A BMP-FGF Morphogen Toggle Switch Drives the Ultrasensitive Expression of Multiple Genes in the Developing Forebrain

Borders are important as they demarcate developing tissue into distinct functional units. A key challenge is the discovery of mechanisms that can convert morphogen gradients into tissue borders. While mechanisms that produce ultrasensitive cellular responses provide a solution, how extracellular morphogens drive such mechanisms remains poorly understood. Here, we show how Bone Morphogenetic Protein (BMP) and Fibroblast Growth Factor (FGF) pathways interact to generate ultrasensitivity and borders in the dorsal telencephalon. BMP and FGF signaling manipulations in explants produced border defects suggestive of cross inhibition within single cells, which was confirmed in dissociated cultures. Using mathematical modeling, we designed experiments that ruled out alternative cross inhibition mechanisms and identified a cross-inhibitory positive feedback (CIPF) mechanism, or “toggle switch”, which acts upstream of transcriptional targets in dorsal telencephalic cells. CIPF explained several cellular phenomena important for border formation such as threshold tuning, ultrasensitivity, and hysteresis. CIPF explicitly links graded morphogen signaling in the telencephalon to switch-like cellular responses and has the ability to form multiple borders and scale pattern to size. These benefits may apply to other developmental systems.


Introduction
The formation of borders between compartments and body parts is crucial for embryonic development [1,2,3,4,5]. A challenge in understanding border formation is the elucidation of mechanisms that convert shallow morphogen gradients into sharp expression domains [3,6]. Such mechanisms fall into two categories: those that involve cell-cell cooperation, such as cell sorting [2,5], and those that do not and are therefore cell-intrinsic. Cell-intrinsic border-forming mechanisms amplify small foldchanges in extracellular morphogen concentration into large fold-changes in target gene expression [7]. Such 'switch-like' behavior, also known as ultrasensitivity, enables cells embedded in a morphogen gradient to convert slight differences in morphogen concentration into sharp gene expression domains.
Extensive studies in many systems [6], including the mammalian spinal cord [8] and syncytial fly blastoderm [6,9], show that ultrasensitivity and border formation can result from complex interactions between a morphogen and its downstream transcription factor network, or within a transcriptional network alone. While such morphogen-transcription networks have been explored, the interactions between extracellular morphogens as a basis for ultrasensitivity has not been described, even though such interactions are common in development [10].
One system patterned by interacting morphogens is the dorsal telencephalon [11], in which cell-intrinsic ultrasensitivity was proposed to mediate border formation between the telencephalic dorsal midline (DM) and cerebral cortex [12]. The DM -located between the cerebral cortices -develops from the roof plate and adjacent tissues to form the choroid plaque, choroid plexus epithelium (CPE), and cortical hem [13] along the mediolateral axis. These tissues produce BMPs -including BMP4 -at high levels [14] to form an activity gradient of BMP signaling [12,15], with BMP-dependent genes Msx1 and Ttr being expressed in the CPE [15], where BMP activity is highest. Msx1 is a high-threshold BMP target gene in many patterning systems [16,17], including the dorsal telencephalon [12,13,14], while Ttr is induced specifically in the CPE at the onset of its definitive differentiation (,embryonic day 11, or E11, in mice) and is stably expressed thereafter [18].
Although Msx1 is restricted to the midline, BMP4 can induce Msx1 expression in dissociated cortical precursor cells (CPCs) in an ultrasensitive fashion [12]. Both in vivo and in vitro, Msx1 ultrasensitivity contrasts with graded changes in nuclear phospho-Smad1, 5, or 8 (pSmad) levels (a direct readout of BMP signaling intensity), within the same cells. This implies that Msx1 ultrasensitivity occurs downstream of pSmad activation. The mechanism underlying this ultrasensitivity, however, remains unknown.
We investigated the influence of FGFs on DM BMP target genes, and found that ultrasensitivity requires cell-intrinsic interactions between the BMP and FGF pathways. Using explants and dissociated cell cultures, we showed that the BMP and FGF pathways mutually inhibit at the single cell level; Epidermal Growth Factor (EGF) acts similarly to FGF. Next, we used modeling to identify experiments that distinguish among different models of cross-inhibition. These experiments identified a crossinhibitory positive feedback (CIPF) mechanism, or ''toggleswitch'', between the BMP and FGF signaling pathways as the basis for ultrasensitivity. We further show how this mechanism is capable of generating multiple sharp borders simultaneously, among other potential advantages.

Exogenous BMP4 upregulates Ttr and Msx1 in midline and cortical cells, respectively
To experimentally study Ttr and Msx1 regulation in the dorsal telencephalon, we used two previously-characterized in vitro systems: a dorsal forebrain explant system and dissociated cultures [12,15,29]. First, we treated dorsal forebrain explants with BMP4. E9.5 explants cultured with BMP4 exhibited marked expansion of Ttr expression towards the RM (n = 19/23 compared to 0/12 BSA-treated controls; Figure 1A,C). The Ttr induction was restricted to the midline, with no expression seen laterally in the cortex. Sections revealed Ttr induction in cells lining the ventricular surface (1-2 cell diameters deep), with the Ttrexpressing cells often bending inward towards the ventricle ( Figure  S1C); these features are characteristic of endogenous CPE. Correspondingly, RT-qPCR analysis revealed that CPE and DM marker gene Lmx1a was also upregulated rostrally in BMP4treated explants ( Figure S1B). Similar findings were obtained from E10.5 explants (n = 14/18 BMP4-treated, n = 0/6 BSA-treated), although midline Ttr induction was more patchy (data not shown).
Interestingly, in response to exogenous BMP4, Msx1 was ectopically induced in the cortex (n = 5/5 compared to 0/4 BSA-treated explants; Figure 1B,D), but not rostrally in the RM. Thus Msx1's ectopic induction to exogenous BMP4 differs from Ttr's response, which ectopically expands rostrally towards the RM, but not laterally into the cortex.
To determine whether these BMP4-mediated responses are cellintrinsic, we applied BMP4 to dissociated midline cells and CPCs. Midline cultures included cells from dorsal and rostral regions, as both regions were clearly competent for Ttr induction ( Figure 1A,C). Both Ttr in midline cells and Msx1 in CPCs were positively regulated by exogenous BMP4 in a concentrationdependent fashion. In midline cultures, (mRNA) Ttr levels peaked at a BMP4 concentration of 16 ng/ml ( Figure 1E). In CPCs, (mRNA) Msx1 levels increased monotonically ( Figure 1F), as reported previously [12,15]. These findings indicate that the Ttr and Msx1 responses to BMP4 are cell-intrinsic.

