of u, from the perspective of the solution landscape. In the ANDAND motif, the attractor basin of LX and LY started to adjoin and occupied the vanishing S attractor basin together (Figure 4C and D). Accordingly, the stem cells cannot maintain themselves and decided to differentiate into either one of the lineages. Moreover, if the cell population possesses the same noise levels in both X and Y, then the fate decisions are unbiased (Figure 4—figure supplement 1B).
Notably, in the ANDAND motif, we observed a brief intermediated stage before S attractor disappears, where all three fates are directly interconnected (Figure 4C second panel and Figure 4D second panel and Figure 4E). To manifest the generality, we globally screened 6231 groups of parameter sets under the ANDAND motif, and this logicdependent intermediated stage can be observed for 82.7% of them (see ‘Materials and methods’; Supplementary file 1), indicating little dependence on particular parameter setting (1.8% in the OROR motif). Unlike the indirect attractor adjacency structure mediated by S attractor (Figure 2D), the solution landscape with fullyconnected structure facilitates transitions between any two pairs of fates. Furthermore, this transitory fullyconnected stage is located between the fateundetermined stage (Figure 4C top panel) and fatedetermined stage (Figure 4C third panel), comparable to the initiation (or activation) stage before the lineage commitment in experimental observations (Brand and Morrissey, 2020; Zhang et al., 2018; Arinobu et al., 2007). Therefore, we suspected that the robust fullyconnected stage in the ANDAND motif may correspond to a specific period in cell fate decisions.
From the standpoint of reprogramming of differentiated cells back into progenitors, in the ANDAND motif, differentiated cells are more capable of maintaining their own fates during the symmetrical increase of the induction signals on both lineages (Figure 4A and C). Whereas in the OROR motif, the attractor basin of LX or LY is progressively occupied by the stem cell fate as u_{x} and u_{y} increase together (Figure 4F and G). In this scenario, the downstream fates are eventually reversed back to the undifferentiated state (Figure 4B and F). Namely, reprogramming engaged in the OROR motif can be accomplished by bidirectional induction of downstream antagonistic fates. In sum, we found that under symmetrical signal induction, the behavior of stem cells is subject to core GRN’s logic motifs. In the ANDAND motif, stem cells prefer to differentiate, while under the OROR motif, the stem cell population inclines to maintain its undifferentiated state.
Section 4: The tradeoff between progression and accuracy of cell fate decisions under the signaldriven mode
According to experimental observations, the majority of fate decisions exhibit lineage preference, also known as ‘symmetry breaking of fate decisions’ (Stanoev and Koseska, 2022; Pei et al., 2020; Psaila et al., 2020; Chen et al., 2018). Take the lineage choices in hematopoiesis as an example, Some HSCs prefer myeloid over lymphoid (Pei et al., 2020; de Haan and Lazare, 2018). This fatedecision bias also further shifts along with aging and infection (Zhang et al., 2018; Reya et al., 2001). In studying this preference in fate decisions, we broke the symmetry in the signaldriven models, by solely increasing u_{x} while keeping u_{y}=0 (Figure 5A). First, it is apparent that the fate decision will significantly steer toward LX along with the increase of u_{x}, regardless of the logic motifs. Ultimately the state spaces contain only LX attractor when u_{x} is sufficiently high (Figure 5B and C and Figure 5—figure supplement 1A). However, the changes in the state space and the solution landscape follow different routes for two logic motifs. In the ANDAND motif, S attractor basin disappears at first, leaving a state space with two differentiated fates (Figure 5D and E). Then the basin of LY attractor shrinks and finally disappears (Figure 5D and E). Whereas in the OROR motif, LY attractor disappears first. Then S attractor, with an enlarged basin, shares the state space with LX attractor. Finally, S attractor basin abruptly disappears by a saddlenode bifurcation (Figure 5F and G).
The distinct sequences of attractor basin disappearance as u_{x} increasing can be viewed as a tradeoff between progression and accuracy. In the ANDAND motif, the attractor basin of LX and LY adjoins (Figure 5D middle panel) when S attractor disappears due to the first saddlenode bifurcation (Figure 5B). Notwithstanding the bias of differentiation toward LX, the initial population still possesses the possibility of transiting into LY (Figure 5—figure supplement 1B). That is, in the ANDAND motif, as the increase of induction signal u_{x}, the ‘gate’ for the stem cell renewal is closed first. Stem cells are immediately compelled to make fate decisions toward either LX or LY, with a bias toward LX but a nonignorable probability of entering LY. Albeit the accuracy of differentiation is, therefore, compromised, the overall progression of differentiation is ensured (i.e., all stem cells have to make the fate decisions downward. This causes the pool of stem cells to be exhausted rapidly). Whereas in the OROR motif, the antagonistic fate, LY, disappears first (Figure 5C). The attractor basin of S and LX are adjacent in the state space (Figure 5F). In this case, the orientation of the fate decisions is generally unambiguous since the stem cell population can only shift to LX, ensuring the accuracy of differentiation. Next, to check if the observed sequences of basin disappearance are artifacts of specific parameter choice, we randomly sampled parameter sets to check the sequence of attractor changes in their state spaces (6207 groups of the ANDAND motifs and 6634 groups of the OROR motifs; Supplementary file 1). We found that 96% ANDAND motifs and 70% OROR motifs exhibit the same sequence of attractor vanishment mentioned above (Figure 5—figure supplement 1C and D and D; see ‘Materials and methods’). These results of the global screen demonstrated that the sequence of attractor vanishment is robust to parameter settings. In sum, we proposed that logic motifs couple the tradeoff between progression and accuracy as a general phenomenon in the signaldriven asymmetrical fate decisions (Figure 5—figure supplement 1E).
Next, we examined the transdifferentiation from LY into LX by increasing u_{x}. In the ANDAND motif, with the induction of X, LY directly transited into LX as the stem cell state disappears before LY (Figure 5D and Figure 5—figure supplement 2A). Intriguingly, for the OROR motif under the same induction, LY population first returned to the S state and then flows into LX (Figure 5F and Figure 5—figure supplement 2B). Namely, different logic motifs conduct distinct trajectories in response to identical induction in reprogramming. The ANDAND motif renders a onestep transition between downstream fates (Figure 5—figure supplement 2C). While in the OROR motif, it is a twostep transition mediated by the stem cell state (Figure 5—figure supplement 2D). This phenomenon suggests the observation that cells may be reprogrammed to distinct cell types depending on the induction dose (Zhao, 2019) is more realizable in the OROR motif. Integrated with the foregoing symmetrical induction, we recapitulated that in the OROR motif, the bidirectional induction or a unilateral induction from a counterpart (e.g., solely induced Y to realize reprogramming of LX to S) confer downstream cell fates to return to the undifferentiated state (Figure 4F and Figure 5F). Whereas in the ANDAND motif, it is substantially more difficult to achieve dedifferentiation. This observation may explain why some cell types are not feasible to reprogram (Shi et al., 2017).
