Overview of the model

The model considers that each cell perceives a level of FGF signaling that is proportional to the area of cell contact with FGF-expressing mesendoderm cells and a level of ephrin signaling that is proportional to the area of cell contact with ephrin-expressing ectoderm cells, as schematized in Fig 1A. These surfaces have been quantified (Fig 1C) and are used to determine the number of receptors in contact with the signals. Ligand-bound FGF receptors (FGFR) and ephrin-bound Eph receptors stimulate the activity of SOS and p120RasGAP, respectively. SOS transforms Ras-GDP into Ras-GTP. Ras-GTP is converted back into Ras-GDP by p120RasGAP and by a cell surface contact-independent GAP activity. Ras-GTP activates the ERK signaling cascade [10,11]. The relation between ERK activity and the level of Ras-GTP is described by a single function, which represents the full Ras-Raf-MEK-ERK pathway. Active ERK in turn phosphorylates the two transcription factors, Ets1/2 and ERF2. Phosphorylated Ets1/2 promotes the expression of the neural immediate-early gene Otx, whereas unphosphorylated ERF2 represses ERK targets [6,12,13]. The model consists in 5 ordinary differential and 13 algebraic equations describing the abovementioned phenomena using Michaelis-Menten and Hill type kinetic expressions. A full description of the model is provided in the Modeling section.

In the 32-cell stage ascidian embryo, neural induction takes place in the ectoderm lineages (anterior (a) and posterior (b) lines) and is dependent on cell-cell contact with FGF-expressing mesendoderm cells (anterior (A) and posterior (B) lines). We mainly focus on the four pairs of a-line cells, namely a6.5, a6.6, a6.7, and a6.8 in the anterior ectoderm (see Fig 1B; [8]). These four cells display different relative areas of cell surface contact with A-line mesendoderm cells that express FGF, with the neural precursor, a6.5, having the largest area of cell surface contact (Fig 1C). Areas of cell surface contact with FGF-expressing cells and ephrin-expressing cells are inversely correlated, such that cells having the largest cell surface contact with FGF-expressing cells have the lowest cell surface contact with ephrin-expressing ectoderm cells (Fig 1C). In the next section, we model ERK activation and Otx expression in these four cells.

Regulation of ERK activity by fractions of cell surfaces exposed to antagonistic signals

Experimental evidence indicates that, in the 32-cell stage ascidian embryo, the level of ERK activation in ectoderm cells is controlled by the extent of cell surface in contact with FGF-expressing mesendoderm cells and with ephrin-expressing ectoderm cells [8]. Because these two contact areas are correlated, the area of cell surface contact with FGF-expressing cells (designated S₁) can also be used to define the area of cell surface contact with ephrin-expressing cells (designated S₂) (Fig 1C). Thus, in terms of modeling, the only parameter that is cell-type-specific is S₁. The steady states solutions of Eqs (1–13) should correspond to the levels of ERK activation measured in the different cell types as anti-dpERK immunofluorescence (IF) signals in their nucleus. Importantly, extracellular FGF and ephrin concentrations are assumed to be the same for all cell types. As shown in Fig 2A, good agreement between observed and modeled ERK activation levels can be obtained for the four cell types when considering the values of parameters listed in Table 1 and assuming a linear relationship between computed normalized ERK activity (Erk*) and dpERK IF signal measured experimentally (Eq (14)).

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Fig 2. Control of ERK activation by cell contact surfaces.

(A) Nuclear dpERK immunofluorescence (IF) signals in the a6.5, a6.6, a6.7 and a6.8 cell types as measured in IF experiments (left) and computed with the model (right, Erk^f values). Each point represents a single cell and modeling results are computed using the measured values of S₁. Means and standard deviations are shown in black. A = 1850 and B = 155.11 in Eq (14). (B) Nuclear dpERK IF signals in individual a-line cells are shown as a function of the relative area of cell surface contact with A-line cells in experiments and in the model. Experimental data are shown in grey (a6.8 cell type), green (a6.6 cell type), blue (a6.7 cell type), magenta (a6.5 cell type), predictions of the model in black. Hill coefficient obtained by fitting the model predictions with a Hill function: 2.41. A = 3000 and B = 324.52 in Eq (14). The area shaded in grey represents the uncertainty on the model prediction. (C) Injected/control ratios of nuclear dpERK signal in dnFGFR half-injected embryos (on the left) and in Eph3ΔC half-injected embryos (on the right). Left and right columns show experimental dpERK IF and computed Erk^f ratios, respectively. Injection of dnFGFR and Eph3ΔC were modeled by considering R_T = 100 and Q_T = 10, respectively. A = 1850 and B = 155.11 in Eq (14).

https://doi.org/10.1371/journal.pcbi.1010335.g002

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Table 1. Default values of parameters.

