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Copyright © 2008 The Biophysical Society. All rights reserved.
Biophysical Journal, Volume 95, Issue 4, 1575-1589, 15 August 2008

doi:10.1529/biophysj.107.120600

Biophysical Theory and Modeling

Positive Receptor Feedback during Lineage Commitment Can Generate Ultrasensitivity to Ligand and Confer Robustness to a Bistable Switch

Santhosh Palani^* and Casim A. Sarkar^*, †, ,  

^* Department of Bioengineering, University of Pennsylvania, Philadelphia, Pennsylvania 19104-6321
^† Department of Chemical and Biomolecular Engineering, University of Pennsylvania, Philadelphia, Pennsylvania 19104-6321

Address reprint requests to Casim A. Sarkar, 240 Skirkanich Hall, 210 S. 33rd St., Dept. of Bioengineering, University of Pennsylvania, Philadelphia, PA 19104-6321. Tel.: 215-573-4072; Fax: 215-573-2071.

The process of cellular differentiation entails a complex series of events through which an uncommitted progenitor can morph into a stable specialized cell. Although many of the critical individual molecular components involved in specific differentiation processes have been identified, the complex interactions and topology of signaling and transcriptional networks can lead to nonintuitive behavior. Mathematical modeling and analysis can provide insights into the system-level properties that arise from such an array of interactions.

In cellular processes in which a binary decision must be made, bistability can be an important system-level property that arises from the corresponding signaling networks. Changes in the system input can toggle a bistable system between two steady states; additionally, the system can display memory by sustaining a high (or low) steady-state response after significant reduction (or increase) in the magnitude of the stimulus 1. Biological examples of bistability include cell-cycle regulation in Xenopus oocytes 2 and Saccharomyces cerevisiae3, self-sustaining biochemical memory 4, synthetic genetic switches 5,6,7,8,9, and differentiation of common myeloid precursors into macrophages and neutrophils 10. Bistability is often accompanied by ultrasensitivity to a stimulus, another common property of nonlinear systems 11,12,13. Since, there is growing evidence that cell differentiation is an all-or-none “switch-like” event, rather than a continuous transition of an unspecialized cell into a mature one 2,14, mathematical modeling of the commitment process is attractive because the switch-like response and cellular memory implicit in the biological process arise naturally in the formulation of such nonlinear models.

Hematopoiesis, the formation of blood cells, takes place in two distinct stages: primitive differentiation and terminal differentiation. During primitive differentiation, a hematopoietic stem cell differentiates into a multipotent or bipotent progenitor cell, which, upon terminal differentiation, gives rise to a mature cell. It has been suggested that primitive differentiation is primarily a stochastic process involving differential expression of several intrinsic transcription factors, whereas terminal differentiation involves both cell-intrinsic and cell-extrinsic factors 15,16. Lineage-specific cytokines (extrinsic) and transcription factors (intrinsic) are believed to be the important molecular components that affect cell survival, proliferation, and commitment during terminal differentiation.

Erythropoietin (Epo) is a hematopoietic cytokine responsible for the proliferation, survival, and differentiation of erythroid cells 17. The Epo receptor (EpoR) has a single transmembrane domain, an extracellular domain for Epo binding, and an intracellular domain for signaling 18. In the absence of ligand, Epo receptors exist predominantly as inactive homodimers on the cell surface 19. Binding of Epo to the receptor homodimer changes the orientation of the receptor subunits, which leads to activation of several signaling cascades including the PI3K/AKT, STAT5-Bcl_XL, and Ras/MAPK pathways 20. Erythroid progenitors lacking functional EpoR do not mature into erythrocytes and show phenotypic abnormalities 21,22.

The zinc-finger GATA-1 is a transcription factor that plays a critical role in erythroid differentiation 23,24. It binds as a monomer to the consensus sequence (A/T)GATA(A/G), which is present in the promoter and enhancer regions of virtually all erythroid-specific genes 25,26,27,28. GATA-1 undergoes several posttranslational modifications (acetylation, phosphorylation, sumoylation, and ubiquitination) that may be critical for its optimal transcriptional activity 29. Analysis of the promoter regions of the EpoR gene shows no TATA or CAAT box, but does reveal the presence of a GATA-1 binding motif, thus providing a meaningful link between a lineage-specific transcription factor and a lineage-restricted receptor 30,31,32. Active GATA-1 also binds to the regulatory region of its own gene, thereby enhancing its total expression through a positive feedback loop 33,34,35,36. Disruption of the GATA-1 gene in murine embryonic stem cells by homologous recombination blocks erythroid development, emphasizing the absolute need for GATA-1 in red blood cell maturation 37.

Common myeloid progenitors give rise to erythroid burst-forming units (BFU-E), the earliest known erythroid precursor cells. BFU-E mature into erythroid colony-forming units (CFU-E); this is accompanied by an increase in EpoR expression and the cells become increasingly dependent on Epo 38,39. EpoR and GATA-1 levels both rise in parallel and reach their maximum during CFU-E maturation into proerythroblasts and their subsequent differentiation into early basophilic erythroblasts 40,41. Both GATA-1 and EpoR levels fall during further maturation from the basophilic stage to the polychromatic stage as cells synthesize large amounts of globins 38,41. Further differentiation from polychromatic erythrocytes to reticulocytes is independent of EpoR and GATA-1, as their levels fall sharply and the cells also show a decrease in globin expression 38,41. Hence, it is during the temporal window from an early CFU-E to a basophilic erythroblast that EpoR and GATA-1 may act in concert to drive commitment of the erythroid precursor to terminal differentiation and induce the synthesis of globins.

