Ethics statement

All work with animals was approved under the license BMWFW-66.018/0006-WF/V/3b/2016 from the Austrian Bundesministerium für Wissenschaft, Forschung und Wirtschaft and procedures were performed in accordance with the relevant regulations.

Model

In order to capture the dynamics of floor plate formation and its relationship with the Shh morphogen gradient within the growing neural tube, we developed a reaction-diffusion model that integrates a thermodynamic description of relevant gene interactions. The model represents a simplified interaction network between three species: F, N and Shh (Fig 1B). F and N represent non-diffusible factors that are linked through effective cross-repressive interactions and define the identity of floor plate or adjacent neural progenitor domains, respectively. Shh represents the diffusible ligand which activates both N and F. The system dynamics is described by: (1) where [F], [N] and [Shh] are the concentrations of the interacting species, D_S is the diffusion constant of Shh, κ_F, κ_N, are uniform activation constants, K_N→F, K_F→N are repressor binding affinities, c_S→F, c_S→N, are morphogen activation strengths relative to uniform activation, and κ_F→S is the activation strength of Shh ligand production by the floor plate. Note that Shh is not produced in the floor plate if either [F] = 0 or κ_F→S = 0. To reduce the number of free parameters, we set the production rates α to 0.1 h^-1, degradation rates γ to 0.72 h^-1 (= 2∙10⁻⁴ s^-1), and D_S = 0.11 μm² s^-1 – these values are of similar orders of magnitude to the values measured for other morphogens (reviewed in [21]) and to inferred values for Shh in other studies [22–25]. The exponents that quantify non-linearity m_N→F, m_F→N were set to 3. All gene expression levels and morphogen concentration depend on position 0 ≤ x ≤ L, and time t ≤ t_end.

We model a one-dimensional tissue that grows from an initial length of L₀ = 100 μm to a default final length L_end = 400 μm at time t_end = 60 h, which is similar to the experimentally measured DV length of the neural tube after 60 hours of development [17]. By default, we implemented linear growth with a default growth rate k_p = 5 μm/h. In order to study how the mode and rate of tissue growth rate affects the model behaviour, we also implemented an exponentially growing tissue, in which L = L₀ exp(t/τ), where τ = 43.3 h (consistent with [18]). Furthermore, we tested the model behaviour for varying linear growth rates. To this end, we set the final tissue length such that it corresponds to growth rates from k_p = 0 μm/h (L_end = 100 μm, non-growing condition) to k_p = 50 μm/h (L_end = 3100 μm).

Simulations are performed on a growing one-dimensional tissue. Growth is implemented by introducing in Eq 1 a new spatial variable for which the domain size is constant [26]. This introduces an additional concentration-lowering term , into each equation, where c is the respective concentration of [F], [N] and [Shh], and is the rate of change of L(t). We also rescale the diffusion constant to D_S/L². The equations are solved on the unit interval 0 ≤ ≤ 1, divided into 100 spatial bins that mimic the discrete cellular structure of the tissue. The integration scheme uses first order finite differences of the spatial derivatives, whilst time steps are handled using the Heun’s scheme, which is a second order method. The solution in absolute units is retained by .

The F and N domains are defined according to the gene expression levels, such that [F] > [N] defines the F domain, and [F] < [N] the N domain. Throughout the text, we refer to the F domain also as FP or FP domain.

The initial conditions of the model are such that at t = 0, N is expressed uniformly across the tissue ([N]_init = 10 a.u. for 0 ≤ x ≤ L_end), reflecting the transient expression of N in future FP cells, while there is no initial expression of F (i.e. [F]_init = 0) [5]. To represent and evaluate the temporal requirements for Shh secreted by the notochord [27–29], we consider different combinations of initial and boundary conditions: 1) a transient burst of Shh at position x = 0 within the tissue ([S]_init = 100 a.u. at x = 0, and [S]_init = 0 for 0 < x ≤ L_end) with double reflective boundary conditions = 0; 2) no Shh present in the tissue at t = 0, but a constant flux of Shh j_Shh through the ventral end of the neural tube = −j_Shh, with reflective boundary condition at the dorsal end = 0; 3) similar to the previous condition, but the flux of Shh j_Shh is abruptly removed at a specific time t_off. After the flux is removed, the boundary conditions are double reflective. Throughout the text and figures, we refer to [S]_init also as S_init.

As a proxy for stochastic effects resulting from cell division, we additionally simulate the deterministic system described in Eq 1 with fluctuations in [F] and [N] as an Ornstein–Uhlenbeck process. We used the discretization following [30]: , where η stands for [F] or [N], α(t) is a Gaussian white noise with zero mean and unit variance, σ_η is the magnitude of noise, and τ_η is the correlation time of noise. The noise is included in [F] and [N] only when the respective expression levels are higher than 1 a. u.