All the units reported in the table are arbitrary. Parameter values were obtained by manual fitting to get best agreement with experimental observations. The values of the parameters of the two modules of the model were fitted separately. For the first module (model of ERK activity), fitting was performed manually to reproduce the experimental data obtained in ephrin inhibited embryos (S1B and S1C Fig), to set the values of the following parameters: V_s, K₁, K_s, K_b, K_erk, K_d, [FGF]. Fitting was then performed on the experimental data obtained in wild type embryos (Fig 2A and 2B) to set the values of the remaining parameters: V_rg, K₂, K_rg, K_e, [ephrin]. Finally, to fix the values of the parameters in the model of Otx expression (k_MM3, K_MM3, v_MM4, K_MM4, k_MM1, K_MM1, v_MM2, K_MM2, v_b, v_o, K_a, K_i, k) fitting was done on Fig 3D.

https://doi.org/10.1371/journal.pcbi.1010335.t001

The computed Erk* activities are equal to 0.0021 ± 0.0013, 0.0078 ± 0.0038, 0.0551 ± 0.0096 and 0.1787 ± 0.0201 in the a6.8, a6.6, a6.7 and a6.5 cell types, respectively (S1A Fig). Each value was computed as the average between the steady state solutions obtained for the 25 values of the area of surface contact (S₁) measured experimentally. Computed values reproduce the observed moderately nonlinear relation between ERK activation and cell surface in contact with FGF-expressing mesendoderm cells. The nonlinear relationship between S₁ and ERK activation can also be compared with experiments where cell surfaces and dpERK were both measured at the individual cell level (Fig 2B, n_H = 2.41). The model predicts a modest level of ERK activation, defined between 0 and 1, in the maximally activated a6.5 cell, ~0.18 (S1A Fig). This value stems from the assumed relatively low concentration of extracellular FGF (K_D/5) and, to a lesser extent, from the presence of ephrin (see below).

The model was next challenged to predict the level of ERK activation in b-line cells. b-line cells are in contact both with A-line mesendoderm cells that express a uniform level of the ligand FGF9/16/20 and with B-line mesendoderm cells that express different levels of FGF9/16/20 (S2A Fig) [8]. In order to account for different levels of FGF expression, equations consider the existence of two distinct populations of FGF receptors, as described in the Modeling section (see section ‘Model of Erk activity in b-line cells’). Model predictions recapitulate experimental observations (S2B Fig) keeping the same set of values of parameters. Only b6.5 cell displays a high level of ERK activity, which is mainly due to the large population of FGF receptors in contact with A-line mesendoderm cells.

In order to fit the computed values to the experimental ones, the fluorescence background needs to be adjusted (see Modeling section). Much of the uncertainty when comparing Erk* to measured dpERK IF signals stems from the background level of fluorescence in experimental measurements. Experimentally determined values are scattered around computed ones (Fig 2). It remains unknown how much of the variation in measured ERK levels within each cell type are true to the system or are inherent to the experimental technique. The larger dispersion of experimental than theoretical values may reveal the existence of cell-to-cell heterogeneities that are not considered in the model or reveal intrinsic noise due to molecular fluctuations (see discussion).

FGF and ephrin control the activation of the ERK pathway during early neural induction and could thus both govern the emergence of the neural cell fate. Because of the inverse relationship between the cell surface contact with FGF-expressing mesendoderm cells and that with ephrin-expressing ectoderm cells (Fig 1C), the a6.5>a6.7>a6.6>a6.8 profile of ERK activation levels in the 32-cell stage ascidian embryo could rely either on a decreasing gradient of active SOS or on an increasing gradient active of p120RasGAP. The respective contributions of these opposite influences were investigated experimentally by several targeted manipulations of the two pathways and by modeling [8]. It was shown that the profile of ERK activation can be controlled by the gradual activation of FGF receptors, even in the absence of ephrin signals. Ephrin signaling is important to reduce the levels of ERK activation across all ectoderm cells. Modeling predicted a greater influence of SOS compared to p120RasGAP on ERK activation levels. In the following, we validated the model and the parameter values by simulating the various experiments conducted in [8].

The pharmacological inhibitor of ephrin receptors, NVP-BHG712 (henceforth NVP), has been shown to inhibit Eph signals in ascidian embryos [8,14]. When the model equations were solved with a minimal value of ephrin (0.001) to simulate the presence of the inhibitor, the a6.5>a6.7>a6.6>a6.8 profile of Erk* activation in the four cell types persists (S1C Fig), as observed experimentally. Moreover, the model still exhibits a positive correlation between the area of cell surface contact with FGF-expressing cells and ERK activation levels, although the dependence is smoother (S1B Fig, n_H = 1.96). The effect of altering the SOS pathway was investigated experimentally by injecting a dominant negative form of the FGF receptor (dnFGFR) into one of the cells of the two-cell stage embryo. In these conditions, dnFGFR blocks FGF signals in one half of the embryo, while the non-injected side represents the control situation. The effect of dnFGFR was simulated by decreasing the number of FGF receptors (R_T) by a factor of 20 (i.e. R_T = 100). Observed ratios of dpERK IF signals between injected and control sides are shown in Fig 2C. Computations predict that the decrease in the number of FGF receptors leads to a marked decrease in the ERK activation levels in cell types a6.5 and a6.7: Erk* now ranges from ~5 10⁻⁶ in a6.8 cells to ~5 10⁻⁴ in a6.5 cells. Thus, the ERK pathway is practically totally inactive in all cell types when FGFR is inhibited. In a symmetrical manner, the model reproduces the effect of the injection of the dominant negative forms of the Eph3 receptor (Eph3ΔC) (Fig 2C). Best agreement with experimental data is found when dividing the number of ephrin/Eph receptors (Q_T) by 200 (i.e. Q_T = 10). Similarly, the effect of dominant negative p120RasGAP (RGΔGAP) was modelled by considering V_rg = 0.01 instead of 0.4 (S1D Fig).

In summary, simple phenomenological equations describing the regulation of the ERK signaling pathway by FGF receptor-controlled SOS activation and ephrin/Eph-controlled p120RasGAP activation can quantitatively account for experimental observations in the 32-cell stage ascidian embryo. Steady-state solutions of the model confirm that gradual ERK activation levels are controlled by the different cell surface contacts to FGF and ephrin-expressing cells. The relation between these surfaces and ERK activation levels is moderately nonlinear. The model predicts that the ERK pathway is far from being maximal at the 32-cell stage of the ascidian embryo, even in the most-strongly activated a6.5 cell type (S1A Fig).