Recent evidence suggests several modes of cross talk between EpoR signaling and GATA-1 transcriptional activity, and analysis of these interactions may offer insights into the commitment program during erythroid differentiation. In brief, EpoR signaling via AKT can lead to GATA-1 activation; in return, active GATA-1 can upregulate synthesis of both itself and EpoR (Fig. 1). Epo activates AKT by phosphorylating this kinase at Ser-473 in a PI3K-dependent manner 42. The importance of AKT signaling in erythropoiesis was demonstrated in JAK2-deficient fetal liver progenitor cells: erythroid differentiation can be supported in these cells by overexpressing active AKT and it can also be inhibited by downmodulating AKT using RNA interference 43. Active AKT appears to have a significant role in enhancing GATA-1 transcriptional activity by mediating some of its posttranslational modifications, including phosphorylation and acetylation. AKT phosphorylates GATA-1 at Ser-310 and enhances its transcriptional activity in primary fetal liver cells 42. However, mice with a S310A mutation in GATA-1 showed no hematopoietic abnormalities during normal or stress erythropoiesis, indicating that phosphorylation of GATA-1 is dispensable for red blood cell differentiation and may only be required for maximal activity 44.

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Figure 1

Kinetic model of the EpoR/GATA-1 network implicated in erythrocyte progenitor commitment. EpoR and GATA-1 are present at low basal levels before the addition of Epo. Epo binds to EpoR homodimers (EpoRJD), thereby activating AKT through the PI3K/AKT pathway. Doubly phosphorylated AKT (AKTpp) activates GATA-1 directly and indirectly through covalent modifications (modeled here as a single step). GATA-1*, the activated form of the transcription factor, upregulates GATA-1 and EpoR gene expression by binding to GATA motifs present in the response elements of their corresponding genes. The core reaction of the model is the activation of GATA-1 by AKTpp and is highlighted with a light gray background. The reactants in the open boxes comprise one feedback process (with transcriptional strength F₁), a synthesis loop that generates more inactive GATA-1 (substrate), and the species in the dark gray boxes represent a second feedback process (with transcriptional strength F₂), an Epo-regulated loop that makes more AKTpp (enzyme). Both feedback loops provide inputs to the core reaction to form GATA-1* (product). All reactants except Epo and JAK2 are time variant. Reaction sets 1,2,3,4,5,6,11,20,21,23,25,26, 7,8,9,10,12,13,14,15,16,17,22,24, and 18,19 are modeled with mass-action, Michaelis-Menten, and rapid-equilibrium kinetics, respectively. The species names ending with mRNAc and mRNAn denote cytoplasmic and nuclear mRNA, respectively. Double-headed and single-headed solid arrows indicate reversible and irreversible reactions respectively. Dashed arrows specify irreversible reactions (transcription, translation) in which reactants are not consumed. *, p, and pp denote the activated, singly phosphorylated, and doubly phosphorylated forms of species. All reactions going to null denote first-order degradation processes. The values of the rate constants shown in the figure are given in the Supplementary Material, Table S5 (Data S1) .

p300 and CREB binding protein (CBP) acetyltransferases acetylate GATA-1 at lysine residues present in the C-terminal tail of its zinc fingers 45,46,47. In vivo chromatin immunoprecipitation assays show that lysine to alanine mutations at the acetylation residues dramatically impair GATA-1 association with chromatin 48, suggesting that acetylation is critical for GATA-1-mediated gene expression. p300 and CBP also have histone acetyltransferase (HAT) activity and may play a role in enhanceosome stability by acetylating GATA-1 and histones 49,50. AKT phosphorylates p300 at Ser-1834 and this has been shown to be essential for AT, HAT, and transcriptional activity of p300 51,52,53. Interestingly, Ser-1834 lies in the E1A binding domain that is necessary for binding of p300/CBP to GATA-1 45. It has also been suggested that phosphorylation may aid in GATA-1 binding to CBP, since the Ser-310 residue of GATA-1 is within the C-terminal acetylation motif of GATA-1 42. Taken together, these results suggest an additional role for Epo (other than providing survival and proliferation cues) in erythroid precursor commitment and differentiation by activating GATA-1 through the PI3K/AKT pathway and influencing the intrinsic signals that lead to commitment and differentiation.

Based on this experimental evidence, we present a deterministic model of the upregulation and activation of the erythrocyte-specific transcription factor GATA-1, a “master regulator” of erythrocyte commitment. Lineage specification models previously reported suggest that erythrocyte differentiation from erythroid/myeloid bipotent precursor can arise due to the differential expression of antagonistic transcription factors (upregulation of GATA-1 and downregulation of PU.1) driven primarily by cell-intrinsic events 54,55. These models provide insight into the dynamics of a binary cell-fate decision from the viewpoint of “multilineage priming”, auto-stimulation, and reciprocal repression.

The work presented here focuses on erythrocyte commitment rather than differentiation, and examines how both intracellular and extracellular factors may influence the cell-fate decision. As depicted in Fig. 1, the topology of our model captures the essential elements of outside-in signaling (Epo-mediated activation of GATA-1), intracellular signal amplification (GATA-1-mediated upregulation of GATA-1 synthesis), and inside-out signaling (GATA-1-mediated upregulation of EpoR). Using this model, we show that upregulation of EpoR in erythroid precursor cells upon Epo addition can generate ultrasensitivity to ligand as well as robust bistability in GATA-1 expression during commitment, and this may provide “switch-like” differentiation characteristics.

Further analysis of a generalized minimal model confirms that the topological connectivity of the two feedback loops alone is both necessary and sufficient for generating the overall system dynamics. Although there are several ways of achieving bistability 1,56, feedback loops are the most commonly identified mechanism; however, feedback loops that give rise to robust bistability in purely deterministic models have, to date, been shown to be highly cooperative in at least one reaction 57,58,59,60. Here, we present what we believe is a novel way of achieving robust bistability in cell signaling networks without molecular cooperativity through two linked positive feedback loops. This topology may have general implications for cellular decision-making.