Computational screen

A priori, the model in Eq 1 (Fig 1B) may lead to no stable formation of a F domain, or have a F domain that extends across the entire tissue. In order to identify parameter sets that result in biologically plausible FP formation (Fig 1C), we define the following constraints: (i) the emerging pattern has two domains, F and N, with the F domain starting at the ventral end (x = 0), (ii) the Shh profile decays monotonically as a function of position x, (we achieve this by imposing an upper bound on Shh concentration at L_end of 2% of [S]_init, hence avoiding rare cases in which the FP forms at the dorsal end), (iii) [Shh] and [F] concentrations differ at most by an order of magnitude, that is 0.1 ≤ [Shh]/[F] ≤ 10; this is a technical assumption to keep the relative range of [Shh] and [F] bounded, as effectively κ_F→S acts as scaling factor for [F] (see Eq 1) and κ_F→S is varied by 6 orders of magnitude (see below). We have two additional criteria that narrow down the solutions to biologically realistic spatial and temporal scales: (iv) the length of the F domain at t_end is between 5% and 25% of tissue length (consistent with 7.5% of tissue length measured at t = 60 h in [17]); this restricts the space of successful solutions to F lengths from 20 μm to 100 μm at t_end, (v) FP is established between 2.5 h and 20 h; the time of FP establishment T_est is defined as the time at which [F] > [N]. The minimal T_est of 2.5 h avoids oversampling the parameter space in the region of establishment times on the order of minutes, which are inconsistent with experimental results [17]. The maximal T_est of 20 h was set to exclude solutions in which FP never forms.

The computational search of parameter space is performed by random walk in the logarithmic parameter space. The following parameters are varied over 6 orders of magnitude: c_S→F, c_S→N, κ_F→S, K_N→F, K_F→N from 0.005 to 5000, and κ_F, κ_N from 5×10⁻⁶ to 5. The initial parameter set is selected randomly. If this set does not fulfil all success criteria (i)-(v), another random set of parameters is selected. If the selected parameter set satisfies all success criteria, the next set of parameters is generated based on the preceding set by multiplying parameters by random factors drawn from a log-normal distribution with 0 mean and 0.2 standard deviation. This process is performed iteratively until a predefined number of parameter sets is visited. The computational search was started independently 10 times with each search visiting 40 000 parameter sets, and the sets of identified successful solutions were combined.

Mouse lines

All work with animals was approved under the license BMWFW-66.018/0006-WF/V/3b/2016 from the Austrian Bundesministerium für Wissenschaft, Forschung und Wirtschaft. All procedures were performed in accordance with the relevant regulations. The following mouse lines were previously described: Sox2^CreERT2 (JAX 017593, [31]), Shh^CreERT2 (JAX 005623, [32]), Shh^Flox (JAX 004293, [33]), and Shh^- (JAX 003318, [34]). Transgenic strains were maintained on a CD-1 background. Sox2^CreERT2/+ was crossed to Shh^Flox/+ to generate Sox2^CreERT2/+, Shh^Flox/+ mice, which were further crossed to Shh^Flox/+ to generate Sox2^CreERT2/+, Shh^Flox/Flox embryos. Females were injected at E7.5 with 3 mg tamoxifen, and embryos were collected at E10.5. Shh^CreERT2/+ mice were crossed to Shh^Flox/+ mice to generate Shh^CreERT2/Flox embryos. Females were injected at either E5.5 and E6.5 with 3 mg tamoxifen each, at E6.5 with 2 mg tamoxifen, or at E8.5 with 3 mg tamoxifen; and embryos were subsequently collected at E10.5. Shh^+/- embryos were collected at E8.75.

Immunohistochemistry

Embryos were fixed in PBS with 4% PFA for either 2 hours (E10.5) or 1.5 hours (E8.75), embedded in gelatin, and cryosectioned. Transverse brachial sections were pre-blocked in PBS containing 0.1% Triton X-100 (PBST), 1% BSA for 1 hour, then incubated overnight at 4°C in PBST with the following primary antibodies: mouse anti Shh (DSHB, supernatant, 1:70), and sheep anti Arx (R&D systems, 1:100) or goat anti T/Brachyury (R&D systems, 1:250). The following day, sections were washed in PBST, incubated with fluorescently labelled secondary antibodies (Jackson ImmunoResearch; donkey anti mouse IgG-Cy3, 1:1000, donkey anti sheep IgG-FITC 1:250, donkey anti goat IgG-FITC, 1:250) and DAPI (ThermoFisher, 1:100) in PBST for 2 h at room temperature, washed in PBST, and mounted for imaging.