FGF receptor inhibition mimics the effects of exogenous BMP4
The rostral expansion of Ttr in BMP4-treated explants suggests that a suppressor of Ttr expression exists in the rostral midline. FGFs produced in the RM, particularly FGF8, are candidates for mediating this suppression, as FGF8 has been shown to negatively influence the BMP pathway in the dorsal telencephalon [25,26]. To test this idea, we treated explants with 100 nM PD173074, a pan-FGF receptor (FGFR) inhibitor [30] (IC 50 = 21.5 nM, K d = 45.2 nM). These explants displayed rostral Ttr expansion reminiscent of that seen in BMP4-treated explants (n = 4/6; Figure 1I); no such changes were seen in control DMSO-treated explants (n = 0/4; Figure 1G). Additionally, placing FGF8-soaked beads adjacent to the endogenous CPE resulted in consistent Ttr suppression (n = 8/12 compared to 0/12 BSA-soaked controls; Figure S1D). These results suggest that FGF8, and possibly other rostral FGFs, normally suppress CPE fate and Ttr expression. In addition, the similarity between BMP4-and PD173074-induced Ttr responses suggests that individual midline cells can respond identically to either increased BMP or reduced FGF signaling. Restricted Ttr induction towards the RM also supports a biphasic model for rostral FGF functions in DM development -i.e. rostral FGFs first provide competency for DM fates, then inhibit them [11] -as seen for FGFs in the chick midbrain DM [11,31].
We then examined Msx1 expression in PD173074-treated explants. Ectopic Msx1 induction was less extensive with PD173074 than with BMP4, but like BMP4, PD173074 treatment led to ectopic Msx1 induction in the cortex but not the midline (n = 7/8; Figure 1H,J). In addition, ectopic Msx1 expression in cortical regions overlapped with PD173074-and BMP4-treated explants (arrows, Figure 1D and J). These findings reveal an FGFR-mediated suppression of Msx1 in the cortex, possibly mediated by FGFs expressed by cortical cells, such as FGF2 and

Author Summary
During development, morphogen gradients play a crucial role in transforming a uniform field of cells into regions with distinct cell identities (marked by the expression of specific genes). Finding mechanisms that convert morphogen gradients into sharp borders of gene expression, however, remains a challenge. Cellular ultrasensitivity mechanisms that convert a linear stimulus into an on-off target response offer a good solution for making such borders. In this paper, we show how a cross-inhibitory positive feedback or toggle switch mechanism driven by two extracellular morphogens -BMP and FGF -produces ultrasensitivity in forebrain cells. Experiments with cells and explanted brain tissue reveal that BMPs and FGFs cross inhibit each other's signaling pathway. Such cross inhibition could occur through four possible mechanisms. By an iterative combination of modeling and experiment, we show the toggle switch to be the mechanism underlying cross inhibition, the ultrasensitive expression of multiple genes, and hysteresis in forebrain cells. As the toggle switch explicitly links extracellular morphogens to cellular ultrasensitivity, it provides a mechanism for making multiple sharp borders that can also scale with tissue size -an important issue in pattern formation. This might explain the abundance of BMP-FGF cross inhibition during development.
FGF1 [20]. They also indicate that cortical cells can respond similarly to either increased BMP or reduced FGF signaling.
To test if FGFR signaling regulates BMP target gene responses at the single cell level, we treated dissociated midline cells or CPCs with PD173074. We found that FGFR inhibition upregulated (mRNA) Ttr ( Figure 1K) in midline cells and Msx1 ( Figure 1L) in CPCs. This indicated that FGFR signaling, presumably activated by FGFs produced by the cultured cells themselves, inhibits the BMP target genes. Next, we tested whether exogenous FGFs inhibit BMP target genes in dissociated midline cells and CPCs. Given the explant results, FGF8 was used in midline cultures, while CPCs were treated with FGF2. Increasing FGF8 led to decreasing Ttr expression in midline cells ( Figure 1M). Similarly, FGF2 resulted in concentration-dependent Msx1 decreases in CPCs ( Figure 1N). Thus, FGF-mediated inhibition of BMP target genes is intrinsic to both midline and cortical cells.