The core reaction of the proposed erythrocyte commitment model is the activation of GATA-1 by AKT through EpoR signaling (Fig. 1, light gray background). Our model concentrates exclusively on the two positive feedback loops that serve to increase the concentrations of the reactant species (AKTpp and GATA-1) in this core reaction, which leads to greater accumulation of GATA-1*, the activated form of a “master regulator” of erythrocyte-specific genes. The model specifically incorporates the following components/motifs in the feedback loops that may have an effect on the overall system behavior:

We have used a deterministic, ordinary differential equation-based approach to model this signal transduction/transcriptional network. Although this modeling framework represents an ideal approximation of the true intracellular milieu 63, it can still provide useful information regarding the system dynamics, particularly for nonlinear systems of the type studied here 64. In step 1 in Fig. 1, JAK2 binds to the intracellular domain of EpoR to form a receptor-JAK complex (EpoRJ). EpoRJ dimerizes to form EpoRJD in step 3. EpoRJ and EpoRJD undergo constitutive receptor endocytosis (steps 2 and 4). In step 5, Epo binds to the extracellular domain of EpoRJD, forming the activated complex (EpoRJD*) and the endocytosis of the complex is shown in step 6. PI3K is activated (PI3K*) by the complex and is deactivated by a phosphatase (steps 7 and 8). PI3K* converts PIP₂ to PIP₃ in step 9. PIP₃ binds to the PH domain of AKT and phosphorylates AKT on Ser-473 and Thr-308 (steps 11–15). This doubly phosphorylated form of AKT (AKTpp) catalyzes the activation of GATA-1 (step 16). Activated GATA-1 (GATA-1*) is deactivated and degraded in steps 17 and 26, respectively. Monomeric GATA-1* enhances transcriptional synthesis of nuclear EpoR mRNA (EpoRmRNAn) and GATA-1 mRNA (GATA1mRNAn) in steps 18 and 19. The nuclear mRNAs (EpoRmRNAn and GATA1mRNAn) are translocated to the cytoplasm (EpoRmRNAc and GATA1mRNAc, respectively), where they are either translated to their corresponding protein forms or degraded (steps 20–25).

EpoR and GATA-1 are present at basal levels in progenitor cells before the addition of Epo. The basal expression of Epo receptor may be independent of GATA-1 as there is also a Sp1 binding site on the 1.7kb 5′-flanking region of the EpoR gene 31. Based on current evidence, it appears that, as EpoR is transported to the cell membrane, it is rapidly bound by JAK2 and homodimerizes 19,20. Accordingly, we have assumed 90% of EpoR to be initially present in the dimeric state, 9% to be monomers bound by JAK2, and 1% to be free receptors. Activation and deactivation reactions of PI3K, PIP₂, GATA-1, and AKT are assumed to have Michaelis-Menten kinetics. AKT phosphorylation is modeled as a two-step process 65. Double phosphorylation of AKT by 3′-phosphoinositide-dependent protein kinase 1 (PDK1) is necessary for its complete activation 66,67. Dephosphorylation of PI3K*, PIP₃, and AKTpp are implicitly modeled without considering the rate of change of the phosphatases involved. The role of AKTpp in GATA-1 activation is modeled as a single enzymatic step, encompassing both direct (e.g., phosphorylation) and indirect (e.g., acetylation) mechanisms. It is important to note that Epo may activate GATA-1 by AKT-independent mechanisms, but this does not change the qualitative nature of the model (see minimal model below). The mRNA transcription rate is assumed to saturate hyperbolically with active transcription factor concentration, a rapid-equilibrium approximation 68. The rate of translation is approximated to be proportional to the concentration of the cytoplasmic mRNA 69. All degradation reactions are modeled with first-order kinetics.

There are two feedback loops considered in this model. Since GATA-1* positively autoregulates its own transcriptional rate, reactions 19, 21, and 24 drive the first positive feedback loop. This loop increases the concentration of inactive GATA-1 in the cell. The strength of this feedback is governed by the parameter F₁V_19, the maximal transcriptional rate of the GATA-1 gene, shown in reaction 19. GATA-1* is also shown to regulate the synthesis rate of EpoRJD through reactions 18, 20, 22, 1, and 3, which start the second positive feedback loop, whose strength is denoted by F₂V_18, the maximal rate of production of EpoR mRNA. In this loop, GATA-1* upregulates the expression of EpoRJD, which in turn increases the number of complexes formed on the cell surface and leads to the increase in the concentration of activated AKT kinase (AKTpp). In this model, F₁ and F₂ determine the relative strengths of the feedback loops as V₁₈ and V₁₉ are kept equal and constant. The core reaction in the model is the activation of GATA-1 by AKTpp (reaction 16) and the two feedback loops work synchronously to drive this reaction and produce GATA-1*, which in turn drives both of the feedback loops and also regulates the transcription of other erythrocyte specific genes.

The full model, which consists of 18 ordinary differential equations derived from 27 reactions with 44 parameters, is given in the Supplementary Material (Table S1, Data S1 ). To simplify parameter estimation and mathematical analyses, the model was completely nondimensionalized; the nondimensional forms of the differential equations and the parameters are also given in the Supplementary Material (Tables S2 and S3, respectively, Data S1 ). In the nondimensional model, each reactant concentration is normalized by the total concentration of its respective basal inactive form. The species used in the mathematical analyses are Epo receptor homodimer (EpoRJD), complex (EpoRJD*), activated AKT (AKTpp), and activated GATA-1 (GATA-1*). The respective nondimensional forms of these reactants are

Of the 44 parameters present in the model, 29 parameters were incorporated directly from the literature, 8 parameters were refined from values provided in the literature, and the remaining 7 parameters (V₁₆, K₁₆, V₁₇, K₁₇, F₁, F₂, k₂₆) were estimated to fit time course measurements of GATA-1 DNA binding activity during erythroid precursor commitment and differentiation as reported by Dalyot et al. 41. Of these 7 parameters, the steady-state values of the reactants in the model are highly sensitive only to F₁, F₂, and k₂₆. To initially compare the model to these experimental data, a negative feedback loop was added to account for the degradation of GATA-1* after progenitor commitment. This was necessary since the experimental data cover a much broader temporal window of the differentiation process, from GATA-1* production in progenitors to complete GATA-1 degradation in mature erythrocytes. We have assumed that the change in GATA-1 DNA binding activity is due to corresponding changes in the levels of GATA-1*. The fitted parameters were then used in mathematical analyses performed without the negative feedback loop, as our model is only intended to analyze the commitment decision of the progenitor cells much earlier in the differentiation process and not account for larger-scale phenotypic changes that are observed in mature erythrocytes after commitment. Tables S5 and S6 (Data S1) give the values of the estimated parameters and the initial conditions of the reactants in the model. Parameter sensitivity analysis was performed for the Epo receptor dimer, complex, activated AKT, and activated GATA-1 by perturbing all 44 parameters and obtaining the normalized steady-state sensitivities (ranging from 0 to 1). The most sensitive parameters for each of the reactants are given in Fig. S2 (Data S1) .