Section 5: The CIS network performs differently during hematopoiesis and embryogenesis
In prior sections, we systematically investigated two logic motifs under the noise and signaldriven modes in silico. With various combinations of logic motifs and driving forces, features about fatedecision behaviors were characterized by computational models. Next, we questioned whether observations in computation can be mapped into real biological systems. And how to discern different logic motifs and driving modes is a prerequisite for answering this question.
To end this, we first evaluated the performance of different models, specifically in simulating the process of stem cells differentiating towards LX (Figure 6A). Under four models with different combinations of driving modes and logic motifs (Figure 6—figure supplement 1A and B), we assessed the expression level and expression variance (defined as the coefficient of variation) of TFs X and Y among the cell population over time in stochastic simulation. We observed that, under the same logic motifs, different driving modes change in the patterns of expression variance rather than expression levels (Figure 6B and C and Figure 6—figure supplement 1C and D). Overall, under the noisedriven differentiation from S to LX, the variance of expression exhibits a continuous and monotonic trend (Figure 6—figure supplement 1D) for both logic motifs. For different logic motifs, in the ANDAND motif, the expression variance of X (highly expressed in LX) declines (Figure 6—figure supplement 1D top panel). Whereas in the OROR motif, it is the expression variance of Y (low expressed in LX) displays a rising trend (Figure 6—figure supplement 1D bottom panel). Nevertheless, under the signaldriven mode, the expression variance increases and then decreases, exhibiting a nonmonotonic transition due to signalinduced bifurcation. During S to LX differentiation, comparable to the noisedriven mode, it is the expression variance of TF X in the ANDAND motif and TF Y in the OROR motif display a nonmonotonic pattern.
Such disparities between logic motifs originate from the location of S attractor (Figure 6D and Figure 2C). Although the target cell types are the same (LX), the ANDAND motif requests the expression of the TF X to be turned on, while the OROR motif requests the TF Y to be turned off. These key fatetransition genes, namely TF X in the ANDAND motif and TF Y in the OROR motif, both exhibit a sharp increase of variation in response to saddlenode bifurcation driven by u_{x} induction (first saddle node in Figure 5B; second saddle node in Figure 5C). Overall, these computational results suggest that we may be able to distinguish the two driving modes according to the expression variance over time series, then logic motifs can be correspondingly assigned by the expression level of the genes in the target cell types. For instance, if the expression variance of the X gene exhibits a nonmonotonic pattern and X is highly expressed in target cell types, then this cell fate decision can be assigned as the signaldriven fate decision in an ANDANDlike motif.
To support our findings with realworld correspondence, we first focused on the differentiation of CMPs in hematopoiesis (Figure 6E). It is acknowledged that the transcriptional regulation of Gata1PU.1 circuit dominates this cell fate decisions, which conforms to the CIS topology (Figure 6E). Mojtahedi et al., 2016 stimulated murine multipotent hematopoietic precursor cell line EML with erythropoietin (EPO) or granulocytemacrophage colonystimulating factor/interleukin 3 (GMCSF/IL3) to examine the commitment into an erythroid or a myeloid fate, respectively. Based their curated dataset, we found that the expression level of Gata1 (highly expressed in MEPs) gradually increased during EPO induction (Figure 6—figure supplement 2A), while the expression variance exhibits a nonmonotonic trend (Figure 6F top panel). Symmetrically, during GMCSF/IL3 induced differentiation toward GMPs, the expression level of PU.1 (highly expressed in GMPs) gradually increased (Figure 6—figure supplement 2B), while the expression variance also presents a nonmonotonic pattern (Figure 6F bottom panel). The trends shown in the dataset resemble the signaldriven mode with the ANDAND motif. In addition, we quantified the expression of Gata1 and PU.1 via the single molecule FISH dataset (Figure 6—figure supplement 2C; Wheat et al., 2020). They are at low levels in CMPs, corresponding to the expression patterns in the ANDAND motifs (Figure 2E and Figure 6D). Together, we suggested that the Gata1PU.1 circuit performs in an ANDANDlike manner, and this differentiation system (Mojtahedi et al., 2016) is under the signaldriven mode.
Another paradigmatic model of fate decision is the differentiation of embryonic stem cells (ESC). Semrau et al., 2017 found that under the retinoic acid (RA) exposure system in vitro, mouse embryonic stem cells (mESC) differentiated into two lineages: extraembryonic endoderm (XEN)like and ectodermlike. The investigators recapitulated that two clusters of TFs with the CIS topology determined this lineage specification (Figure 6G). We observed that the expression variance in most of these fatedecision TFs (16/22 73%) are gradually increasing during time, and 14% (3/22) of them exhibit nonmonotonic behavior (Figure 6—figure supplement 2D), suggesting the process is more likely driven by noise (Figure 6—figure supplement 2E). Furthermore, we focused on potential key regulators: Gbx2 and Tbx3, the two likely targets of RA that are crucial for this fate decision (Semrau et al., 2017). The expression variances over time of these two TFs are consistently increasing (Figure 6—figure supplement 2D). In addition, their initial expressions are at a high level, in agreement with that of the OROR motif (Figure 6D and H). In short, we proposed that the mESCs differentiation system under RA exposure performs in an ORORlike manner, and its differentiation is under the noisedriven mode in this experimental setting.