Dual control of the expression of the Otx gene by ERK

The Otx gene is a direct target of ERK signals during the initial step of ascidian neural induction [6,15]. Our previous results [8], modeled in the preceding section, have shown that ERK activation in the 32-cell stage ascidian embryo exhibits a relatively smooth profile in response to contact-dependent exposure to FGF signals (Figs 2A and 2B and S1B). In contrast, the transcription of the immediate-early gene Otx is bimodal, with a ON response that is restricted to neural precursors. We hypothesized that the ON/OFF response is, at least in part, due to the dual control of Otx expression [8]: Ets1/2 phosphorylation by ERK is required for its transcriptional activator activity [16], while phosphorylation of the ERF2 repressor by dpERK promotes its nuclear export, thus preventing its binding to DNA [12,13,17] (Fig 1A).

To further validate this hypothesis, we have extended the model of ERK activation studied in the previous section to include the relation between ERK activition, phosphorylated Ets1/2, unphosphorylated ERF2 and Otx expression. We first seek a single set of parameter values accounting for the observed levels of Otx expression in the four cell types. The existence of such a set would corroborate the assumption that the emergence of the neural cell fate in ascidian embryos is controlled by the contact areas with FGF and ephrin-expressing cells. Steady state levels of Otx smFISH spots computed with the model defined by Eqs (1–18), using the parameter values listed in Table 1, are shown in Fig 3A, together with experimental data for comparison (see also S3A Fig for the O values before the linear transformation into Otx smFISH spots). When simulating the inhibition of FGF signaling (R_T = 100 instead of 2000), O values drop to nearly zero in all cell types, which corresponds to the absence of Otx expression observed in dnFGFR-injected half embryos (Figs 3B and S4A). In the model as in the experiments, Otx expression is enhanced by inhibition of Eph signaling. This is noticed in the simulations of invalidation of the expression of ephrin/Eph receptors (Q_T = 10 instead of 2000) or of p120RasGAP enzymes (V_rg = 0.01 instead of 0.4), or by simulating the inhibition of the ephrin/Eph receptors by NVP ([ephrin] = 0.001 instead of 5) (see Figs 3B, 3C and S4). In all cases, these treatments reproduce the observed ectopic activation in a6.7 cells. This suggests that the moderate increase in ERK activation levels that results from the absence of Eph signaling (Figs 2C right and S1D) has a drastic effect on Otx expression in this cell type. In Eph-inhibited embryos, if ERK signaling is reduced by another means (low dose of the U0126 inhibitor of MEK that is simulated by increasing K_erk (0.6 instead of 0.5)), an Otx output similar to the wild-type pattern is recovered (Figs 3C, right column and S4B).

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Fig 3. Control of Otx expression by ERK.

(A) Levels of Otx expression in the a6.5, a6.6, a6.7 and a6.8 cell types as measured by single molecule fluorescence in situ hybridization (smFISH, left) and computed with the model (right, Otx_smFISH). Each point represents a single cell and modeling results are computed using the measured values of S₁. Means and standard deviations are shown in black. C = 66 and D = 2.75 in Eq (18). (B) Otx expression in control, dnFGFR, Eph3ΔC and RGΔGAP injected embryo halves. Left and right columns show experimental Otx smFISH spots and computed Otx_smFISH, respectively. Results for the four cell types are shown in the same columns with different colors. Each dot represents a single cell. Injection of dnFGFR, Eph3ΔC and RGΔGAP were modeled by considering R_tot = 100, Q_tot = 10, and V_rg = 0.01, respectively. In Eq (18), C = 92 for the control embryos and for dnFGFR injected embryos, C = 95 for Eph3ΔC injected embryos and C = 122 for RGΔGAP injected embryos. D = 1.5 for the control and dnFGFR injected embryos, D = 2.71 for Eph3ΔC and D = 1.61 for RGΔGAP injected embryos. (C) Effect of the ephrin/Eph inhibitor NVP on Otx expression in the four cell types. On the left, control; in the middle, NVP-treated embryos; on the right, embryos treated with NVP and with the MEK inhibitor U0126 0.2 μM. In the three cases, experimental Otx smFISH spots are shown on the left and computed Otx_smFISH values on the right. Results for the four cell types are shown in the same column with different colors. Each dot represents a single cell. NVP and moderate U0126 treatment are simulated by considering [ephrin] = 0.001 and K_erk = 0.6, respectively. In Eq (18), C = 94, D = 1.2. For B-C) the data for individual cell-types is shown in S4 Fig for clarity. (D) Otx expression as a function of ERK activation levels (Erk*) in the four cell types (colored points and triangles) and computed with the model (Otx_smFISH, black points and triangles). In the two cases, dots indicate the control while triangles indicate the ephrin-inhibited embryos. The experimental data were fitted by a Hill function, best-fit Hill coefficient = 6.09. For the experimental points, the value of Erk* corresponding to each cell was obtained by inversion of Eq (14), with A = 3200 and B = 0. The experimental value of O corresponding to each cell was obtained by inversion of Eq (18), with C = 120 and D = 0. Modelled Otx outputs were computed using the experimentally measured Erk* estimates as inputs. (E) Levels of Otx expression in the a6.5, a6.6, a6.7 and a6.8 cell types when unfertilized eggs have been injected with the ERF2-morpholino to prevent translation of ERF2. Injection of this morpholino is modeled by considering I as a constant equal to 0.01 in Eq (15). C = 75 and D = 73.4 in Eq (18). Modeling results for control embryos are the same as in panel (A). (F) Otx expression (O) as function of Erk*, computed with the model considering the presence (blue line, Hill coefficient = 6.3) or the absence (orange line, Hill coefficient = 2.6) of I (repressor). The Hill coefficients were computed using relation (25). Dashed vertical lines represents the mean values of Erk* for each cell type.