To further validate this fully parameterized EpoR/GATA-1 model (including the negative feedback loop), it was used to perform kinetic simulations of total (nuclear + cytoplasmic) EpoR and GATA1 mRNA levels. These simulations were then compared to experimental measurements of EpoR and GATA-1 mRNA levels 71, which were obtained from the NCBI Gene Expression Omnibus (GEO) database (No. GDS2431) and which represent experimental data entirely independent from those used for parameter fitting. The comparison of model and experiment is given in Fig. S1 (Data S1) and suggests that the model is capable of making accurate predictions.

To ascertain what topological features in the EpoR/GATA-1 model are responsible for its robust bistability, the model was systematically reduced to a minimal form by stepwise elimination of various linear topological motifs, including EpoR homodimerization, PI3K/AKT cascading, multisite phosphorylation, and individual transcription and translation steps (data not shown). Conversely, both feedback loops were critical for robust bistability (see Results).

The minimal model (see Fig. 6) consists of the following reactions. The cell-surface receptor and the inactive lineage-specific transcription factor (InactiveTF) are expressed at basal (ligand-independent) levels in the naïve cell. After addition of ligand, a fraction of the cell-surface receptors become complexes (step 2) and transmit a downstream signal to enzymatically activate the transcription factor (step 4). Constitutive receptor endocytosis, complex internalization, and InactiveTF degradation reactions are shown in steps 1, 3, and 8, respectively. The active transcription factor (ActiveTF) can then upregulate the expression of both receptor and inactive transcription factor (steps 6 and 7, respectively). ActiveTF can be deactivated or degraded (steps 5 and 9, respectively). The activation of transcription factor by complex and its deactivation are modeled as single enzymatic steps and are assumed to have Michaelis-Menten kinetics. Complex internalization and all degradation reactions are modeled to have first-order kinetics. The transcription and translation reactions are modeled as a single step, where the rate of protein formation is assumed to saturate hyperbolically with the concentration of active transcription factor. The state of the system is represented by the concentration of ActiveTF; high levels denote the on-state (committed state) and low levels denote the off-state (naïve state).

The dimensional and nondimensional forms of the minimal model, each consisting of four differential equations, are provided in the Supplementary Material (Tables S7 and S8, Data S1) . The species present in the minimal model are ligand (L, time invariant), receptor (R), complex (C), inactive transcription factor (ITF), and active transcription factor (ATF). The nondimensionalization was performed in a manner analogous to the EpoR/GATA-1 model (Tables S9 and S10, Data S1) :

Stimulus/response plots have been used to predict bistability, hysteresis, and ultrasensitivity in molecular networks 2,5. The system is induced over a wide range of input stimuli and the corresponding responses are obtained after the system reaches steady state. The state of our EpoR/GATA-1 network is represented by the concentration of GATA-1*; high levels (obtained from both accumulation and activation of GATA-1) denote the on-state (committed state) and low levels denote the off-state (uncommitted state). In these simulations, Epo was considered to be the stimulus, and the responses of important downstream effectors activated by the ensuing signals were analyzed. In Fig. 2, the steady-state values of the nondimensionalized reactants (RJD, C, App, GA) are plotted against the concentration of Epo normalized to its dissociation constant (K_d=58pM 72). When [Epo]=0, the system is in the off-state, with RJD at its basal steady-state value of 0.45, and C, App, and GA all at zero, as there are no complexes. As the Epo concentration increases from 0, the steady-state value of the Epo receptor dimer decreases (Figure 2A) as a result of complex formation (Figure 2B) and there is a subsequent marginal increase in App and GA (Figure 2CD). As the concentration of Epo is further increased to 0.96 K_d, the number of complexes formed increases, but this is still not sufficient to maintain the positive feedback loops and the system remains in the off-state. Only when the input stimulus exceeds 0.96 K_d does the system switch to the on-state, as the complexes can then generate enough AKTpp for GATA-1* levels to exceed the threshold concentration needed to sustain the feedback loops. Therefore, the system exhibits ultrasensitivity for a small perturbation in the concentration of Epo about 0.96 K_d. The on-state is accompanied by a large burst of GATA-1*, an event known to precede the accumulation of various erythroid specific genes 41. The system continues to remain in the on-state with further increase in Epo levels.

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Figure 2

Nondimensionalized steady-state response plots: (A) Epo receptor dimer [RJD]. (B) Epo-Epo receptor complex [C]. (C) Activated AKT [App]. (D) Activated GATA-1 [GA]. The stimulus, Epo, is normalized to its K_d value and each downstream effector is normalized to the total concentration of its respective basal inactive form. The plots show that, for the fitted values of F₁ (0.04) and F₂ (0.123), the system is ultrasensitive to Epo and exhibits bistability for a wide range of Epo concentrations (0.008–0.96 K_d).

To explore whether this network can exhibit memory, the system was taken to the on-state by increasing the concentration of Epo to its K_d value. The stimulus was then reduced to 0.96 K_d and the system was allowed to reach steady state. It can be seen from the plots that the system remains in the on-state as the active positive feedback loops can sustain the system in the committed state. Thus, the downstream effectors in the system exhibit hysteresis with respect to cytokine stimulus. As the Epo concentration is further reduced from 0.96 K_d to 0.008 K_d, the steady-state value of RJD increases since less complexes are formed and, in turn, there is a reduction in the levels of App; nonetheless, the number of complexes is still sufficient to sustain the feedback loops and to maintain high levels of GATA-1*. As Epo levels are reduced below 0.008 K_d, the system switches back to the off-state due to a lack of sufficient Epo-mediated signaling. When the system is in the committed state, removal of the stimulus below the threshold level does not immediately bring the system back to the off-state, instead exhibiting bistability over a large range of stimulus concentration. This bistable expression of GATA-1* can reduce the sensitivity of the system to noise by necessitating a high Epo concentration to initially achieve the on-state and, thereafter, by providing marked robustness to the active state. Though the on-state is still maintained when the stimulus level is reduced ∼120-fold below the threshold concentration, further decreases in Epo concentration drive the system back to the off-state, suggesting that it is not completely irreversible. This is in accord with the phenotypic change observed after commitment during which the cell becomes increasingly independent of EpoR signaling and GATA-1 levels start to fall. The high expression of GATA-1 achieved by Epo induction at commitment can initiate chromatin rearrangements and expression of lineage specific genes, thereby “locking” the cell in the mature state. The steady states plotted in Fig. 2 are only the stable values; the unstable steady states are omitted, as they are not experimentally accessible.