Section 6: The chemicalinduced reprogramming of human erythroblasts (EB) to induced megakaryocytes (iMK) is the signaldriven fate decisions with an ORORlike motif
The foregoing cell fate decisions initiate from pluripotent cells in mice (mESC, CMP) corresponding to a typical ‘downhill’ in Waddington’s metaphor. In 2006, Yamanaka et al. accomplished the reprogramming from mouse fibroblasts into iPSC state via the noted ‘OSKM’ factors, representing ‘uphill’ in Waddington’s metaphor (Shi et al., 2017; Hamazaki et al., 2017; Abdallah and Del Vecchio, 2019). Likewise, transdifferentiations from one lineage to another have been realized by overexpression or chemical inductions (Xu et al., 2015), whether they correspond to direct ‘trespassing’ of the ridge or an ‘upanddown’ through the peak in Waddington landscape are still elusive. We then applied our models to reprogramming systems, with a primary focus on hematopoiesis. Qin et al., 2022 recently achieved the direct chemical reprogramming of EBs to iMKs using a foursmallmolecule cocktail (Figure 7A). Investigators presented that EBs underwent an induced bipotent precursor for erythrocytes and MKs (iPEM) to finally desired iMKs. It is acknowledged that the FLI1KLF1 circuit with the CIS topology dominates this fatedecision process (Orkin and Zon, 2008; Palii et al., 2019). To deduce the logic motif of the FLI1KLF1 circuit, we quantified the expression patterns of FLI1 and KLF1 based on published singlecell RNAseq data (Qin et al., 2022). We can observe the fate transition from the EB population (FLI1^{low}, KLF1^{high}) to the iMK population (FLI1^{high}, KLF1^{low}) (Figure 7B). According to their expression level, the cell populations can be primarily classified into three clusters. In addition, both FLI1 and KLF1 are highly expressed in the intermediate cell population suspected to be the progenitors of iMKs and EBs (Qin et al., 2022). Namely, the pattern of expression level is concordant with the OROR motif in our framework (Figure 6D).
Next, to investigate the driving force of this reprogramming system, we simulated the fate transition from EB (corresponding to LX, blue) to iMK (LY, purple) under both driving modes. Under the noisedriven mode, we assumed that the reprogramming system facilitated the noise levels of both TFs. For simplicity, the starting cell population (EBs) was assigned symmetrical high noise levels. While under the signaldriven mode, we assumed that the foursmallmolecule cocktail upregulated the expression of FLI1 (highly expressed in iMKs). Then, we simulated the transition from EBs to iMKs by lifting the basal expression level of FLI1, corresponding to parameter u_{y} in model (Figure 7C). The bifurcation diagrams indicate that the signaldriven fate transition is mediated by the iPEM state (Figure 7C). In particular, overexpression of FLI1 renders sequential saddlenode bifurcations. Thus, EBs are converted to iPEMs before steering toward the terminal iMK state (Figure 7C and D), which is consistent with the experiment’s findings.
Furthermore, we next assessed the dynamic behaviors in models of this system under different driven modes with the OROR logic. As discussed before, there is no discernible difference in the expression levels between the two driving modes. Overall, the common tendency is an upregulation in KLF1 and a downregulation in FLI1 (Figure 7—figure supplement 1A). We then quantified the expression variances during this fate transition by the model. Under the noisedriven mode, expression variances of FLI1 and KLF1 would gradually decrease and increase, respectively, until stabilizing (Figure 7—figure supplement 1B). Under the signaldriven mode, the expression variance of FLI1 would first decline and then remain nearly constant, while KLF1 would exhibit a nonmonotonic pattern (Figure 7E left panel). From the view of modeling, the nonmonotonic pattern presented by KLF1 originates from the rapid shutoff during the transition from iPEM state to iMK state (second saddle node in Figure 7C).
Accordingly, we next quantified the expression variances in the real dataset over time. Impressively, the pattern emerging from the data accommodates the hypothesis of the signaldriven mode (Figure 7E). Altogether, we proposed that this reprogramming system (Qin et al., 2022) is the signaldriven process underlying the ORORlike motif. Moreover, the high expression level of FLI1 induced by small molecules is the key driving force of the fate transition, as suggested by the properties of the OROR motif. We underlined that it is a classical twostep fatedecision process mediated by the upstream progenitor state, which is in agreement with the phenomenon articulated by Qin et al., 2022.
We then searched for genes with similar patterns of expression variance to those of KLF1 and FLI1, with a hypothesis that genes possessing comparable expression variance patterns, especially TFs, may synergistically perform fatedecision related functions. Thus, we applied the fuzzy cmeans algorithm (Kumar and Futschik, 2007) to cluster genes based on their expression variances, rather than expression levels. In total, we observed 12 distinct clusters of temporal patterns (Figure 7—figure supplement 1C; Supplementary file 2). We focused primarily on clusters 5 and 10, where KLF1 and FLI1 are found respectively (Figure 7F). To testify our hypothesis, we conducted an enrichment analysis using the gene set in clusters 5 and 10. Functions associated with specific cell types are significantly enriched (Figure 7G). In particular, cluster 5 with FLI1 is largely related to plateletrelated functions, whereas cluster 10 with KLF1 is largely related to the energy metabolism of blood cells. Furthermore, we filtered out 15 TFs in clusters 5 and 10 (11 in cluster 5; 4 in cluster 10). Next, by harnessing the TF interaction database (see ‘Materials and methods’; Supplementary file 3), we collected 21 regulons associated with 10 TFs to construct the TF regulatory Network (Figure 7H). Intriguingly, in the original article, genes were classified into six ‘Supermodules’ according to the patterns of expression levels. Genes in cluster 5 are located in Supermodules 1 and 3, representing a decreasing tendency for expression level. Meanwhile, genes in cluster 10 are distributed in Supermodules 2 and 5, and their expression levels raise from low to high (Figure 6D from Qin et al., 2022). Of note, most of the TFs filtered by expression variance patterns appear in the GRN constructed in the original article (8/10, 80%; Figure 6G from Qin et al., 2022). As a result, we underscored that EGR3 and ETS1, which did not appear in the GRN of the original article, have been suggested to play important roles in the transdifferentiation. In addition, we observed that MYC and FOS possess the largest connectivity in our 10node GRN, suggesting that these two TFs as hubs are essential regulators of this reprogramming system.
Together, in mapping our framework to the real reprogramming system, we assigned the cell fate decisions of EBs to iMKs to the signaldriven mode incorporated the ORORlike motif, by comparing the expression and expression variance patterns measured from the real dataset with pseudodata produced by our models in a ‘topdown’ fashion. According to the model, the reprogramming is primarily driven by induced upregulation of FLI1. Additionally, from the view of expression variance, we recapitulated a concise 10node GRN and identified some TF nodes not previously recognized as major regulators, like EGR3 and ETS1.
Discussion
Comprehending the driving forces behind cell fate decisions is crucial for both fundamental scientific research and biomedical engineering. Recent advances in data collection and statistical methods have greatly enhanced our understanding of the mechanisms that regulate cell fate. However, despite the widespread use of the Waddington landscape as a metaphor in experiments, there has been little examination into whether the fate decision observed in a particular experiment corresponds to a stochastic shift from one attractor to another on the landscape (i.e., noisedriven) or an overall distortion of the landscape (i.e., signaldriven). The application of appropriate computational models in systems biology can aid in uncovering the underlying mechanisms (Cahan et al., 2021; del Sol and Jung, 2021). One of the most representative work is that Huang et al., 2007 modeled the bifurcation in hematopoiesis to reveal the lineage commitment quantitatively. Compared to simply modularizing activation or inhibition effect by employing Hill function in previous work, our models reconsidered the multiple regulations from the level of TFCRE binding.