BMP signaling downregulates FGF target responses
We next investigated whether BMP4 can inhibit FGF responses. We first confirmed that FGF8 and FGF2 positively upregulate the FGF target gene, (mRNA) Spry1 [22,32], in dissociated midline and cortical cells, respectively (Figure 2A,D). Correspondingly, the FGFR inhibitors PD173074 and SU5402 decreased endogenous Spry1 expression in midline cells ( Figure S1E). When BMP4 was administered, (mRNA) Spry1 levels were downregulated in a dosedependent fashion in both cell types ( Figure 2B,E). Thus, BMP4 can downregulate an FGF target gene in dissociated midline and cortical cells. We also treated CPCs with the BMP receptor inhibitor LDN193189 [33] (IC 50 = 5 nM). The treatment resulted in dose-dependent decreases in (mRNA) Msx1 levels, while (mRNA) Spry1 levels increased ( Figure S1F). These results with LDN193189 -the converse of those obtained with BMP4provide further support that BMP signaling inhibits the FGF target gene Spry1.
As an additional test for BMP4 inhibition of the FGF pathway, we examined how BMP4 affected FGF-stimulated cell proliferation [34,35]. FGF2 is a known mitogen for CPCs in culture [20].
We found that FGF8 also acted as a concentration-dependent mitogen for dissociated midline cells ( Figure S1E). When BMP4 was coapplied with FGF2 or FGF8, FGF-induced proliferation decreased in a dose-dependent fashion in both cell types ( Figure 2C,F). Thus, BMP4 inhibits FGF-driven proliferation as well as Spry1 expression. Taken together, the experimental data indicate that BMP and FGF signaling inhibit each other's target responses, and that this mutual or cross inhibition is intrinsic to both midline and cortical cells.

Models of BMP-FGF cross inhibition
How might such BMP-FGF cross inhibition occur? One possibility is that FGF directly inhibits Ttr and Msx1 and BMP4 directly inhibits Spry1 ( Figure S2A). Inhibition could also occur upstream at the level of signaling pathway components or other genes that themselves regulate target responses ( Figure S2A). Such upstream inhibition might lead to feedforward and feedback loops (e.g. Figure S2B) and complicated response dynamics. To investigate the behaviors of such systems, we turned to mathematical modeling (Text S1 for rationale and details). In our single cell models, signaling pathways are represented by extracellular morphogens (FGF and BMP), intermediate signals (FGF and BMP intermediates, or F I and B I ), and target responses (F T and B T ). Inhibitory links between the pathways were then introduced, resulting in 81 possible configurations ( Figure S3). Models were reduced to ordinary differential equations, with interactions represented by Hill functions [36], then grouped by similarity in their steady-state behaviors and topology ( Figure S3, Numerical Methods, Text S1 sections 2 and 3).
This grouping resulted in four classes of models described by generalized equations (Table 1): two non-feedback classes, with FGF-to-BMP inhibition occurring at or upstream of B T , one feedforward, and one feedback. We refer to these classes as: 1) simple target inhibition (STI), 2) simple upstream inhibition (SUI), 3) coherent feedforward (CFF), and 4) cross-inhibitory positive feedback (CIPF). Representative models that captured the basic response dynamics of each class are shown in Figure 3A (equations 1-4, and Text S1 for modeling rationale). For simplicity, and because our work focuses on BMP targets (B T ), interaction and nodes that have no influence on B T (e.g. F T ) are omitted from these depictions. Later, we will argue that the particular selections of inhibitory connections in the representative models are likely to be well justified (see Discussion).
Both CFF and CIPF are known motifs. CFF can provide for a ''sign-sensitive delay'' that protects outputs against transient activation spikes [37]. CIPF, first identified by Monod as the theory of double bluff [38], is also known as mutual negative feedback [39], double negative feedback [40], or the ''toggle switch'' motif [36]. It operates during cell fate specification in many developmental systems (e.g [10]). In these systems, CIPF serves to compare two inputs, ultimately turning on targets for the stronger one while turning off those for the weaker input. Depending on the relative strengths of the inputs, CIPF can therefore toggle between two mutually-exclusive sets of target genes.
The BMP-FGF cross inhibition models generate distinct responses One way to compare the different models is to examine the dose-response relationships between BMP and its targets (B T ) in the presence or absence of a fixed amount of FGF. Under these circumstances, three features of the B T response are potentially informative: maximal levels, EC 50 values, and sensitivity. The sensitivity could be either linear (hyperbolic) or ultrasensitive (sigmoidal) to varying degrees, as quantified by its apparent Hill coefficient, or nH. Changes in these response features were evaluated over a wide range of parameter space and across different ''contexts'' in which different links within the models were made nonlinear to different degrees (Figures 3B,C, S4, Text S1 sections 3 and 4, Materials and Methods for curve fitting).
While CIPF and STI produced consistent response changes across contexts, CFF and SUI produced more context dependent response changes (Figures 3B,C,E, S4B). Notably, only CFF and CIPF created or enhanced ultrasensitivity ( Figure 3B,C,E). CIPF always increased ultrasensitivity even with all links linear and more so with non-linear inhibitory links (Figures 3B,E, S4B). With CFF and CIPF parameters that increased ultrasensitivity, FGF decreased B T levels at low BMP concentrations and had negligible effects on B T at high BMP concentrations. Such selective B T suppression at low BMP concentrations invariably resulted in more sigmoidal dose-response curves with higher EC 50 values ( Figures 3C, S4E). This was the invariant pattern by which ultrasensitivity emerged or increased with CFF and CIPF in the presence of FGF.

Dependence of CPC ultrasensitivity on FGF signaling supports CFF or CIPF
To distinguish the models, we performed dissociated culture studies. In the absence of FGF8, maximal Ttr expression in midline cells occurred at 16 ng/ml BMP4 ( Figure 1K). With FGF8 (8 ng/ ml), maximal (mRNA) Ttr levels did not change, but the EC 50 increased to ,31 ng/ml BMP4 ( Figure 4A). The effects argue against an STI model, but are consistent with SUI, CFF, or CIPF ( Figure 3B,E).
We then evaluated Msx1 ultrasensitivity in dissociated CPCs. Our previous Msx1 studies utilized CPCs cultured with FGF2 and EGF [12]. Like FGF2, EGF signaling had an inhibitory effect on Table 1. Generalized equations for each class of models.