The steady-state response plot of GA (Figure 2D) shows that the Epo concentration has to be >0.96 K_d for the system to be in the on-state. Is there a way to attain the on-state for values of Epo less than the threshold concentration? Given the memory implicit in this network, we hypothesized that transient pretreatment of cells with high concentrations of Epo should influence their commitment decision since the switch to the on-state is determined by the number of complexes needed to sustain the positive feedback loops. If the cell were pretreated with a high concentration of Epo for a fixed amount of time, it should still be possible to achieve the on-state even if the Epo concentration was then reduced to a level lower than the threshold concentration (0.96 K_d), since there would be an appropriate accumulation of multiple activated species during pretreatment. To test this using our model, the concentration of Epo during pretreatment was fixed at its K_d value and was then reduced to the value given on the x axis in Fig. 3. The minimum pretreatment time required for the system to attain the on-state for a range of constant Epo concentrations lower than the threshold concentration is plotted in Fig. 3. The corresponding plots of RJD, C, App, and GA requirements to achieve the on-state for lower Epo concentrations are similar and are given in the Supplementary Material (Fig. S4, Data S1) . For Epo concentrations >0.96 K_d, the threshold concentration, the cell does not require pretreatment for commitment. As the Epo concentration is reduced from 0.96 K_d to 0.008 K_d (a range that corresponds precisely to the bistable window in Fig. 2), the pretreatment time required to accumulate sufficient GATA-1* to attain the on-state increases exponentially. Reducing the Epo concentration below 0.008 K_d, does not bring the system to the on-state for any pretreatment time, as the system is in the monostable off-state below this Epo concentration (see Fig. 2).

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Figure 3

Minimum pretreatment time required for the system to attain the on-state for a range of Epo values lower than the threshold concentration (0.96 K_d). Epo is normalized to its K_d. The pretreatment concentration of Epo is kept at its K_d value and is thereafter reduced to the value given on the x axis. The plot suggests that for Epo concentrations >0.96 K_d, the cell should not require pretreatment and, for values less than the threshold concentration, the pretreatment time increases dramatically with decreases in Epo concentration. For Epo values <0.008 K_d, the system will always remain in the monostable off-state for any pretreatment time (cf. Fig. 2).

The EpoR/GATA-1 network consists of two positive feedback loops that coordinate to create a burst of GATA-1*, an event critical for erythrocyte commitment. The first feedback loop is the transcription of GATA1 by GATA-1*, which increases the concentration of inactive GATA-1 (substrate), and the second feedback loop is the transcription of EPOR by GATA-1*, which leads to an increase in the levels of AKTpp (enzyme) in the presence of Epo. Parameters F₁ and F₂ govern the maximum transcriptional rates of GATA1 and EPOR, respectively, and hence represent the strength of the GATA-1*/GATA-1 and GATA-1*/EpoR/AKTpp feedback loops, respectively. The parameter-fitted values of F₁ and F₂ are 0.04 and 0.123, which correspond to a steady-state value of GA=295 as seen in Figure 2D. In addition to other epigenetic factors, one possible explanation for the difference in the transcriptional rates of GATA1 and EPOR could be the distinct mechanisms by which GATA-1 binds to its consensus sequence present in the promoter regions of these genes 32,34). It should be noted that the two positive feedback loops are interdependent (linked via the GATA-1 activation reaction; reaction with light gray background in Fig. 1) and are necessary for the commitment decision to accumulate GATA-1*. When F₁=0, the cell cannot make more inactive GATA-1, and can only activate the existing low levels of GATA-1, so the system stays in the off-state for any value of F₂>0; similarly, when F₂=0, the cell cannot make enough surface complexes to activate GATA-1 via AKTpp, so the system remains in the off-state for any physiologically reasonable value of F₁>0. For very large values of F₁, however, high levels of GATA-1* can be achieved, albeit in a manner that does not impart memory to the system (the stimulus/response plot in this case is hyperbolic and monostable everywhere).

Figure 4A shows a three-dimensional (3D) plot of the steady-state value of GA as a function of F₁ and F₂ when the Epo concentration is equal to its K_d. Changing the values of F₁ and F₂ can switch the system from the off-state to the on-state as well as change the set point of the reactants—specifically GA—in the on-state. As seen from the plot; if either F₁ or F₂=0, the system is always in the off-state, regardless of the strength of the other feedback process. For the estimated value of F₁=0.04, as we increase F₂ from 0, the system remains in the off-state until F₂ reaches 0.118. Any increase of F₂ over 0.118 causes the system to switch to the on-state with a GA set point value of 295. Further increasing F₂ does not change the value of GA and the system remains in the on-state. For the estimated value of F₂=0.123, as we increase F₁ from 0, the system remains in the off-state until F₁=0.01, at which point the system switches to the on-state with a low GA set point value. As we further increase F₁ to 0.04, the system stays in the on-state and increases the GA steady-state value to the estimated value of 295. When F₁ is increased beyond 0.04, the steady-state value of GA increases and saturates at an F₁ value of 1.2.