Our computational investigations have emphasized the importance of the combinatorial logic in connecting gene expression patterns with the driving forces underlying cell fate decisions. Utilizing a representative network topology known as the CIS, our analysis demonstrated how both driving forces and regulatory logic jointly shape expression patterns during fate transitions. In turn, mean and variance in gene expression patterns can reveal logic motifs and driving forces. Our analytical framework promotes the interpretability of fate decisions and can be employed to speculate on the driving factors of fate decisions using a ‘topdown’ approach, thereby providing a reference for investigating the causality of fate decisions and experimental validation.
The roles of noise as a possible driver of fate transitions are intriguing. By our models, the relationship among noise configuration, logic motifs, and fatedecision bias has been unveiled, and we noticed the opposite fate bias for the ANDAND and OROR motifs. Conversely, on the demand of the same bias, the progenitors with different logic motifs tend to employ a different profile of noise level (Figure 3E and F). Therefore, we suggested that under the noisedriven mode, the logic motif works like a ‘broker’ to shape the fate preferences. Based on the assumption that the preference of fate decisions is the result of evolution and adaptation, we posited that if an organism has a functional demand for a specific bias, the noise profile will be iterated via logic motifs. In turn, changes in noise levels will be mediated by logic motifs to shape the differentiation bias. One intuitive example is the significant shift in differentiation preferences of HSCs over aging (LópezOtín et al., 2013; Beerman and Rossi, 2015; Dorshkind et al., 2020). It has been reported that aging HSCs show different levels of DNA methylation and epigenetic histone modifications from young HSCs (de Haan and Lazare, 2018; Kovtonyuk et al., 2016; Pang et al., 2017). How changes in epigenetic level shape the noise profile of the cell population and further affect the shift of fatedecision preference is a fascinating question. We underscored that the dissection of logic motifs underlying associated GRNs is a prerequisite for answering that question.
With the everexpanded of singlecell sequencing data, characterizing genes by their mean expressions does not make the most of highthroughput datasets. As an intrinsic characteristic in central dogma, expression variance has been utilized to locate the ‘critical transitions’ in complex networks (Wang and Chen, 2020; Scheffer et al., 2012), e.g., identify the critical transitions in diseases like lymphoma (Liu et al., 2012). Instead of differentially expressed genes, RosalesAlvarez et al., 2022 harnessed differentially noisy genes to characterize the functional heterogeneity in HSC aging. Our work presents that the patterns of expression variance can also be used to indicate driving forces and key regulators during the fatedecision processes. Nevertheless, compared to traditional gene expression, the interpretation of expression variance patterns is generally not intuitively accessible (Grün et al., 2014). Additionally, extra a priori knowledge is needed to filter out the cluster of interest. To this end, our framework enables researchers to locate functional clusters via mathematic model based on appropriate hypothesis. Notably, if the genes that constituting the CIS network are not specified, we can conversely leverage the patterns of temporal expression variance to nominate key regulators in a modelguided manner. Collectively, our framework provides a mechanistic explanation for expression variance patterns and qualitatively characterize key expression variance patterns to locate core regulators of fate decisions without reliance on a priori knowledge.
Comparing to tuning noise, altering signals is a more accessible approach for experimentally manipulating cell fates. When the basal expressions of two lineagespecifying genes grow symmetrically, we have shown that opposite trends of fate transitions occur under two logic motifs: In the ANDAND motif, it promotes differentiation, whereas the OROR motif stabilizes stem cell fates. This is reminiscent of the ‘seesaw’ model where maintenance of stemness can be achieved by overexpression of antagonistic lineagespecifying genes (Shu et al., 2013). Our model suggests that restoring stemness by inducing two antagonistic lineagespecifying genes is more likely under the ORORlike motif (Figure 4—figure supplement 1C). This is in concert with our analysis that mESC differentiation system performs in an ORORlike manner. In addition, Mojtahedi et al., 2016 found that, under simultaneous induction of two antagonistic fates (EPO and GMCSF/IL3), although differentiation was delayed, it eventually occurred. This observation is consistent with models in the ANDAND motif, and we suggested that the core regulatory circuits in hematopoiesis performs in an ANDANDlike manner. More experimental validations would be needed to validate this hypothesis that ‘the seesaw model prefers the OROR motif.’ Conversely, insight from the ‘seesaw’ model also provides a candidate approach to further testify logic motifs underlying GRNs in experiments. For instance, stem cells with an ANDANDlike circuit are expected to display differentiation rather not maintenance in response to bidirectional induction.
In quantifying the signaldriven landscape changes, the solution landscape enables intuitive interpretation even in highdimension GRNs (Yin et al., 2020; Sáez et al., 2022; Corson and Siggia, 2012). In this work, we used both the state space and the solution landscape, in order to relate them for further investigations involving more than two TFs. Interestingly, from the perspective of the solution landscape, we found a robust fullyconnected stage in the ANDAND motif. We envisioned that this period corresponds to the priming stage of differentiation. Notably, this fullyconnected stage was not found in the OROR motif, suggesting that the necessity for priming during differentiation may be subject to the logic motifs of core GRNs.
Actual cell fate decisions are seldom purely unbiased. Under the asymmetrical signaldriven mode, we summarized the progressionaccuracy tradeoff in cell fate decisions: If a large number of cells are ensured to differentiate, then concessions have to be made in the accuracy of differentiation, and vice versa (Figure 5—figure supplement 1E). An intuitive example is the largescale apoptosis occurs daily in hematopoiesis. Hence, maintaining homeostasis in vivo inevitably requests cells to respond rapidly to differentiation. A recent study reported that in response to systemic inflammation by polymicrobial sepsis, pool of CMPs is rapidly depleted to accelerate the production of downstream cell fates (Fanti et al., 2023). This is concordant with our result that Gata1PU.1 circuit in hematopoiesis performs in an ANDANDlike manner. On the other hand, in embryonic development like C. elegans, the accuracy of cell fate decisions is considerably emphasized. How this nature of differentiation has been adapted in evolution is an interesting question. Our work highlights that this property is associated with the logic motifs of GRNs, suggesting the emphasis on progression or accuracy may be embedded in the logic motifs of core GRNs.