Models
Generalized B T solutions where, X (F ) is a monotonically increasing function of FGF where, X (F ) is a monotonically increasing function of FGF where  Figure S1G). Notably, EGF is expressed in the antihem and may form a rostro-lateral to caudal-medial gradient [42]. With FGF2 and EGF, Msx1 induction by BMP4 displayed ultrasensitivity (nH = 3.7; red curve in Figure 4B), as described previously [12]. In the absence of FGF2 and EGF, however, Msx1 induction followed an ideal hyperbolic curve (nH = 1.0; blue curve in Figure 4B). The dose-response curve with FGF2 and EGF differed in four ways from those without them: 1) Msx1 mRNA levels were reduced at low BMP4 concentrations, 2) maximal (mRNA) Msx1 levels at high BMP4 concentrations were unchanged, 3) the EC 50 increased (1.5 to 8.3 ng/ml), and 4) marked ultrasensitivity emerged. These characteristics precisely matched the CFF (with nonlinear inhibitory links) and CIPF models (compare Figures 3C and 4B).

Presence of hysteresis implicates CIPF
To further distinguish among models, we tested for a property of positive feedback systems known as hysteresis, a form of cellular memory or bistability [43,44]. Cells display hysteresis when their dose-response curves differ depending on whether they start in an 'on' or 'off' state (e.g. whether CPCs start with Msx1 highly expressed or not, Figure S4D). Importantly, hysteretic responses, unlike irreversible ones [43], turn off after stimulus removal ( Figure S4D). We examined all four models for their ability to produce hysteresis or any form of bistability.
The STI, SUI, and CFF models, unlike CIPF, yielded steady state solutions for B T that were amenable to analysis. When tested mathematically, STI, SUI, and CFF always produced identical onand off-curves for B T . However, CIPF -which was examined with simulations and monotone stability analysis [43] -produced different on-and off-curves under certain conditions ( Figures 3D,  S4E, S5). Thus, amongst the four cross inhibition models, only CIPF is capable of generating hysteresis. It is possible to produce bistable responses with other motifs, such as auto-regulatory feedback, but these are unlikely in this system (see Figure S4E and Text S1 sections 5 and 6 for discussion on feedback and other models).
Further analysis showed that hysteresis occurs only if at least one of the CIPF loop links was nonlinear, and both links were roughly matched in strength ( Figure S5B-D, Text S1 section 5). This requirement for a ''balanced'' CIPF loop can be explained by considering cases in which F I or B I is far stronger than the other. When F I dominates, CIPF reduces to SUI, whereas when B I is too strong, CIPF reduces to the BMP core pathway; and neither SUI nor the BMP core pathway alone can generate hysteresis. A balanced CIPF loop was also required to produce ultrasensitivity, and the magnitudes of increased sensitivity and hysteresis (i.e. size of the hysteresis 'window') correlated strongly ( Figure S5C-E). Thus, a balanced nonlinear CIPF loop is required for hysteresis and increased ultrasensitivity.
To test for hysteresis, CPCs were cultured for two hours with high BMP4 (64 ng/ml) to induce the Msx1 'on' state [12], or with BSA to maintain CPCs in the Msx1 'off' state. After washing out the BMP4 or BSA thoroughly, media alone (no BMP4) or BMP4 at different concentrations (4-64 ng/ml) was reapplied before harvesting CPCs two days later ( Figure 4C). In CPCs exposed to BSA, then low BMP4, Msx1 expression remained low ( Figure 4D, blue line). However, in CPCs exposed to high BMP4, then no or low BMP4 after washout, (mRNA) Msx1 levels remained high ( Figure 4D, red line), as previously observed [12]. Since Msx1 did not turn off after BMP4 removal, Msx1 appeared irreversible rather than hysteretic. However, another explanation was persistent BMP signaling after washout. Persistent signaling was possible, since slow dissociation of BMP and other TGF-beta molecules from their receptors can lead to prolonged signaling even after free extracellular ligand is removed [45,46,47].
To address the possibility of persistent signaling, we modified the above experiment in two ways. First, we cultured the cells for a longer period after washout before harvesting (four days; Figure 4E). In CPCs treated with high BMP4, then no or low BMP4 after washout (0-4 ng/ml), (mRNA) Msx1 did indeed return to low baseline levels ( Figure 4F, red line). Thus, Msx1 induction was not irreversible. When comparing the CPCs initially treated with BSA (Msx1-off) or high BMP4 (Msx1-on), (mRNA) Msx1 levels were higher in the Msx1-on CPCs regardless of the BMP4 concentration that was reapplied after washout (compare red and blue lines in Figure 4F). In other words, we observed hysteresis.