https://doi.org/10.1371/journal.pcbi.1010335.g003

In fact, a highly nonlinear relationship between Otx expression and ERK activation is observed in the experiments and in the model. In Fig 3D, values of Erk* and O corresponding to the experiments are inferred by inversion of Eq (14) and (18), respectively. Experimental values (colored dots and triangles) are in good agreement with model results (black dots and triangles) and can be fitted by a sharp Hill function (Hill coefficient = 6.09). For each cell type, inhibition of the ephrin pathway by the injection of Eph3ΔC shifts the representative points to the right. While the effect of this shift on Otx expression is negligible for the a6.5, a6.6 and a6.8 cell types that are quite far from the threshold, it moves the a6.7 from the steep part of the curve to the plateau. Thus, Otx expressions in the a6.7 and a6.5 cell types become similar. This threshold-like behavior involves a repressor of Otx expression, ERF2 [8]. The key role of this transcriptional repressor is visualized in Fig 3E–3F. Using the values of parameters calibrated on experimental data (Table 1), the levels of active, phosphorylated Ets1/2 activator (green curve in S3B Fig) and of active, unphosphorylated ERF2 repressor (red curve in S3B Fig) are plotted as a function of Erk*. These curves are sharp (n_H = 6.3 for the two curves) despite the Michaelis-Menten kinetic functions, because the phosphorylation and dephosphorylation reactions of the two transcription factors are not far from saturation (K_MM1 = K_MM2 = K_MM3 = K_MM4 = 0.05), a situation close to zero-order ultrasensitivity [18], although different means of introducing ultrasensitivity between ERK and Otx are also compatible with our data (S7 and S8 Figs and see section ‘Model predictions’). As a consequence of the combination of these two relatively sharp functions, the relation between Otx expression and ERK activation is characterized by a Hill coefficient of 6.3 (Fig 3F). Interestingly, when the effect of the repressor is not considered in the model (I = 0), the curve is not only shifted to the left because of the smaller effective binding constant of the activator, but also becomes much smoother (Hill coefficient = 2.6). This theoretical result is in line with the overall increase and smoother behavior of Otx expression in embryos injected with an ERF2 anti-sense morpholino oligo (compare Fig 3A and 3E). We thus conclude that the sharp relation between expression of the Otx gene and ERK activation is possible because the ERK pathway both promotes the expression of Otx and mitigates the repression of this expression.

Sensitivity analysis

We next conducted a sensitivity analysis to assess the robustness of the model and to identify the parameters most affecting the model’s behavior. Fig 4A shows the predicted levels of ERK activation in the different cell types when randomly changing the values of the parameters linked to the FGF pathway (on the left) or the ephrin pathway (Fig 4A, middle) in the range defined by the standard value ± 20%. The spread in the level of ERK activation obtained in the first case is much larger than the spread obtained in the second one, suggesting that parameters linked to the FGF pathway have a higher influence on the level of ERK activation than parameters linked to the ephrin pathway. However, the average value remains practically unchanged in each cell. Along the same lines, changing all parameters ± 20% results in a robust profile of Otx expression (Figs 4A, left and S7F).

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Fig 4. Sensitivity analysis.

(A) On the left: levels of ERK activation (Erk*) in a6.5, a6.6, a6.7 and a6.8 cell types computed with the model when randomly varying the values of the parameters linked to the FGF pathway ([FGF], K_d, K₁, K_s, V_s, R_T) in the interval: standard value ± 20%. Middle: Erk* levels in the different cell types obtained when randomly varying the values of the parameters linked to the ephrin pathway ([ephrin], K_e, K₂, K_rg, V_rg, Q_T) in the interval: standard value ± 20%. Right: levels of Otx computed with the full model randomly varying the values of the parameters linked to the FGF and ephrin pathways, as above, as well as the other parameters (all the parameters in panel D plus K_b and K_erk). (B) Erk* values computed when varying the value of one parameter at a time by ± 20%. Shown are the ratios between the computed values and the value obtained with the default values of parameters indicated in Table 1. Parameters affecting the SOS pathway are indicated in blue, while those affecting the p120RasGAP one are indicated in red. Results above S₁ and S₂ were obtained by considering S₁ and S₂ as separate inputs, thus disregarding Eq (12). These changes can also be regarded as changes in receptor densities (see Eq (6) and Eq (10)). Results for S₂(S₁) were obtained by increasing/decreasing the value of the slope in Eq (12) by ± 20%, and then adjusting the intercept to best fit with experimental data. (C) O values computed when varying the value of one parameter at a time by ±20%. Results are obtained in the same way as for panel (B). (D) O values computed when varying the value of one parameter at a time by ± 20%. All panels show the ratios between the computed values and the value obtained with the default values of parameters indicated in Table 1 with S₁ set as the average value measured for each cell type. Shaded grey areas represent 20% and 40% changes in the levels of ERK activity/Otx expression obtained by changing the values of the parameters of the model.