We classified three examples of cell fate decisions based on patterns of expression and expression variance. In hematopoiesis, we took fate choice between erythroid and myeloid as a paradigm, and assigned it an ANDANDlike motif under the signaldriven mode. In embryogenesis, we suggested the fate decision in RA exposure system is an ORORlike motif under the noisedriven mode. In reprogramming, the chemicalinduced transdifferentiation is the signaldriven fate decisions incorporated an ORORlike motif. For simplicity and intuitiveness, we devised our model with two symmetrical combinations of regulatory logic (ANDAND/OROR). Albeit there are merely four types of cell fate decisions in consideration, our framework enables to be generalized and expanded to accommodate multinode GRNs and complex logic combinations. Plenty of studies zoomed in one particular fatedecision events. However, from the standpoint of systems biology, we underlined that classification of fate decisions is a vital step for further investigation, as is the case for the typing of cells and tumors. Theoretically, appropriate classification of fatedecision systems enables the enrichment of common properties. So, accumulated knowledge can be inherited to new fatedecision cases. Taking reprogramming as an example, Zhao, 2019 recapitulated five kinds of trajectories in chemicalinduced reprogramming. We suggested that the reprogramming trajectory is coupled with the logic motifs. On one hand, it is possible to answer why a certain reprogramming system exhibit a particular trajectory. On the other hand, it is possible to postulate achievable reprogramming according to the logic motifs of core GRNs (e.g., the ANDAND motif is more likely to enable direct conversion; model 4 mentioned in Zhao, 2019). Recently, synthetic biology has realized the insertion of the CIS network in mammalian cells (Zhu et al., 2022). One of the prerequisites for recapitulating the complex dynamics of fate transitions in synthetic biology is systematical understanding of the role of GRNs and driving forces in differentiation. And the logic motifs are the essential and indispensable elements in GRNs. Our work also provides a blueprint for designing logic motifs with particular functions. We are also interested in validating the conclusions drawn from our models in a synthetic biology system.
Limitation of this study
Although our framework enables the investigation of more logic motifs, we chose two classical and symmetrical logic combinations for our analysis. Future work should involve more logic gates like XOR and explore asymmetrical logic motifs like ANDOR. The gene expression datasets analyzed here are only available for a limited number of time points. Though they meet the need for discerning trends, it is evident that the application to the datasets with more time points will yield clearer and less ambiguous changing trends to support the conclusions of this paper more generally. Notwithstanding the fact that the CIS network is prevalent in fatedecision programs, there are other topologies of networks that serve important roles in the cellstate transitions, like feedforward loop, etc. The framework should further incorporate diverse network motifs in the future. In addition, for simplicity and intuition, we here considered signals as uncoupled and additive effects in ODE models, due to feasible mapping in real biological systems, such as ectopic overexpression.
Materials and methods
Derivation of the CIS network
Request a detailed protocolIn GRN, each TFs is represented by a node and the edge between nodes represents the regulatory relationship. In our work, we considered a GRN comprised of 2 TFs (i.e., X, Y), formulated with 2 ODEs describing the change of each TF. Define ([X], [Y]) to be the concentration of TF X and Y, the ODE model is described as follows:
where H_{X}([X], [Y]) is the production rate of TF X that combines the effects from both activators and suppressors of the X and d_{X} is the decay rate from the X and Y, which integrate degradation and dilution.
The exact form of H_{X}([X], [Y]), H_{Y}([X], [Y]) are derived based on the simple GRN with nodes X and Y via a set of molecular interactions between these TFs themselves, genes that encode for them, and the mRNAs (Abdallah and Del Vecchio, 2019).
TFs in a GRN act as multimers to implement regulatory interaction. In our model, we treated TFs X and Y as acting in their homomultimer forms, X_{n1} and Y_{n2}, respectively (i.e., n_{1} TF X monomers reversibly form an activated homomultimerized form X_{n1} and n_{2} monomers of TF Y to reversibly form activated homomultimer Y_{n2}). Of note, n_{1} and n_{2} here are able to be further generalized beyond the number of binding elements (Nam et al., 2022; Santillán, 2008). The multimerization biochemical reaction of TFs X and Y can be represented as follows,
There are many diverse mechanisms of regulation (Lambert et al., 2018; Balsalobre and Drouin, 2022). For simplicity, our model considered transcriptional regulation as a major mechanism since it is feasibly characterized in experiments and can well represent the interaction between genes in CrossInhibition with Selfactivation (CIS) network. To activate downstream transcription, TFs bind with their CisRegulatory Elements (CREs). We denoted D_{X} for the nobound CREs of TF X and D_{Y} for the TF Y in an independent manner. Here, we posited different TFs bind to exclusive, nonoverlapping CREs to regulate target genes. Hence, there are totally eight binding patterns described as follows,
Next, we modeled the biochemical reactions of transcription and translation. Transcription is an elaborate process involving many steps, including initiation, elongation, and termination of an mRNA transcript. Likewise, translation includes peptide formation and elongation, followed by protein folding before function. However, to evaluate the behavior of GRN concisely, we treated transcription and translation as singlestep reactions and then use lumped rates to encompass the time it takes for all steps in the elaborate machinery to complete (Abdallah and Del Vecchio, 2019). Hence, the transcription process can be represented as follows,
Here, we distinguished the basal rate of transcription of a gene without activation or repression (termed as constitutive transcription). The basal transcription rate is shown in the left panel in Equation 8.
Though we considered eight configurations, these ultimately led to two mRNA transcripts, m_{X} and m_{Y}. Translation can also be represented by a onestep process from these transcripts to their protein products. The onestep translation biochemical reactions are described as follows,
Next, we considered the decay of these proteins as well as their respective mRNA species. In general, these proteins and mRNA species will undergo decay to some extent which is a combination of both degradation and dilution.
here, the left and middle panels represent the degradation of TFs X and Y as well as mRNA transcripts m_{X} and m_{Y}. The right panel represents the dilution of TFs X and Y due to cell division. In general, the degradation rate γ can be determined using the following relations,
where t_{1/2} represents the halflives of each protein or mRNA transcript, t_{doubling} represents the cell division rate. For protein products X and Y, we used the notation d to illustrate the total rate of decay rather than two separate parameters in our model, i.e., d_{1} = δ_{x} + β, d_{2} = δ_{y} + β, respectively. In general, t_{1/2} and t_{doubling} are robust and stationary, thus we treated these paramaters as constant in our model, i.e., d_{i} = Const, for i = 1, 2.