For the second modification, we repeated the two-day culture studies, but also included the BMPR inhibitor LDN193189 during and after BMP4 washout to block persistent BMP signaling ( Figure 4G). In CPCs initially exposed to high BMP4, LDN193189 caused (mRNA) Msx1 to return to low baseline levels when no or low BMP4 was reapplied (0-4 ng/ml; Figure 4H, red line). Thus, persistent BMPR signaling contributed to the maintained Msx1 expression observed initially ( Figure 4D, red line). As in the fourday cultures, there were two distinct curves that depended on initial conditions -i.e. whether CPCs were initially Msx1-on or Msx1-off (compare red and blue lines in Figure 4H). Thus, the response again displayed hysteresis. Collectively, the two lines of evidence for hysteresis strongly implicate CIPF as the mechanism underlying BMP-FGF cross inhibition in CPCs. Except for a few parameter sets, CIPF consistently increases nH, even under all linear conditions. (C) Representative BMP dose-response simulations with and without FGF (solid and dashed lines, respectively) for context 1 (inhibitory strength 88%, nH = 1) and context 3 (inhibitory strength 65%, nH = 4). Upon FGF addition, CFF (context 3 only) and CIPF lead to B T suppression at low BMP concentrations, while B T maximum levels are maintained at high BMP concentrations. This leads to increased EC 50 values and the emergence of ultrasensitivity. (D) BMP dose-response simulations (inhibitory strength 76%, nH = 4) with different initial conditions (B T starts 'off' in blue or 'on' in red). Based on initial conditions, CIPF produces different B T response curves, thus displaying hysteresis, while STI, SUI, and CFF produce identical curves. (E) Summary of modeling results. Grid coloring indicates the presence (black), absence (white), or a less than 10% presence of the property (grey) indicated on the left: e.g. B T maximum levels decrease with the STI model in all contexts and is shown by black boxes for contexts 1, 2, 3, and 4. See Table S4 in Text S1 for summary values, Tables S1, S2, and S3 in Text S1 for parameter values. See also Figure S4. doi:10.1371/journal.pcbi.1003463.g003 with and without FGF8 (red and blue lines, respectively; RT-qPCR). FGF8 shifts the (mRNA) Ttr peak from 16 to 32 ng/ml BMP4, but maximal (mRNA) Ttr levels and its apparent ultrasensitivity are unchanged. (B) Msx1 responses of E12.5 CPCs to BMP4 with and without FGF2 and EGF (red and blue lines, respectively, curve fit to a Hill equation; RT-qPCR). Msx1 responds linearly to BMP4 (nH = 1.0) in the absence of FGF2 and EGF. With FGF2 and EGF, Msx1 induction becomes ultrasensitive (nH = 3.7) due to its selective suppression at low BMP4 concentrations, as predicted by the CFF and CIPF Msx2, like Msx1, is a pSmad-dependent BMP target gene expressed in the telencephalic DM [16,48]. In CPCs treated with increasing BMP4, but no FGF2 or EGF, Msx2 expression was linearly sensitive and increased monotonically ( Figure 5A). With FGF2 and EGF, Msx2 induction by BMP4 was ultrasensitive to a similar degree as Msx1. Msx2 expression was also hysteretic in two-day washout studies in the presence of LDN193189 ( Figure 5A).
While Msx2 and Msx1 responses were qualitatively similar, the Msx2 EC 50 (22 ng/ml BMP4) was significantly lower than that of Msx1 (32 ng/ml BMP4) in the same cells ( Figure 5B). In traditional views of morphogen action, EC 50 values and border positions are inversely related -e.g. a lower EC 50 shifts borders away from a morphogen source, thus creating a larger expression domain for a positively-regulated target gene. Published images of the dorsal telencephalon suggest that the Msx2 domain is indeed larger than the Msx1 domain [48]. To verify this, we stained adjacent sections from Msx1 (Msx1-nlacZ) embryos which express nuclear lacZ in the Msx1 domain [49], with antibodies that detect lacZ or MSX1/2 proteins. The results suggest that the E10.5 Msx2 expression domain extends ,100 mm farther than that of Msx1 (8 sections from 2 embryos; Figure S6A).
For morphogen thresholds ,1.5 fold apart (22 vs. 32 ng/ml BMP4 for Msx2 and Msx1, respectively; Figure 5B) and separated by 100 mm, the length scale of an exponential morphogen gradient (the distance over which morphogen concentration falls by 12e 21 , or ,63%) would be ,270 mm. This value agrees well with the dorsoventral pSmad gradient in the E10.5 dorsal telencephalon [15], whose best-fit exponential curve had a length scale of ,290 mm (see Materials and Methods). Thus, Msx1 and Msx2 EC 50 values in vitro correlate well with their expression borders in vivo. How might CIPF produce different borders or EC 50 values with the same BMP and FGF gradients? We found that it is possible to produce two distinct EC 50 values and borders, when BMP and FGF drive two different intracellular CIPF loops with common elements. The two CIPF loops can differ in two ways: either at the level of the CIPF loop -through differential regulation of the intermediates (B I and F I ) by BMP and FGF or differential loop inhibition between the intermediates -or downstream of the CIPF loop. Our simulations show that the first category produced B T responses with separate EC 50 values (Figures 5C, S6B-C, Text S1 section 7), while differences downstream of the CIPF loop did not ( Figures 5D, S6B,D).