https://doi.org/10.1371/journal.pcbi.1010335.g004

To investigate in more detail the sensitivity of the model to individual parameter change, the influence of varying the value of each parameter by ±20% on the steady states of ERK activity (Erk*) and Otx expression (O) in the four cell types is presented in Fig 4B and 4C (see also Table 1 for the definition of each parameter). It is clear that ERK activation (Fig 4B) and Otx activation (Fig 4C) are more sensitive to changes in the values of parameters affecting the SOS pathway (indicated in blue) than the p120RasGAP one (indicated in red). This confirms and extends our previous conclusion that the control exerted by the FGF-receptor pathway is stronger than that exerted by ephrin/Eph [8]. The model is also quite sensitive to the value of K_erk that represents the fraction of Ras-GTP leading to half-maximal activation of the ERK pathway. Given that this parameter controls the entire ERK pathway, which is modeled by a single Hill function Eq (1), it is intuitively expected that its value strongly affects the final value of Erk*. Expression of Otx is dramatically affected in the a6.7 cell type when parameters promoting ERK activation are modified (Fig 4C), in agreement with the position of this cell on the (Erk*, O) curve discussed in the previous section. Globally, the model is very robust to changes in the value of parameters describing the relation between Erk* and O (Fig 4D), except again for the a6.7 cells. For this cell type, values of O are strongly influenced by the rates of phosphorylation (k_MM3) and dephosphorylation (V_MM4) of the Ets1/2 activator. It should be noted, however, that this sensitivity analysis does not explore situations in which the relation between Erk* and Otx expression is not ultrasensitive since 20% changes around the standard K_MMi values (0.05) still correspond to a zero-order regime. Most importantly, the overall a6.5>a6.7>a6.6>a6.8 profile of ERK activation and Otx expression is maintained in all tested parameter ranges (e.g Fig 4A). Thus, the mechanism of neural induction controlled by the cell contact area is robust against variations in the values of parameters.

Model predictions

Effect of ephrin-Eph signaling.

In this section, we use the model that has been calibrated on experimental data and shown to display robustness against changes in the values of parameters to make theoretical predictions. We investigated the sensitivity of ERK activation levels with respect to changes in the area of cell surface exposed to the ligands and to changes in ligand concetration in different conditions (in the presence or in the absence of ephrin). Because curves with a large steepness indicate a highly sensitive input-output relation, we used the Hill coefficient of the curves (see Modeling section for the definition) to measure sensitivity. The procedure is schematically represented in Fig 5A. In our previous study [8], we showed that ERK signaling is more sensitive to the cell surface contact to FGF-expressing cells (S₁) than to FGF concentration. Quantification of this statement, in terms of differences between Hill coefficients, is shown in Fig 5B keeping the relation between the area of each cell surface in contact with FGF expressing cells (S₁) and with ephrin expressing cells (S₂) observed in embryos (Fig 1C). For each column, we use the default values of all parameters (Table 1), except the one that is indicated. In the same line, we now investigate if ERK signaling is more sensitive to cell surface contact with ephrin-expressing cells (S₂) than to ephrin concentration itself. To this end, we computed the Hill coefficients of the Erk* vs S₂ relation and of Erk* vs [ephrin]. As shown in Fig 5C, ERK activation levels are more sensitive to cell surface contact to ephrin-expressing cells than to ephrin concentration.

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Fig 5. Model predictions related to the factors affecting Erk activity.

(A) Graphical explanation of the numerical procedure. We consider two generic curves A and B with Hill coefficients n_A and n_B respectively. The shape of both curves depends on the value of a parameter p that can range from 0 to 1. On the left, the curves A and B are shown when the value of the parameter p is equal to 0.1. In this condition the difference of the two Hill coefficient n_B-n_A is large. In the middle, the curves A and B are shown when the value of the parameter p is equal to 0.9. In this condition the difference of the two Hill coefficient n_B-n_A is smaller. The right panel is the heatmap showing the differences between the Hill coefficients of curves A and B (n_B-n_A) computed for all values of p between 0 and 1. (B) Erk* values are more sensitive to S₁ than to [FGF]. Heatmaps show the difference between the Hill coefficient of the curve Erk*(S₁) and the Hill coefficient of the curve Erk*([FGF]). Hill coefficients obtained with default values of the parameters are 1.47 for Erk*([FGF]) and 2.39 for Erk*(S₁). To compute Erk*([FGF]) we fixed the value of S₁ = 0.3. In the two panels, S₁ and S₂ are related by Eq (12), which reflects the relation between the portions of cell surfaces exposed to FGF and to Ephrin, respectively. (C) Erk* values are more sensitive to S₂ than to [ephrin]. Heatmaps show the difference between the Hill coefficient of the curve Erk*(S₂) and the Hill coefficient of the curve Erk*([ephrin]). Hill coefficients obtained with default values of the parameters are 1.17 for Erk*([ephrin]) and 1.57 for Erk*(S₂). To compute Erk*([ephrin]) we fixed the value of S₁ = 0.3. In the two panels, S₁ and S₂ are related by Eq (12), which reflects the relation between the portions of cell surfaces exposed to FGF and to Ephrin, respectively. (D) The presence of ephrin increases the steepness of the relation between Erk* and S₁. Heatmaps show the difference between the Hill coefficient of the curve Erk*(S₁) computed in the presence of ephrin and the Hill coefficient of the curve Erk*(S₁) computed in the absence of ephrin ([ephrin] = 0.001). The Hill coefficient obtained with standard values of the parameters for Erk*(S₁) in the absence of ephrin is 1.85. (E) The presence of ephrin increases the steepness of the relation between Erk* and [FGF]. Heatmaps show the difference between the Hill coefficient of the curve Erk*([FGF]) computed in the presence of ephrin and the Hill coefficient of the curve Erk*([FGF]) computed in the absence of ephrin ([ephrin] = 0.001). The Hill coefficient obtained with standard values of the parameters for Erk*([FGF]) in the absence of ephrin is 1.42. To compute Erk*([FGF]) we fixed the value of S₁ = 0.3. For all panels, the Hill coefficients were computed varying the values of the parameters from 0 to 1 (K_b, K₁, K₂, V_s, V_rg; left panels) or from 0 to 2000 (K_s, K_rg, K_d, K_e; right panels). Values of the Hill coefficients were obtained by curve fitting. The standard values of the parameters (Table 1) are highlighted with black boxes.