We have described the reactions that comprise the endogenous components of our GRN. In our work, fate decisions are classified into two modes, i.e., driven by the noise of intracellular gene expression or driven by extracellular signals. Under the noisedriven mode, a Gaussian white noise is added to the concentration of TFs to illustrate the random intracellular noise. Whereas if driven by signals, the expression of a gene will be affected by extracellular molecular cocktails, physical stimulation, etc. Therefore, we added the production rate of the TF’s mRNA from the ectopic DNA to generalize our model.
here u_{x}, u_{y} represent the additional mRNA species m_{X} and m_{Y} via ectopic overexpression at rates u_{x} and u_{y}, respectively. The biochemical reactions take place on a faster timescale. This enables us to derive the change in concentration of species from a biochemical reaction based on the law of mass action. For example, for multimerization reaction of X in Equation 6, once equilibrium, we can get
Also, for TF X bind with its CRE D_{X}, we can get
here, we define ${K}_{1}\triangleq \frac{{p}_{1}}{{q}_{1}}$ as a constant, i.e., chemical equilibrium constant. Furthermore, we define ${K}_{1}\triangleq \frac{{p}_{1}}{{q}_{1}},i=x,y,\mathrm{1,2},\cdots ,8$ for biochemical reactions in Equation 6 and Equation 7, respectively.
Then, we constructed an ODE model for the GRN. For a given species S, the principle to get a single equation is as follows:
To explain how this principle assists us to build the ODE model, we took species m_{X} as a running example. Summarizing all of these biochemical reaction rates involving m_{X} we can get the change in concentration [m_{X}],
Doing so for our biochemical reaction network yielded the equations of our ODE model,
The 14dimension ODE model of the GRN contains overwhelming parameters that make it impractical to analyze for cell fate. To reduce the dimensionality of the model, we assumed the multimerization, DNA binding/unbinding and mRNA dynamic occurs sufficiently faster than protein production and decay, the temporal derivatives of the respective species can be set to 0, indicating that the species concentration reaches its quasisteady state (Abdallah and Del Vecchio, 2019). Thus, we can get
To better demonstrate the deviation of the model, we take TF X as a running example. Since GRN is symmetric, we can deviate TF Y’s equation in the same way. Once the biochemical reaction reaches its equilibrium, using the law of mass action we can get
Noticing that $\left[\dot{{m}_{X}}\right]$ = 0 in Equation 18, the species [m_{X}] can be represented by CRE terms,
In our wellstirred system, for TF X’s CREs, the conservation law holds, i.e., the total CRE concentration of X, as a constant, equals the sum of CRE concentration that is bound by X, by Y, and by X and Y. By employing so and combining with Equation 19 we can get a representation of [D_{X}],
Substitute [D_{X}] in Equation 20 with Equation 21, we can get
Then finally we reach to the point that the deviation of [X]. In Equation 18 we substituted [m_{X}] with Equation 22, and got
here, we define $\gamma}_{0}^{x}={\alpha}_{x}\frac{{D}_{TX}{\kappa}_{x}}{{\eta}_{x}},{\gamma}_{1}={\alpha}_{x}^{1}\frac{{D}_{TX}{\kappa}_{x}}{{\eta}_{x}},{\gamma}_{2}={\alpha}_{x}^{2}\frac{{D}_{TX}{\kappa}_{x}}{{\eta}_{x}},{\gamma}_{3}={\alpha}_{x}^{3}\frac{{D}_{TX}{\kappa}_{x}}{{\eta}_{x}$ as relative protein generation rates, ${k}_{1}={K}_{1}{K}_{x},{k}_{2}={K}_{2}{K}_{y},{k}_{3}={K}_{x}{K}_{y}{K}_{1}{K}_{5}$ as the weight of each TF binding patterns, as ectopic DNA (to simplify the notation, we may drop the tilde of ${\stackrel{~}{u}}_{i},i=x,y$), then we can get
Similarly, for TF Y, we can get
Ultimately, by combining Equation 24 and Equation 25, we reduced our 14dimension model to a 2dimension dynamic system,
Stochastic simulations
Request a detailed protocolIn Equation 22, parameters of the ANDAND motif (Figure 2C top panel) are: (${r}_{0}^{x}$ , ${r}_{0}^{y}$ , r_{1}, r_{2}, r_{3}, r_{4}, r_{5}, r_{6}) = (0, 0, 0.95, 0, 0, 0.95, 0, 0), (k_{1}, k_{2}, k_{3}, k_{4}, k_{5}, k_{6}) = (1.5, 0.8, 1, 1.5, 0.8, 1) and (d_{1}, d_{2}) = (0.55, 0.55). Parameters of the OROR motif (Figure 2C bottom panel) are: (${r}_{0}^{x}$ , ${r}_{0}^{y}$ , r_{1}, r_{2}, r_{3}, r_{4}, r_{5}, r_{6}) = (0.15, 0.15, 0.95, 0, 0.95, 0.95, 0, 0.95), (k_{1}, k_{2}, k_{3}, k_{4}, k_{5}, k_{6}) = (2.1, 1.7, 1.8, 2.1, 1.7, 1.8) and (d_{1}, d_{2}) = (0.55, 0.55). In both the ANDAND and OROR motifs, (n_{1}, n_{2}) = (2, 2). We simulated noise in the model using a Langevin equation (Foster et al., 2009):
Where ${F}_{i}\left(x\right)$ is the deterministic function of Equation 22. ${W}_{i}$ is a Wiener process which introduces additive noise with no dependence on the state x. We integrated the Wiener process over the interval which gives us $\sqrt{\u2206t}{\xi}_{i}$ and ${\xi}_{i}$ is a Gaussian white noise with zero mean and given variance (σ^{2}). Thus we can compute the new state of the dynamical systems using the Runge–Kutta method.
Parameter screening
Request a detailed protocolWe conducted global parameters screening of the fully connected stage and ProgressionAccuracy tradeoff in our models. First, to collect parameter sets with 3 SSSs, we used Latin hypercube sampling (LHS) to screen kseries parameters symmetrically (i.e., k_{1} = k_{4}, k_{2} = k_{5}, k_{3} = k_{6}) ranging from 0.001 to 5 both in the ANDAND and OROR motifs. We ultimately collected 6,231 sets for the ANDAND motif and 6682 sets for the OROR motifs (Supplementary file 1).
To analyze the sequence of vanishing SSSs. We further filtered parameter sets with 2 SSSs remained as increasing ux (6207 sets for the ANDAND motif; 6634 sets for the OROR motif). For each SSS, we quantified the Minus of [X] and [Y] (Figure 5—figure supplement 1C, D).