Discussion
A variety of mechanisms have been proposed to explain how shallow morphogen gradients are converted into sharp borders of gene expression and cell fate. In the dorsal telencephalon, previous work implicated cell-intrinsic BMP-driven ultrasensitivity in this conversion [12]. Here we show that this form of ultrasensitivity arises from cross inhibition between BMP and FGF (and EGF) signaling pathways. Such inhibition manifests as cross-inhibitory positive feedback (CIPF), a mechanism characterized by ultrasensitivity (Figures 3-5, S4, S5), hysteresis (Figures 3, 4, 5, S4, S5) and an ability to generate different thresholds (EC 50 values) within a gradient (Figures 5, S6).
The complete elimination of Msx1 and Msx2 ultrasensitivity in the absence of FGF2 and EGF (Figures 4, 5) indicates that CIPF alone can account for Msx1 and Msx2 ultrasensitivity. In midline cells, however, FGF elimination caused a left-shift of the bimodal Ttr induction by BMP4 without eliminating ultrasensitivity ( Figure 4A). Consistent with this, FGFR inhibition shifted the rostral Ttr boundary in forebrain explains, without making a more diffuse boundary or extending expression into the cortex (Figure 1). This implies that mechanisms in addition to CIPF might contribute to sharp Ttr borders. The relatively late induction of Ttr in vivo (two days after Msx genes) and its apparent irreversibility [18] suggest that regulation of Ttr is more complex than Msx genes, which are direct and immediate BMP targets. The differences between Ttr and Msx1 in the BMP4-and FGFR inhibitor-treated explants also raise the possibility that Ttr and Msx1 are regulated by different FGFs (e.g. FGF8 and FGF2). Although changes in cell composition after two days in culture may contribute to observed findings, previous two-day cultures on more naïve E10.5 CPCs yielded similar ultrasensitive Msx1 responses [12], and BMP2 (which has very similar effects to BMP4 on CPCs) did not significantly alter cell type ratios in comparably-staged rat CPC cultures in the presence of FGF2 even after eight days [50,51]. While no single parameter set from the simulations and parameter searches can be considered representative, these modeling techniques enable investigation of complex models such as CIPF and CFF that are not otherwise amenable to analytical techniques, which can be sampled over large areas of parameter space (Tables S1, S2, S3, S4, S5 in Text S1) and crossvalidated with other methods (Table 1).

Molecular basis for BMP-FGF CIPF in the forebrain
BMP-FGF cross inhibtion, particularly specification of mutually exclusive BMP-and FGF-dependent cell fates, has been reported in diverse developmental contexts (Table S6 in Text S1, e.g. [52]). The most well-studied molecular mechanism proposed to explain FGF inhibition of BMP signaling has been the ability of FGFactivated ERK to phosphorylate and trigger the degradation of BMP-activated pSmads [27,28]. This mechanism, however, is unlikely to account for CIPF in the forebrain for two reasons. First, there is no apparent pSmad ultrasensitivity to BMP in CPCs. In vivo, pSmad1/5/8 immunoreactivity declines gradually and smoothly with distance from the DM [15] while MSX1 and MSX2 borders are sharp ( Figure S6, [16,48]). In CPCs in vitro, in the presence of FGF2 and EGF, nuclear pSmad levels exhibit a graded relationship to BMP4 dose while Msx1 and Msx2 inductions are ultrasensitive (Figures 4, 5, [12]). Ultrasensitivity must then be generated downstream of pSmad. Second, the occurrence of Msx1 and Msx2 EC 50 values at different BMP4 doses suggests a mechanism downstream of pSmad as well ( Figure 5B). As Msx1 and Msx2 share the pSmad activation pathway, the points of FGF inhibition into the BMP pathway required for separate EC 50 values ( Figure 5C,D) probably lie downstream of pSmad (e.g. a Smad-induced gene or Smad coactivator complex). Although less is known about mechanisms underlying BMP inhibition of FGF signaling, the smooth gradient of phospho-ERK in the developing cortex [24] suggests that this inhibition may similarly occur downstream of ERK activation.

CIPF and pattern formation
Cross-inhibition (or cross-repression) is, of course, not new in developmental biology (Table S7 in Text S1). Specifically, cross inhibition in the form of a toggle switch (i.e. CIPF) is thought to underlie the generation of sharply-bounded domains of mutuallyexclusive cell fates in diverse contexts e.g. [53,54,55], including the two well-studied systems of the Drosophila embryo and mammalian spinal cord [8,9]. In these systems, patterning emerges from the collaboration between transcription factor morphogens (Bicoid-Caudal in the syncytial Drosophila embryo) or a single extracellular morphogen and a transcriptional network (Sonic Hedgehog in the mammalian spinal cord) [6,8,9]. Both architectures contain a toggle switch sub-motif similar to BMP-FGF CIPF, which can generate ultrasensitivity, hysteresis (buffering noise), and multiple sharp borders [6,8,9,56]. The Bicoid-Caudal system also reduces variation in border position and scales borders to tissue size [6,9,56], which we reason below should apply similarly to BMP-FGF CIPF.
What is unique about BMP-FGF CIPF -compared to Shh, Bicoid-Caudal, and other defined patterning systems -is that a cellular-level toggle switch is explicitly driven by and dependent on multiple extracellular morphogens. BMP-FGF CIPF therefore provides a direct and explicit link between tissue-level patterning by antagonistic morphogens and cellular-level ultrasensitivity. Unlike the antagonistic Bicoid-Caudal system, BMP-FGF CIPF occurs in a non-syncytial system that may apply broadly to vertebrates, given the prevalence of BMP-FGF cross-inhibition in vertebrate development (Table S6 in Text S1). Furthermore, the current study defines new requirements for CIPF-mediated toggle switches (nonlinearity and loop balance) that likely apply to the Drosophila embryo, mammalian spinal cord, and other crossinhibition systems belonging to the CIPF class of models (Table S7 in Text S1).
In the dorsal telencephalon, DM rostral border position would be determined by BMPs interacting with rostral FGF8 and related FGFs (Figures 6A, S1A). Mediolaterally, the same BMPs would interact with FGF2 and FGF1 in the cortex, and possibly with EGF from the antihem [42]. Along both axes, interactions between opposing BMP and FGF/EGF gradients would determine cellular ultrasensitivity thresholds and gene expression borders ( Figure 6). These border positions coincide with ''equivalence'' points -i.e. points at which BMP and FGF signaling (or B I and F I ) balance each other. The effects of BMP4 or FGFR inhibitors on forebrain explants (Figures 1, S1) can then be understood in terms of shifts in equivalence points.
The ability of morphogen-driven CIPF to specify different equivalence points for different target genes ( Figures 5, S6B,C) provides a straightforward mechanism for establishing multiple sharp borders, a general problem in morphogen-mediated patterning [5]. So far, only a few solutions have been discovered for making multiple borders, including the generation of sequential thresholds in protein modification [57], a temporal overshoot mechanism [58], and time-dependent changes in cell competence [4].
A second useful property of CIPF driven by opposing morphogen gradients would be its ability to scale pattern to tissue size ( Figure 6B). Pattern scaling is important in development, as size variations naturally arise as functions of time, genetic background, and environment [4]. As others have pointed out, mechanisms that assign positional values based on the ratio between signals emanating from opposing sources, rather than as a function of a single signal, have an inherent tendency to scale [59,60], although not necessarily with equivalent accuracy at every location. While no studies have investigated scaling of pattern in the forebrain DM, the forebrain provides a promising avenue for future studies. The forebrain grows rapidly during early development, and scaling needs to occur on a spatiotemporal scale. Additionally, mouse and human microcephaly and megalencephaly cases in which patterning appears to be maintained represent a potentially rich area for research into the toggle switch mechanism and brain scaling.
Notably, for morphogen-driven CIPF to specify border positions, it is not necessary for both morphogens to be graded. For example, a BMP gradient superimposed on a uniform FGF field would also produce sharp boundaries at locations corresponding to BMP-FGF equivalence points. This scenario may apply to mediolateral borders in the dorsal telencephalon, where FGF2 production appears to be relatively uniform [20]. Referring to FGF as a ''morphogen'' in this scenario departs from conventional usage (which presumes a graded distribution), but perhaps in patterning systems driven by collaborations between diffusible signals, such a departure is justified.