https://doi.org/10.1371/journal.pcbi.1010335.g005

The model can also be used to gain a deeper understanding of the role of ephrin, which was shown to prevent ectopic expression of Otx in the a6.7 cell by decreasing ERK activity below the threshold level in this cell type. Simulations reveal that the presence of ephrin also increases the steepness of the relation between Erk* and the cell surface contact to FGF-expressing cells, S₁ (Fig 5D). Interestingly, the steepness of the relation between Erk* and [FGF] is also increased by the presence of ephrin, although this effect is less pronounced (Fig 5E). In the two situations, the increase in steepness of the Erk* ([FGF]) (or Erk*(S₁)) relation due to the presence of ephrin is maximal in the absence of ephrin-independent RasGAP activity (K_b = 0). The increased steepness of the Erk*([FGF]) relation is however not observed at low values of K_rg, which defines the sensitivity of p120RasGAP to ephrin-bound receptors (Fig 5E right). For small values of K_rg, p120RasGAP activation is actually insensitive to ephrin, which makes the computation physiologically meaningless. Results thus show that ephrin signaling reinforces the sensitivity of the cells to changes in cell surface contact with FGF-emitting cells.

Comparison between FGF>FGFR >SOS and ephrin>Eph>RasGAP signaling.

As qualitatively discussed previously [8] and indicated above (Fig 4), the model predicts (Fig 6A) that ERK signaling is more sensitive to changes in the cell surface contact to FGF-expressing cells (S₁) than to the cell surface contact to ephrin-expressing cells (S₂) when considering these two surfaces as independent parameters.

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Fig 6. Model prediction.

(A) Erk* values are more sensitive to S₁ than to S₂. Heatmaps show the difference between the Hill coefficient of the curve Erk*(S₁) and the Hill coefficient of the curve Erk*(S₂). Erk*(S₁) was obtained by fixing the value of S₂ = S* (S* = 0.3) and letting S₁ vary between 0 and 1-S*. The same procedure was used to compute Erk*(S₂). The Hill coefficients obtained with standard values of the parameters are 1.99 and 1.04 for Erk*(S₁) and Erk*(S₂), respectively. (B) Erk* values are more sensitive to [FGF] than to [ephrin]. Heatmaps show the difference between the Hill coefficient of the curve Erk*([FGF]) and the Hill coefficient of the curve Erk*([ephrin]). To compute Erk*([FGF]) and Erk*([ephrin]) we fixed the value of S₁ = 0.3. The standard values of the parameters are highlighted with black boxes. (C) ERK activation levels (Erk*) shown as a function of FGF concentration (in red) or ephrin concentration (in orange). Values of the parameters used: V_s = V_rg = 1, K₁ = K₂ = 0.5, K_rg = K_s = 1200, [FGF] or [ephrin] = 5, K_d = K_e = 25, R_T = Q_T = 2000, S₁ = S₂ = 0.5, K_b = 10⁻⁶. Values of the Hill coefficients were obtained by curve fitting. (D) Upper panel: Effect of changing the k_MM3 in Eq (16) on the relation between the fraction of phosphorylated activator, A_p and active ERK, Erk*. Lower panel: Heatmaps showing the Hill coefficients of the relation between Otx and Erk* when changing the values of the k_MMi in Eqs (16–17). (E) Relation between the level of Otx expression predicted by the model (O) and cell surface in contact with FGF-expressing mesendoderm cells (S₁) in the presence (blue curve, Hill coefficient = 14.2) or in the absence (orange curve, Hill coefficient = 5.7) of the repressor I. Dashed vertical lines represents the mean values of S₁ for each cell type. The Hill coefficients were computed using relation (26).

https://doi.org/10.1371/journal.pcbi.1010335.g006

Simulations shown in Fig 6B indicate that, in line with this property, Erk* is more sensitive to [FGF] than to [ephrin]. It is thus clear that there is an asymmetry in the respective influences of the FGF>FGFR>SOS and of the ephrin>Eph>RasGAP pathways in controlling the possible outcome of the neural fate. Simply stated, the upregulation of one of the pathways is not equivalent to the downregulation of the other. Interestingly, even in the absence of the ephrin-independent RasGAP activity (K_b), the Erk* vs [FGF] curve is slightly steeper than the Erk* vs [ephrin]. Indeed, Fig 6C indicates Hill coefficients of 2.13 and 1.80 respectively for the Erk* vs [FGF] and Erk* vs [ephrin] relations. For these curves, the values of all parameters have been taken the same for the FGF and the ephrin pathway, with K_b~0 and S₁ and S₂ are both equal to 0.5. Even in these completely symmetrical conditions, the domain of values covered by the fraction of Ras-GTP (T) differs when varying FGF or ephrin concentrations (S5 Fig). In consequence, Erk* can never reach zero when increasing [ephrin] and can never reach 1 when increasing [FGF], which slightly modifies the steepness of the curves (Fig 6C).