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Figure 4

Effect of the two positive feedback loops on the on-state GA value. (A) Steady-state GA values as a function of F₁ and F₂; Epo is kept at its K_d value. For the estimated values of F₁ (0.04) and F₂ (0.123), the system is strongly F₁-limited. (B) Corresponding phase diagram of the 3D plot showing the off-state region, the F₁-limited on-state, and the F₂-limited on-state. Increasing the values of F₁ and F₂ increases the on-state set point of GA in the F₁-limited and F₂-limited regions, respectively. The EpoR/GATA-1 system is likely to behave as an F₁-limited system due the high GA values required to be F₂ limiting.

A top view of the 3D plot is given in Figure 4B to address the effect of changes in F₁ and F₂ on the set point of GA in the on-state. In this phase diagram, the on-state is divided into two regions: F₁-limiting, where an increase in F₁ (but not F₂) will increase the set point of GA in the on-state, and F₂-limiting, where an increase in F₂ (but not F₁) will increase the set point of GA in the on-state. It can also be seen that the critical value of F₂ above which the system attains the on-state slightly decreases as we increase F₁ and the critical line eventually asymptotes at F₂=0.05 for very high F₁. The EpoR/GATA-1 system is likely to always be F₁-limited because of the extremely high GATA-1 levels required to be F₂-limited. Steady-state response plots of GA for several values of F₁ and F₂, spanning both F₁- and F₂-limited regions, are given in Fig. S5 (Data S1) .

The steady-state response plot in Figure 2D shows the wide range of Epo concentrations in which GATA-1* exhibits bistable expression for the fitted F₁ and F₂ values. To understand the influence of the two positive feedback loops in defining the bistable window, we plotted the monostable (either ON or OFF) and bistable (ON and OFF) GATA-1* regions as a function of Epo concentration and feedback strength (Fig. 5). Here, the F₁ and F₂ values are chosen to cover both the F₁- and F₂-limiting regions as shown in Figure 4B. In Figure 5A, a log-log plot of F₁ versus [Epo]/K_d, with F₂ constant (0.20), shows the regions of monostable and bistable expression of GATA-1*. At low F₁ values, the system only achieves bistability for a narrow range of Epo concentrations. As we increase F₁, the bistable window increases and remains constant for larger F₁ values. The increase in the bistable window is only due to the decrease in the bistable-on to monostable-off transition concentration, as the bistable-off to monostable-on threshold concentration remains constant for all values of F₁. This reveals that the Epo concentration at which the system initially switches to the on-state is independent of F₁. However, F₁ governs the extent of memory in the system by changing the Epo concentration at which the system switches from the on-state to the off-state. Figure 5B shows a semilog plot of F₂ versus [Epo]/K_d, with F₁ constant (1.8). For values of F₂ <0.08, the system remains in the off-state for all Epo concentrations. The system attains the on-state for higher F₂ values and also exhibits bistability for a wide range of Epo concentrations. In contrast to Figure 5A, the bistable window in Figure 5B shifts as we increase F₂ due to a substantial decrease in the bistable-on to monostable-off transition concentration as well as a smaller decrease in the bistable-off to monostable-on threshold concentration. This indicates that F₂ plays a role in determining the Epo concentration at which the system reaches on-state as well as in governing the magnitude of memory in the system.

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Figure 5

Bistable expression of GATA-1*. (A) A log-log plot showing the change in the bistable expression of GATA-1* for varying F₁ (F₂=0.20). The threshold Epo concentration needed to achieve the on-state appears to be independent of F₁. The maximum bistable window achievable is dependent on F_2; however, the width of the bistable window is F₁-dependent for lower F₁ values. (B) A semi-log plot showing the bistable expression of GATA-1* for changing F₂ values (F₁=1.8). There appears to be a threshold F₂ value below which the system is purely monostable. Increasing F₂ increases the width of the bistable window and, to a lesser extent, decreases the threshold Epo concentration required to reach the on-state. (C) Narrow bistable expression of GATA-1* for various F₂ values when F₁=0. The ON* state denotes an on-state due to activation but no accumulation. This state would not commit a cell to differentiate due to low basal levels of GATA-1. When F₂=0, the system is monostable for any F₁ value (e.g., the x axis in Figure 5B).

Since basal levels of inactive GATA-1 are low, the system needs both F₁ (for accumulation of GATA-1) and F₂ (for activation of GATA-1) to attain the on-state (accumulation of activated GATA-1). For systems having a high basal expression level of inactive transcription factor or lineage-specific receptor (though neither is the case for the erythrocyte differentiation problem), it becomes relevant to examine how bistability may be achieved. Can such systems potentially attain the on-state* (activation, no accumulation) even if F₁=0 or F₂=0? We tested this using our EpoR/GATA-1 model, with F₁ fixed at 0. Figure 5C shows the bistable expression of the active transcription factor in the absence of F₁. For a given F₂, the Epo concentration at which the system attains the on-state does not change when compared with Figure 5B, but the respective concentration at which the system switches back to the monostable off-state is increased dramatically, thus narrowing the bistable window, or memory, in the system. A system that has feedback 1 (upregulation of GATA-1 by GATA-1*) but no feedback 2 (upregulation of EpoR by GATA-1*) does not exhibit bistability for any value of F₁, confirming the observation that autoregulating positive feedback loops without cooperativity do not show bistability in deterministic models 57,58; bistability in a system lacking feedback 2 can be recovered by incorporating the need for transcription factor dimerization for activation (data not shown). In summary, this shows that the bistability and ultrasensitivity achieved in the EpoR/GATA-1 model were primarily due to the presence of feedback 2, and that feedback 1 only plays a role in increasing the extent of memory in the system. The values of F₁ and F₂ may also change during the differentiation process, thus dynamically modulating the robustness of the system, though this time dependence was not considered here.