Ethics statement
All animal studies were performed in accordance with protocol # 2001-2304 approved by the Institutional Animal Care and Use Committee (IACUC) of the University of California, Irvine. All experiments were conducted in accordance with protocol # 2001-1024 approved by the Institutional Biosafety Committee (IBC) of the University of California, Irvine. All surgeries were performed on euthanized animals with all efforts made to minimize suffering. Animals were euthanized with carbon dioxide from compressed gas cannisters, with secondary physical method of cervical dislocation to ensure euthanasia.

Mice
Noon of vaginal plug date was day 0.5; developmental stages were confirmed by crown-rump measurement. CD1 mice (Charles River Laboratories, Wilmington, MA) were used for wild-type studies. Msx1-nLacZ mice [49] were mated with CD1 for expression analysis and were genotyped by Xgal staining of limb buds [12]. Real-time semiquantitative RT-PCR RNA preps, cDNA syntheses, PCR quality controls, experimental runs, and statistical methods were performed as described previously [13]. Primers and amplicons were verified by melting curve analysis, agarose gel electrophoresis, and tested for amplification efficiency; amplicons were verified by sequencing. cDNA samples were validated for reverse transcription (RT) reaction efficiency and minimal genomic DNA contamination (cDNA/genomic target ratio .10 5 ) and run in duplicate or triplicate for 40 cycles; cyclophilin A (CYPA) and 18S reference primers were included in runs (used for normalization to control for variations between wells and cell populations), except for explant studies (CYPA only). Mean, SEM, SD, and p values (two sample, two-tailed t-tests) were calculated from cycle threshold (dCt) values (Ct gene of interest -Ct reference ) and plotted as normalized ddCt values (upregulation is positive and downregulation is negative). In Figures 1, 2, 4, and 5, relative values were normalized to control.

Dissociated cell cultures
Midline cells and CPCs were isolated and dissociated from E12.5 CD1 dorsal telencephalon as described previously [12], then plated at 50,000 cells/ml (unless otherwise indicated) in defined media [61]. In previous studies [12], we observed ultrasensitivity in E10.5 cells as well as E12.5 cells. As ultrasensitivity is higher in E12.5 cells and border refinement remains ongoing and continues beyond E12.5, this time point is more amenable and practical experimentally. Midline cultures, in which contamination with cortical cells was likely, were analyzed exclusively for midlinerestricted genes (Ttr, Msx1, Spry1) to maintain specificity for midline cells.  (Figures 1, 2) were harvested 40 hrs after initial treatment, and 2) midline cells treated with SU5402 or PD173074 ( Figure S1) were harvested 12 hrs after initial treatment. For washout experiments (Figures 4, 5), 64 ng/ml BMP4 was applied for 2 hrs, then aspirated and washed three times with fresh media containing 20 ng/ml EGF, 10 ng/ml FGF2, and 2 ug/ml heparin, with or without BMP4 or LDN193189; time points correspond to hours after initial BMP4 application.

Explant cultures
Dissections were performed as described [12,13] in ice-cold L-15 with 2% glucose. Dorsal forebrains from embryos were placed ventricular surface down on the dull surface of 8 mm pore polycarbonate membranes (Whatman, Clifton, NJ) floating on DMEM/F-12 with 20% calf serum, sodium pyruvate, nonessential amino acids, and penicillin/streptomycin. After 1 hr, 50 ng/ml BMP4 was added for three days, or 100 nM PD173074 was added for two days (Figure 1). Explants were processed for Ttr ISH or X-gal staining. For FGF8 bead studies ( Figure S1), heparin acrylic beads (Sigma) were soaked in 10 ml of 100 ng/ml FGF8 or BSA, rinsed briefly in PBS, and placed on explants using pulled flamepolished microcapillary pipettes.