Model of ERK activation

We assumed that FGF and ephrin act as short-range ligands, with ephrins being membrane-tethered ligands, and that the concentration of extracellular FGF ([FGF]) between mesendoderm and ectoderm cells is uniform. This hypothesis is motivated by the observation that all a-line ectoderm cells are in contact with A-line mesendoderm cells that express uniform levels of FGF (see S1A Fig of [8]). Similarly, we assumed the concentration of extracellular ephrin ([ephrin]) to be constant. In line with experimental observations [8], we considered that the density of FGF receptors on the plasma membrane of ectoderm cells is uniform. In the absence of experimental data, we similarly considered the density of Eph receptors on the plasma membrane of ectoderm cells to be uniform. Thus, the number of receptors in contact with the ligand is different for each ectoderm cell, since it is proportional to the area of cell surface in contact with FGF- or ephrin- expressing cells.

The activation (phosphorylation) of the extracellular signal-regulated kinase, ERK, is induced by Ras-GTP (see Fig 1A) via a cascade of phosphorylations that is not considered explicitly. Assuming that the total amount of Ras-GDP and Ras-GTP is conserved, we described the relation between the fraction of active, doubly phosphorylated ERK (Erk*) and the fraction of Ras bound to GTP (T) as a Hill function, i.e.: (1)

Erk_max is the maximal level of ERK activation (Erk^A), n is the Hill coefficient and K_erk is the fraction of Ras-GTP leading to half maximal ERK activation.

The evolution equation for the fraction of Ras-GTP is (see Fig 1A): (2) where V_SOS([FGF]) is the conversion rate from Ras-GDP to Ras-GTP (mediated by the FGF-dependent SOS activity), V_p120RasGAP([ephrin]) is the conversion rate from Ras-GTP to Ras-GDP (mediated by the ephrin-dependent p120RasGAP activity) and is the conversion rate from Ras-GTP to Ras-GDP mediated by ephrin-independent RasGAP activity. We will now describe these three processes.

Model of Otx expression

Otx expression is regulated by dpERK (Fig 1A). A minimal enhancer of the Otx gene, named the Otx a-element, contains three GATA and two ETS (E26 transformation-specific)-binding sites required for neural-specific expression in the ectoderm lineages via Gata.a and Ets1/2 transcription factors [6,24].

The Ets2 repressor factor 2 (ERF2) [29] recognizes the same binding site as the transcriptional activator Ets1/2 and its activity is negatively controlled by ERK via nuclear export [12,13,17]. Thus, we considered that Otx expression is enhanced by the stimulation of an activator (A; e.g. Ets1/2) and is inhibited by a repressor (I; e.g. ERF2). The phosphorylation of both the activator and the repressor is controlled by ERK. Thus, we modeled Otx expression with the following evolution equation: (15) where v_b is the basal Otx expression rate, (v_b+v_o) is the maximal Otx expression rate and k is the degradation rate for Otx. K_a is the half saturation constant for the activator A_p while K_i is the dissociation constant for the repressor. In Eq (15), it is considered that ERF2 (I) is a competitive inhibitor of Ets1/2 (A). Thus, there is an effective half-saturation constant that depends on the concentration of I, which is equal to . When I is large, this effective constant is much larger than K_a and, when I is small, this effective constant approaches K_a.

The evolution equations describing the phosphorylation and dephosphorylation of Ets1/2 and ERF2 are: (16) (17) where A_p represents the fraction of active, phosphorylated activator; (1−A_p) the fraction of inactive, unphosphorylated activator; I the fraction of active, unphosphorylated repressor and (1−I) the fraction of inactive, phosphorylated repressor. For the two phosphorylation reactions, the rate constants k_MM3 and k_MM1 of A and I phosphorylation are multiplied by the fraction of doubly phosphorylated ERK (Erk*). v_MM4 and v_MM2 are the maximal rates at which A_p and I_p are dephosphorylated. K_MM3 and K_MM1 are Michaelis-Menten constants for the phosphorylation of A and I, respectively, and K_MM2 and K_MM4 are Michaelis-Menten constants for the dephosphorylation of I_p and A_p, respectively. K_MM3 and K_MM4 are normalized with respect to the total activator concentration, while K_MM2 and K_MM1 are normalized with respect to the total repressor concentration.

Since the quantity measured experimentally is the counts of smFISH Otx spots, to compare the simulations results with experiments we considered that the level of counts of smFISH Otx spots (Otx_smFISH) is a linear function of the level of Otx expression (O): (18) where C is the maximal number of smFISH Otx spots and D is the basal smFISH Otx spot count. Taking into account that the value of O computed with the model was close to zero in the a6.8 cell type, we chose D as the Otx smFISH count in the a6.8 cell type in each experiment. C was defined empirically in order to obtain the best fit to the data.

To obtain an expression for Otx expression (O and Otx_smFISH) as a function of the inputs of the signaling cascade (S₁) we combined the models of ERK activity and of Otx expression presented in the previous sections. We computed ERK activity using Eq (1) and we used this as the input (Erk*) to compute the fraction of active Ets1/2 (A_p) and active ERF2 (I) using Eqs (16) and (17). Then we computed the level of Otx expression by solving Eq (15). All equations are solved at steady state.