The EpoR/GATA-1 model exhibits ultrasensitivity and bistability for a wide range of Epo, F₁, and F₂ values. The structural aspects of the EpoR/GATA-1 model are the two linked positive feedback loops, receptor homodimerization, PI3K/AKT signaling pathway, double phosphorylation of AKT, and the transcription and translation steps. We systematically developed and tested various submodels of the parent model to identify the dispensable steps and to obtain a generalized minimal model that still retains the ultrasensitivity and robust bistability of the parent model (data not shown). This analysis revealed that the two positive feedback loops were both necessary and sufficient for recapitulating the overall system behavior of the full EpoR/GATA-1 model. This reduced lineage-specific receptor/transcription factor model (Fig. 6) includes only four time-dependent species: receptor (R), complex (C), and inactive (ITF) and active (ATF) transcription factor. This model was solved analytically and the exact solution for each of the four reactants was determined (Table S11, Data S1) . The steady-state response plots for these reactants for selected values of F₁ and F₂ are given in Fig. S6 (Data S1) . The bistability plots of ATF (Fig. S7, Data S1) in the minimal model closely mimic those in the EpoR/GATA-1 model (Fig. 5).

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Figure 6

Generalized minimal model for lineage commitment. Receptor and Inactive Transcription Factor (InactiveTF) are present at basal levels before the addition of ligand. Ligand binds to Receptor to form Complex and activates InactiveTF to form Active Transcription Factor (ActiveTF). ActiveTF upregulates the levels of InactiveTF and Receptor through two positive feedback loops. Reaction sets 1,2,3,8,9, 4,5, and 6,7 are modeled with mass-action, Michaelis-Menten, and rapid-equilibrium kinetics, respectively. Double-headed and single- headed solid arrows indicate reversible and irreversible reactions, respectively. Dashed arrows specify irreversible transcriptional activation and translation reactions (modeled as a single step). All reactants except ligand are time variant. All reactions going to null denote first-order degradation processes. The values of the rate constants shown in the figure are given in Table S12 (Data S1) . (Inset) A further simplified schematic of the minimal model highlighting the two feedback loops. L, R, ITF, and ATF denote the nondimensional forms of Ligand, Receptor, InactiveTF, and ActiveTF, respectively. Basally expressed R converts basally expressed ITF to ATF only in the presence of L. ATF upregulates itself by inducing the expression of both ITF (with transcriptional strength F₁) and R (with transcriptional strength F₂). It should be noted that positive feedback to ITF is intrinsically regulated, whereas positive feedback via R (to activate ITF) is dependent on the external stimulus L.

Unlike the EpoR/GATA-1 model, the minimal model can be solved analytically, which can prove useful in understanding the contributions of each of the two positive feedback loops to the overall behavior of the system. The solution curves of ATF are plotted against L (normalized to its K_d) for various values of F₂ holding F₁ constant at 20 as shown in Figure 7A. The solid lines and the dotted lines denote the stable and unstable roots, respectively. For low values of F₂ (0.01), the system has only two real roots (one stable and one unstable) and is purely monostable for all ligand concentrations. As F₂ is increased to 0.1, the expression of the ActiveTF becomes narrowly bistable, with the endpoints of this bistable window defined by two saddle-node bifurcations that appear to depend on F₂ but not on F₁ (see below). The degree of bistability increases dramatically as F₂ is increased to 1 and then to 10. It can be seen that, for these larger F₂ values, the solution curves intersect to form a transcritical bifurcation. The transcritical set point, which constrains the maximum theoretical value of ATF, seems to be independent of F₂. The transcritical bifurcation also divides the bistable window into F₁-limiting (right of the transcritical bifurcation point; constant on-state value) and F₂-limiting regions (left of the transcritical bifurcation point; variable on-state value). For low F₂ values, the bistable region is completely F₂-limited and as we increase F₂, the bistable region becomes increasingly F₁-limited. Finally, it can also be observed that the threshold ligand concentration to achieve the on-state decreases as we increase the F₂ value.

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Figure 7

Bifurcation analysis for the minimal model. (A) Solution curves obtained from the analytical solution of nondimensionalized Active Transcription Factor (ATF) are plotted against ligand concentration normalized to its K_d (L) by varying F₂ (0.01, 0.1, 1, and 10); F₁ is held constant at 20. The solid lines and the dotted lines denote the stable and unstable roots, respectively. For low F₂ values, the system is purely monostable. As we increase F₂, two saddle-node bifurcations emerge and they determine the width of the bistable window and the threshold ligand concentration necessary to reach the on-state. As we further increase F₂, the solution curves intersect to form a transcritical bifurcation. The saddle-node set points and the transcritical set point (maximum achievable on-state value) are F₂- and F₁-dependent, respectively. (B) F₁ is varied (0, 4.3, 20, and 200); F₂ is held constant at 10. The system can achieve bistable expression of ATF even when F₁=0. As we increase F₁, the bistable window increases and reaches a maximum width, which is determined only by the F₂ value. Increasing F₁ further increases the on-state set point value of ATF but has no effect on the threshold ligand concentration required to reach the on-state.

Figure 7B shows the bifurcation diagrams for ATF plotted against L by varying F₁ and keeping F₂ constant at 10. Unlike the previous case (Figure 7A), the system can achieve bistability over a narrow range of L by forming two saddle-node bifurcations even when F₁ is zero. As we increase F₁, the size of the bistable window and the on-state set point value both increase. At a critical value of F₁ (here 4.3), the maximum bistable window is achieved, coincident with the appearance of an apparent subcritical pitchfork bifurcation at the lowest value of L at which the system is still bistable. At this F₁ value, the set point of the ATF in the bistable region is completely F₁-limited. As F₁ is increased beyond 4.3, the solution curves form a transcritical bifurcation similar to that seen in Figure 7A. As we further raise the value of F₁ to 20 and then 200, the value of L at which the transcritical bifurcation occurs shifts from low to high, making the bistable region increasingly F₂-limited. This is in contrast to Figure 7A, where the transcritical bifurcation point moves from right to left and the bistable region becomes increasingly F₁-limited as we increase F₂. Importantly, increasing F₁ augments the maximum on-state set point value of ATF but has no effect on the threshold ligand concentration necessary for achieving the on-state.

By comparing the bifurcation plots of ATF in Fig. 7 with various plots of activated GATA-1 (Figure 2D, Fig. 5, and Fig. S5 in Data S1 ), the following conclusions can be deduced for the EpoR/GATA-1 system: the width of the bistable region and the range of GATA-1* values in the on-state can both depend on F₁ (under F₁-limited conditions) and/or F₂ (under F₂-limited conditions); the maximum GATA-1* value in the on-state is determined by F₁; the threshold Epo concentration at which the system switches to the on-state is dictated by F₂; the maximum bistable window achievable is set by F₂; and, the system requires an F₂ value above a critical threshold to exhibit bistability.