In situ hybridization, histology, and imaging
These were performed and imaged as described previously [13]. Comparative images and intensity measurements, tissue processing, assays, image acquisition, and processing were performed in parallel on sections from comparable rostrocaudal levels with identical camera settings and enhancements. Parallel image enhancements were limited to levels, brightness, and contrast in Photoshop. Unless indicated, presented images are representative of multiple sections from at least two embryos.

Proliferation assay
Proliferation studies were performed with a WST1 proliferation assay kit (Clontech, Mountain View, CA). Cells were plated in 96 well plates at matched densities from 25,000-50,000 cells/well. 24 hrs post-plating, BMP4, FGF2, FGF8, and/or EGF at the indicated concentrations was added to media ( Figure S1). WST1 was added 36 hrs later at a 1:10 ratio, and spectrophotometer readings taken 3 hrs post-WST1 addition.

Immunohistochemistry and immunocytochemistry
Immunohistochemistry was performed as described [12]. Primary CPCs were isolated at E12.5, plated in chambers containing FGF2, FGF2+PD173074, and no FGF media. BMP4 was added 24 hours after plating, and then cultured for 48 hours. Cells were fixed in 4% paraformaldehyde/PBS for 15 mins, permeabilized with 0.3% Triton for 5 mins, and washed in PBS. Cells were blocked with 5% BSA at room temperature for 1 hour, followed by primary antibody in 1% BSA and incubated overnight at 4C. PBS washes were followed by secondary antibody in 1% BSA for 2 hours at room temperature, Hoechst counterstaining, mounting with Vectashield. The following antibodies were used: anti-MSX1/2 (mouse monoclonal antibody against chick Msx1/2; 1:350 dilution; 4G1; Developmental Studies Hybridoma Bank, University of Iowa), and secondary (Alexa555 goat anti mouse); 1:500 dilution.

Curve fitting
For Figures 3, 5 and other modeling results, Hill coefficients (nH) and EC 50 values were obtained by fitting the B T response to a Hill equation using the Mathematica FindFit function. The maximal values were measured by computing the B T response at an extremely high BMP dose to account for asymptotic behavior. For Figure 3B, the plotted points were obtained by dividing the property value (e.g. nH) of the BMP response in the presence of FGF with that in the absence of FGF. 2000 parameter points were simulated, and for each point, all three properties (maximal levels, EC 50 , and nH) were assessed for all four models (STI, SUI, CFF, and CIPF).
In Figures 4 and 5, nH and EC 50 values were obtained using the curve fitting function of Deltagraph 6.0. The length scale of the pSmad gradient in the dorsal telencephalon was based on previous data [15]. MS Excel software was used to fit the data to an exponential curve with unknown backgound, c~Ae {lx zb, where c is the concentration in ng/ml, x the position in mm, l (lambda) the length scale, A the highest concentration without b, the background.

Numerical methods
Each of the models was defined by a set of ordinary differential equations. The form of these equations and their rationale is similar to previous modeling approaches [4,10,36,62]. The equations for each model are described briefly, but for modeling details and analysis, please see Text S1.
The four models shown in Figure 3 are represented by the following equations: To reduce the number of parameters and simplify analysis, these equations were non-dimensionalized (see Text S1 equations 7-10), which were then used for modeling and simulations. Figure S1 Additional evidence for BMPs and GFs affecting DM gene expression. (A) Forebrain schematic (dorsal view, anterior towards the top) showing the DM region where BMP4 is produced (orange), which forms between the two cortical hemispheres (green). FGF2 is produced in the cortex, whereas FGF8 is produced in the rostral midline (blue). (B) RT-qPCR of the rostral and dorsal halves of explants. DM-specific genes are upregulated in their endogenous (dorsal) and ectopic domains (rostral) in BMP4-treated explants compared to BSAtreated controls (n = 4 each condition; see also Figure 1c) (C) Sections of whole-mount explants shown in Figure 1C;   Figure 3. (B) Representative BMP dose-response simulations with a nonlinear BMP core pathway (nH = 10 for BRB I or B I RB T ) and with or without FGF (solid or dashed lines, respectively). See Figure 3 and Table S2 in Text S1 for inhibitory link values and other details. (C) B T responses with all links linear (context 1), but with increasing inhibitory strengths (95, 97, and 98% represented by increasing line thickness). Other parameters match those in Figure 3C. Increasing inhibitory strength leads to increased ultrasensitivity with CIPF (nH = 2, 2.5, and 3), but not with CFF or the other models, which remain linearly sensitive. (D) Schematics of different types of memory; red and blue lines indicate responses that start either on or off, respectively. From left to right -1) no memory, in which responses do not depend on starting condition; 2) hysteresis, in which responses depend on starting condition, but can return to 0; and 3) irreversibility, in which the response, once 'on', never returns to 0. (E) Comparison between CIPF (top row) and auto-regulatory positive feedback (bottom row), with increasing feedback strength from left to right. Increasing CIPF feedback strength increases its bistability window, but never produces irreversibility, as B T can always returns to 0. Auto-regulatory positive feedback can generate irreversibility, depending on feedback strength. Text S1 Supplementary information text. Provides (mostly) mathematical and computational analyses that support the main text. Contents include:

Supporting Information
N Tables S1, S2, S3, S5 listing parameter values of simulations in N Table S4: a summary of parameter exploration results shown in Figure 3.
N Table S6: examples of BMP-FGF inhibitory interactions in development. N