Equations are solved analytically at steady-state and solutions evaluated for all investigated sets of values of parameters with codes written in Python. All the codes and the experimental data are available at https://github.com/rossanabettoni/Model-of-neural-induction-in-the-ascidian-embryo.

Model of ERK activation in b-line cells

b-line ectoderm cells are in contact both with A-line mesendoderm cells that express a uniform level of FGF and with B-line mesendoderm cells (B6.1, B6.2, B6.3 and B6.4) that express different levels of FGF (S2 Fig) [8]. Therefore, Eqs (5) and (6) do not appropriately describe activation of the FGF pathway in these cells. To model activation of the ERK pathway in the b-line cells, we considered two distinct populations of FGF receptors. The first population encompasses the receptors located on the surface in contact with the B-line cells. For each b-line cell, the number of FGF-bound receptors at the interface with cell B6.i (with i ranging from 1 to 4) is given by: (19) where S_1B6.i is the area of the b-line cell surface in contact with cell B6.i and [FGF]_B6.i is the modelled level of extracellular FGF concentration at this interface.

The second population of FGF receptors encompasses those located on the surface portion in contact with A-line mesendoderm cells. On this surface portion, the number of FGF-bound receptors is given by Eq (5). Thus, the total number of bound FGF receptor for a given b-line cell is: (20)

Values of S₁ and S_1B6.i are given on github (see above). The level of extracellular [FGF] at each interface was estimated from the experimentally measured gene expression levels of the FGF9/16/20 ligand (see S1A in [8]). As these data only allow comparison with respect to the concentration of FGF perceived by the A-line mesendoderm cells for which [FGF] was taken equal to 5 (Table 1), we consider [FGF]_B6.1 = 5, [FGF]_B6.2 = 0.7, [FGF]_B6.3 = 0.08 and [FGF]_B6.4 = 0.7. For this analysis, we disregarded Eq (12) and entered S₁ and S₂ measurements as independent variables. Experimental measurements for S₁ and S_1B6.i and S₂ are indicated in S2A Fig.

Cooperativity in Otx activation

Cooperativity in the Ets1/2-dependent Otx activation could also contribute to the ultrasensitivity in the Otx response to ERK activation. Indeed, the Otx a-element contains two ETS-binding sites [6].

We modelled cooperativity in Ets1/2 binding introducing a Hill coefficient equal to 2 in the evolution equation for Otx: (21)

Eq (21) and the following parameters (K_MM1 = K_MM2 = K_MM3 = K_MM4 = 0.3, k_MM1 = k_MM3 = 14) were used to obtain the results presented in S7 Fig. This figure illustrates that zero-order ultrasensitivity in Eqs (16) and (17) can be replaced by cooperativity in Eq (15).

Experimental methods

All experimental results, except the data obtained in b-line cells, are taken from our previous work [8]. Experimental measurements of cell-surface contacts and dpERK IF levels in b-line cells were obtained as described previously [8].

Estimation of uncertainties

Ratios between anti-dpERK immunofluorescence levels in treated and non-treated cells (for Figs 2C and S1C)

The ratio between the level of dpERK IF signals in the injected side of the embryo and the level of dpERK IF signals in the control side of the embryo (for a single cell) is:

With uncertainty: where and are equal to ΔB. In Figs 2C and S1D the uncertainties Δr for the single cells are not shown.

The average of the ratios for each cell type was computed as:

With corresponding uncertainty: (23) in which ΔErk_M^f(inj) and ΔErk_M^f(control) are given by Eq (22).

This uncertainty is indicated on the figure panels as the model error bars.

Otx smFISH spot counts

Given a complete set of parameter values including S₁ and S₂, there is no uncertainty on the deterministically modeled value of Otx expression (O). However, to compare with experimental data, we computed the average of 25 values of O (O_M), corresponding to 25 couples of (S₁, S₂) values. The uncertainty on this computed average activity (ΔO_M) is the standard deviation.

Experimentally, Otx expression is quantified by the Otx smFISH spot counts. As described above (Eq (18)), we assumed that the number of fluorescent spots is a linear function of O: where D is the basal value of Otx smFISH spot counts.

In the absence of other information, we postulated that the uncertainty on D (ΔD) is the same as the standard deviation of the experimental values of Otx smFISH spot counts measured in the a6.8 cells. We considered that there is no uncertainty on C.

Thus, the uncertainty ΔOtx_smFISH is equal to ΔD for each cell, i.e. for each couple of (S₁, S₂) values. In the Figs 3A, 3B, 3C, 3E, S4A, S4B, S7A and S7C the uncertainties ΔOtx_smFISH for the single cells are not shown, for clarity reasons.

The average Otx smFISH spot counts for a given cell type is computed as: with the corresponding uncertainty: (24)

This uncertainty is indicated on the figure panels as the model error bars.

Estimation of the Hill coefficients

When not mentioned explicitly, Hill coefficients were obtained by fitting the curves with Hill functions.

Since the fit of the curve Otx expression (O) as a function of Erk* in the absence of the repressor (Fig 3F) was poor, we used the fact that Hill coefficients can be calculated in terms of potency as: (25) where Erk*₉₀ and Erk*₁₀ are the levels of ERK activity needed to produce 90% and 10% of the maximal Otx expression, respectively.

Similarly, we computed the Hill coefficient between Otx expression (O) and S₁ in Fig 6E as: (26) where S₁₉₀ and S₁₁₀ are the fractions of cell surface in contact with FGF needed to produce 90% and 10% of the maximal Otx expression, respectively.