EpoR and GATA-1 are both essential for erythrocyte precursor commitment and differentiation, and we present here a deterministic model that bidirectionally links the lineage-specific receptor and transcription factor. Based on recent biochemical data 42,43,45,53, we chose the PI3K/AKT cascade as the signaling pathway that connects EpoR and GATA-1. The model accounts for basal expression of EpoR and GATA-1, Epo binding to EpoR to activate the PI3K/AKT pathway, activation of GATA-1 by phosphorylated AKT, positive autoregulation of GATA-1 expression by GATA-1*, and upregulation of EpoR expression by GATA-1*.

To gain mechanistic insights into system behavior, we chose to focus on this small set of critical molecular effectors implicated in erythropoiesis. However, it should be noted that our explicitly modeled topology represents only a fraction of the full regulatory network and, therefore, inferring cell fate from the level of a single metric (e.g., GATA-1*) represents an approximation of a high-dimensional attractor 55,73. Signaling pathways that were excluded from our model include JAK2/STAT5/Bcl_XL, which provides antiapoptotic signals during erythrocyte differentiation 74, and Ras/MAPK, which is involved in cell survival 75, cell-cycle regulation 76, and the degradation of DNA-bound GATA-1 77. Also, the JAK2/STAT5 pathway activated by Epo can initiate a negative feedback loop on the PI3K/AKT pathway by activating SOCS proteins that can suppress Epo receptor signaling 78,79. Our model, despite neglecting these additional complexities, can nevertheless effectively capture the system dynamics observed in multiple indepedent experimental data sets (see Fig. S1, Data S1 ).

Through steady-state response plots (Fig. 2) and bistability plots (Fig. 5), it was revealed that the EpoR/GATA-1 network can exhibit ultrasensitivity and bistability. Since these properties may play important roles in erythrocyte commitment, it was informative to probe the role of positive feedback in such a topology. As shown in Figure 7A, positive receptor feedback can ultrasensitize the system to ligand and can generate a considerable memory effect once the on-state is achieved. Other transcription factors (e.g., GATA-3 80) are believed to be intracellularly amplified through a classical autoregulatory positive feedback loop: synthesis of the new transcription factor is followed by dimerization (or higher order oligomerization) and the complex is then transcriptionally active. If this is sufficient for programming cell fate, why, then, might a transcription factor such as GATA-1 have evolved to upregulate a lineage-specific receptor as well?

The answer may lie in the different modes of activation. Whereas the dimerization step closes the positive feedback loop for some transcription factors, experimental evidence suggests that GATA-1 binds DNA as a monomer 34,81,82 and shows no detectable DNA-binding ability before the addition of Epo 41. Thus, EpoR signaling may be necessary to close the GATA-1 autoregulatory loop by activating the transcription factor via AKT. By upregulating EpoR to increase its own activation, GATA-1 can effectively mimic the molecular cooperativity of other transcription factors in generating robust network bistability without employing any cooperative reactions. (The importance of cooperativity in achieving bistability is restricted to the class of deterministic models discussed in this work; it is indeed possible to achieve steep sigmoidal responses through nonidealities such as molecular crowding 83, stochastic focusing 84, and dimensionally restricted reactions 63.)

Two unique elements of the EpoR/GATA-1 model should be highlighted. First, by decoupling the synthesis and activation steps in the positive GATA-1 autoregulatory loop, a cell may be able to independently tune the switching threshold, the on-state expression level, and the extent of memory in the network by separately modulating F₁ and F₂ (e.g., epigenetically). Second, there is an external checkpoint (Epo) that modulates this autoregulatory loop. This is attractive because it provides a novel and meaningful link between canonically extrinsic (cytokine) and intrinsic (transcription factor) signals in regulating not only cell survival but also maturation.

The strengths of the positive feedback loops are governed by the rates of transcription of GATA1 (F₁) and EPOR (F₂). The estimated values of F₁ and F₂ are 0.04 and 0.123, respectively. The difference in the rates of transcription of EPOR (chromosome 19p13.2) and GATA1 (chromosome Xp11.23) may be due to the distinct binding mechanisms of GATA-1 to these promoters, dissimilarities in the ease of accession of the GATA-1 binding sites, and the recruitment of other cofactors that may regulate EPOR and GATA1 expression differently. GATA-1 also interacts with other factors, notably the ubiquitous transcription factor Sp1, erythroid restricted factor EKLF, and friend of GATA-1 (FOG1) that may alter its transcriptional activity among the various GATA-1 regulated genes 29.

Though treated as constants in our model, F₁ and F₂ may also change temporally during commitment and differentiation due to additional biophysical (e.g., chromatin remodeling) and biochemical (e.g., cofactor upregulation/downregulation) processes. Accordingly, the values of F₁ and F₂ may also vary substantially between primary cells and immortalized lines, and may even differ among cell lines, depending upon how far a cell line is from commitment toward the erythrocyte lineage, relative expression of GATA-1 cofactors, basal levels of EpoR and GATA-1 expression, and expression of antagonistic transcription factors driving other lineages. Cell-specific feedback strengths that differ significantly from those used in our models may serve to attenuate or amplify the actual effects on the network.

Finally, system-level properties such as bistability and ultrasensitivity that may be generally applicable to lineage commitment can be experimentally corroborated. Pretreatment of progenitor cells with ligand, as outlined in the Results section (Fig. 3), can be performed to show expected hystereses in transcription factor activation and lineage commitment. Additionally, the models elucidate how the steady-state response profiles of activated transcription factor can be influenced by F₁ and F₂, and these can be experimentally validated by using pharmacological inhibitors or RNA interference to exogenously manipulate the values of F₁ and F₂. Although the results presented here are motivated by the EpoR/GATA-1 network and its critical role in erythropoiesis, it will be interesting to see whether similar topologies are uncovered in other cell systems that enable their hosts to make robust decisions in response to external stimuli.