Saddle points and saddle dynamics
Request a detailed protocolSet an autonomous dynamical system (Yin et al., 2021)
where $F:{R}^{n}\to {R}^{n}\text{is a}{C}^{r}\left(r\u2a7e2\right)$ function, and a point $x}^{\ast}\in {R}^{n$ is called a stationary point (or equilibrium solution) of Equation 28 if $F\left({x}^{\ast}\right)=0$. Let $J\left(x\right)=\nabla F\left(x\right)$ denote the Jacobian of $F\left(x\right)$ . For a stationary point $x}^{\ast$ , taking $x={x}^{\ast}+y$ in Equation 28, we have
where $\parallel *\parallel $ denotes the norm induced by the inner product. The associated linear system
is used to determine the stability of $x}^{\ast$ . Depending on the eigenvalues of $J\left(x\right)$ with positive, negative, and zero real parts, we can define unstable, stable, and center manifolds of the Jacobian $J\left(x\right)$ spanned by the corresponding eigenvectors, as ${W}^{u}\left({x}^{\ast}\right),{W}^{s}\left({x}^{\ast}\right)\text{and}{W}^{c}\left({x}^{\ast}\right)$ .
From the primary decomposition theorem, $\mathbb{R}}^{n$ can be decomposed as a direct sum:
A hyperbolic stationary point is called a saddle if ${W}^{u}\left({x}^{\ast}\right)$ and ${W}^{s}\left({x}^{\ast}\right)$ are nontrivial. The hyperbolic stationary point $x}^{\ast$ is called a sink (source) if all the eigenvalues of $J\left({x}^{\ast}\right)$ have negative (positive) real parts. The index of a stationary point $x}^{\ast$ is defined as the dimension of the unstable subspace ${W}^{u}\left({x}^{\ast}\right)$ .
The HiOSD method (Yin et al., 2019) is designed for finding index$k$ saddles of an energy function $E\left(x\right)$ . For gradient systems, $F\left(x\right)=\nabla E\left(x\right)$ , and the Jacobian $J\left(x\right)={\nabla}^{2}E\left(x\right)=G\left(x\right)$ , where $G\left(x\right)$ denotes the Hessian of $E\left(x\right)$ . The $k$saddle $x}^{\ast$ is a local maximum on the linear manifold ${x}^{\ast}+{W}^{u}\left({x}^{\ast}\right)$ and a local minimum on ${x}^{\ast}+{W}^{s}\left({x}^{\ast}\right)$ . So, the highindex saddle dynamics (HiSD) for a index k saddle (ksaddle) is a transformed gradient flow
where ${P}_{V}$ denotes the orthogonal projection operator on a finitedimensional subspace $V$. Here, ${P}_{{W}^{u}\left(x\right)}F\left(x\right)$ is taken as an ascent direction on the subspace ${W}^{u}\left(x\right)$ and $F\left(x\right){P}_{{W}^{u}\left(x\right)}F\left(x\right)$ is a descent direction on the subspace ${W}^{s}\left(x\right)$ . The subspace ${W}^{u}\left(x\right)=span{\{v}_{1},\dots ,{v}_{k}$ } where ${v}_{i}$ is the unit eigenvector corresponding to the smallest $i$th eigenvalues, which can be obtained by many methods such as minimize the Rayleigh quotients. According to the above the HiSD for a ksaddle (kHiSD) is:
which coupled with the initial condition
Where $I$ is the identity operator and β, γ>0 are relaxation parameters.
Similarly, the GHiSD (Yin et al., 2021) for a ksaddle (kGHiSD) of the dynamical system (28) has the following form:
which coupled with the initial condition(34). The kGHiSD can be accelerated by the Heavy Ball method.
Constructing the solution landscape
Request a detailed protocolFor a given set of parameters, we can use kGHiSD to find each order saddle point of the dynamical system, and then construct the solution landscape of the system. The solution landscape is a pathway map consisting of all stationary points and their connections (Yin et al., 2020; Yin et al., 2022). Solution landscape is a new tool used to describe the dynamic behavior of stationary points in a dynamic system which can show the connection and transfer path between stationary points. In general, we can use the downward search algorithm and the upward search algorithm to construct the solution landscape, which starting from the ksaddle points and the stable points respectively. Because of the symmetry and other prior knowledge of the dynamic system used in our paper, The saddle points of the system tend to occur in the range of ${\left[0,{x}_{max}\right]}^{2}$ , where the ${x}_{max}$ means the maximum coordinate of the stable point. We grid the range, select each grid point as the initial state, and find saddle points of corresponding order (k = 1,2) using kGHiSD. Taking the saddle points found before as the initial states, we disturb its unstable directions to search other stationary points of lower index, establish the connection relationship between stationary points and finally construct the solution landscape. In addition, we use geometric minimum action method (gMAM) (VandenEijnden and Heymann, 2008) to find the minimum action path between the stable points, use the obtained action to represent the stability of the stable points, and represent them at different heights in the solution landscape.
Screening the fullyconnected stage
Request a detailed protocolAs shown in the case of u = 0.0565 in Figure 4D, the fully connected stage (FCS) has three stable points and three 1index saddle points to connect the stable points with each other. Therefore, we posited that finding a fully connected stage can be equivalent to finding three different 1index saddle points in this system.
First, build a set of parameters with three stable points. In two different logics, 10,000 sets of dynamic parameters are randomly generated respectively, and all stable points of the system corresponding to each set of parameters are found. If there are three stable points, the corresponding dynamic parameters are recorded as a new set of parameters for the subsequent search of 1index saddle points.
Second, finding the 1index saddle points in a system with three stable points. According to the above results, we can find that saddle points tend to appear in the range of ${\left[0,{x}_{max}\right]}^{2}$ , where the ${x}_{max}$ means the maximum coordinate of the stable point. So we can grid the range in the same process as before, and search 1index saddle points with each grid point as the initial structure. In order to find the relevant saddle points as detailed as possible, denser grid points were used as the initial structure of the system without three 1index saddle points to find the 1index saddle points.
Due to the calculation error of numerical calculation itself, the saddle points found by saddle point dynamics may have the following problems:
The same saddle points found from different initial structures were mistaken for two.
Find a negative or infinite saddle point.
The saddlepoint judgment is inaccurate due to the influence of zero eigenvalue.
We can set corresponding judgment conditions to solve the above problems.
In addition, since there is also a parameter u representing the foreign signal in the system, we can let u vary to find if there are three 1index saddle points. At the beginning, u = 0 is set, and then u is changed according to the number of saddle points found and the corresponding judgment conditions, so as to find the system with three 1index saddle points. We use the idea of adaptive step size to speed up the search process in the algorithm of finding three 1index saddle points in the system. The specific process is as follows:
Set the initial u = 0, du = 1.
Saddle Dynamics is used to find the 1index saddle points in the parameters.
When the number of 1index saddle points is equal to 0: there is only one stable point in the system (u = 0.12 in Figure 4F), it indicates that u is too large and needs to be reduced to make u = u du.
When the number of 1index saddle points is equal to 1: the system has two stable points (u=0.12 in Figure 4C), it indicates that u is too large and needs to be reduced so that u = u du.
When the number of 1index saddle points is equal to 2: the system has two kinds of cases, one of which needs to make u increase (u = 0 in Figure 4C). In this case, the coordinate of stable point near y = x is smaller than the midpoint of two 1index saddle points, so we let u = u + du, and let du = 0.1 * du. In the other case, there are two possibilities. One is like u = 0 in Figure 4.F, and the other is like u = 0.0595 in Figure 4F. In this case, there is no stable point near y = x or the coordinate of the stable point is larger than the midpoint of two 1index saddle points.
When the system finds that the number of 1index saddle points is equal to 3: we need to judge whether there are three nondegenerate saddle points that are different from each other. If the condition is sufficient, it indicates that we have found the fully connected solution landscape; otherwise, let u = u + du, du = 0.1 * du, and continue searching.
When the number of 1index saddle points is greater than 3, it indicates that u is too large and needs to be reduced to make u = u du.
If three 1index saddle points are not found after 100 changes of u, it is considered that there is no fully connected solution landscape in the system.
We, respectively, used the above algorithm to search for the fully connected solution landscape under two different logics and obtained the following results:
Logic  # of 3 SSS  # of FCS  Ratio 

ANDAND  6231  5151  82.7% 
OROR  6682  120  1.8% 
It can be seen from the simulation results that the fully connected solution landscape can appear in most ANDAND motifs, but almost not in the OROR motifs. Due to the error of numerical calculation itself, the obtained results may have a certain deviation. In addition, in our search process, due to the limitation of computing resources and the complex nature of some systems, in order to take into account the majority of cases, some systems with full connectivity have not been found. Subsequent manual search found that the fully connected solution landscape is existed in some systems which not found under the ANDAND motifs previously (17.3%). Therefore, the obtained proportion can be further improved.
Data processing of singlecell data
Request a detailed protocolThere are totally four public datasets used in our work (related to Figure 6F and H and Figure 7 and Figure 6—figure supplement 2C). In mice’s hematopoiesis (Mojtahedi et al., 2016), expression of genes is quantified as 2^{LOD−Cq}−1 (LOD: limit of detection; Cqvalue is the cutoff of amplification signal in singlecell qPCR data). Thus, we can compute the coefficient of variation (CV) over time. In embryogenesis (Semrau et al., 2017), public dataset used in our work was generated by singlecell SMARTseq2 (GEO: GSE79578). We utilized preprocessed data to compute the CV over time. In reprogramming, public dataset was generated by 10 x Genomics (GEO: GSE207654). Counts Per Million (CPM) matrix are used to compute the CV. We assigned time points by the Leiden clustering algorithm with parameters consistent with the original paper (see methods in Qin et al., 2022).
Fuzzy Cmeans clustering
Request a detailed protocolCV of 2000 high variable genes given by original paper (Qin et al., 2022) over four time points were computed. We filtered out 1677 genes by removing NA due to missing values. Noise of 1677 genes were then grouped into 12 clusters using the Mfuzz package in R with fuzzy cmeans algorithm (Kumar and Futschik, 2007; Supplementary file 2).
GO enrichment analysis
Request a detailed protocolGO and pathway enrichment analyses were performed using Metascape (Zhou et al., 2019).
Integration of human TFtarget interactions
Request a detailed protocolA collection of 456,698 TFtarget interactions for 1541 human TFs and their mode of regulation (MoR, activation or repression) were integrated by merging a builtin collection of human regulons from the DoRothEA R package (version 1.8.0) (GarciaAlonso et al., 2019) and human TRRUST database (Han et al., 2018). To be specific, resource from DoRothEA includes 454,504 TFtarget interactions for 1541 human TFs, retrieved from (1) literaturecurated resources, (2) ChIPseq binding data, (3) TFBS (TF binding sites) predictions, and (4) transcriptional regulatory interactions inferred from published gene expression profiles. Human TRRUST database is a manually curated database of transcriptional regulatory networks, whose current version contains 8427 TFtarget regulatory relationships of 795 human TFs with annotation of MoR (activation, repression, or unknown). For our purpose, we excluded TFtarget interactions with unknown MoR, yet 207 TFtarget pairs still remained to show conflict MoRs in records from literature, according to TRRUST database. To integrate data from both resources, when a conflict occurred in TRRUST database, we retained the MoR recorded in DoRothEA resource for the same pair of TFtarget, but omitted the conflict records if the TFtarget pair was not included in DoRothEA resource. As for TFtarget pairs without conflict MoRs in TRRUST database but shows conflicts across the two resources, records in TRRUST database were taken as more credible (Supplementary file 3).
Integration of human TF list
Request a detailed protocolIn addition to the 1541 human TFs we obtained from the collection of TFtarget interactions mentioned earlier, 1639 human TFs reported by Lambert et al., 2018 and 1564 by Ng et al., 2021 were used as complements, ending up with a total of 2051 human TFs (Supplementary file 3).
Data availability
All data analysed during this study are included in the manuscript and supporting files.

NCBI Gene Expression OmnibusID GSE79578. Dynamics of lineage commitment revealed by singlecell transcriptomics of differentiating embryonic stem cells.

NCBI Gene Expression OmnibusID GSE207654. Direct chemical reprogramming of human cord blood erythroblasts to induced megakaryocytes that produce platelets.
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Article and author information
Author details
Funding
National Key Research and Development Program of China (2021YFF1200500)
 Zhiyuan Li
National Natural Science Foundation of China (12225102)
 Zhiyuan Li
National Key Research and Development Program of China (2021YFA0910700)
 Zhiyuan Li
National Natural Science Foundation of China (12050002)
 Zhiyuan Li
National Natural Science Foundation of China (12226316)
 Zhiyuan Li
The funders had no role in study design, data collection and interpretation, or the decision to submit the work for publication.
Acknowledgements
We thank Y Zhao for advice on model application on real biological processes; D Grün for insightful and generous feedback on noise interpretation; J Zhang, Y Li, and X Pei for providing original rawdata processing R scripts; Z Zhou for illustrations and useful discussion; and the entire Zhiyuan laboratory for support and advice. This work was supported by the National Key Research and Development Program of China (No. 2021YFF1200500, 2021YFA0910700), and the National Natural Science Foundation of China (No.12225102, 12050002, and 12226316). It is also supported by grants from the PekingTsinghua Center for Life Sciences.
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