Imaging and quantification of mouse sections

Immunostained sections were imaged using a Zeiss LSM800 Axio Observer Z1 confocal microscope with a Plan-Apochromat 40x/1.2 water objective, at 0.8x zoom, with a Z-stack composed of 8 Z-planes 0.6 μm apart. Subsequent image analysis was performed using Fiji [35]. To measure the FP size at E10.5, first the Arx-positive FP area was quantified in maximum intensity projections using the “Threshold” function in default mode. Because the neuroepithelium, including the FP, is a monolayer, we obtained the FP length by dividing the measured area by the mean apicobasal thickness measured at the ventral midline, which was 17 ± 0.44 μm (mean ± SEM, n = 19 sections). The Shh fluorescence intensity (FI) profiles were quantified using the workflow described in [36]. Briefly, a 14 μm-wide region of interest (ROI) was aligned along the apical surface of the neural tube, starting from the ventral midline, to obtain a mean pixel intensity calculated across the 14 μm ROI width as a function of ventral-to-dorsal position. The Shh profiles were subsequently analyzed in Python. For every profile, background, defined as the minimum FI within 10–90% of DV length, was subtracted and the FI was smoothed with a moving average filter (window size of 5 μm). The mean FI profile was calculated for each genotype. Then, x = 0 was defined as the position of the mean Shh intensity peak located at the dorsal edge of the floor plate. x = 0 for Sox2^CreERT2/+, Shh^Fl/Fl and Shh^CreERT2/Fl profiles was set to the same position as in their respective control mean profiles. Shh background subtraction was also performed on the mean profiles. Shh profiles were normalized to the mean maximum intensity of the control group. To measure the Shh FI in the notochord at E8.75, the T-positive signal was used to outline the notochord in maximum intensity projections. The background FI was measured outside the notochord and subtracted.

Floor plate formation independent of floor plate-derived Shh relies on strong network interactions

In order to understand how FP size is determined, we performed a computational screen to identify parameter sets that lead to FP formation using the reaction-diffusion model described above. This model incorporates the genetic interactions that influence the specification of FP identity and tissue growth (Fig 1B). The gene regulatory network starts from an initial state with high N expression, no F expression and responds to an initial burst of Shh to activate F (Fig 1C and 1D). We used a set of criteria to define the formation of an experimentally plausible F domain at the end of the simulation at t = 60 h (Fig 1D) as described above. Based on these criteria, we identified 169 979 successful parameter sets out of 400 000 visited parameter sets. Within this set of successful solutions, FP sizes followed a half-normal distribution, such that FP sizes between 20 μm and 40 μm accounted for 66% of successful solutions, while 4% had FP size ≥ 80 μm (Fig 1E).

To understand how FP size depends on model parameters, we examined the parameter distributions for subsets of networks that produced FP sizes within a defined range. The distributions of model parameters for FPs of different sizes showed that the model can produce all possible FP sizes over broad parameter ranges (S1 Fig). In particular, contrary to our expectation that the strength of activation of F by Shh should influence FP size, the distribution of c_S→F did not vary with FP size (S1 Fig). This suggests that FP size is indirectly affected by other model parameters. To understand the sensitivity of FP size to model parameters, we performed sensitivity analysis focusing on a subset of networks that yield floor plates with a relative size of 20% of DV length. We measured how the relative size of the FP changed upon 10-fold up- and down-regulation of each parameter relative to its mean value for that subset, while all other parameter values were held constant. The sensitivity analysis revealed that FP size was highly sensitive to changes in all parameters, except κ_F→S, for which the dependence was weak (Fig 2). In particular, FP size increased when F was strongly activated and N was deactivated or repressed. Conversely, FP size decreased when N was strongly activated and F was deactivated or repressed. These results were independent of the exact value of the diffusion coefficient (S2A Fig). Furthermore, the robustness of the model to changes in κ_F→S values was also preserved in the presence of noise in gene expression (S2B–S2E Fig). Altogether, this indicated that FP size is influenced by all interactions within the network, but is least sensitive to Shh production in the FP. This suggests that the FP may form in a manner that is largely independent of the Shh that it produces.

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Fig 2. FP size sensitivity to perturbations of model parameters.

Mean relative FP size at the end of the simulation upon perturbation of the indicated model parameter for a subset of 214 randomly selected networks with relative FP size of 20%. The parameters are modified in 40 equally distributed logarithmic steps from 0.1 to 10-fold of their value. The shaded regions are SE.

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To further investigate how Shh production in the FP affects FP size, we set the production term κ_F→S to 0, which corresponds to a case in which there is no floor plate-derived Shh (Shh^FP). We then compared the full model to the perturbed case by quantifying the change in FP size: , where is FP size in the perturbed case. In most cases, the perturbed networks exhibited one of three types of responses, in which the FP size was either maintained, decreased, or completely lost (Fig 3A). This indicates that the FP in these networks was insensitive, partially sensitive or completely sensitive to its own production of Shh, respectively. In a small fraction corresponding to ~1.5% of all cases, the FP size increased when the production term κ_F→S was set to 0 (Fig 3A). This could result from a situation in which Shh produced in the FP is required to maintain N expression, thereby restricting the expansion of FP dorsally. Together, this indicates that the effect of Shh^FP on FP size depends on the values of the other network parameters.

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Fig 3. FP formation can occur by different mechanisms, dependent or independent of FP-derived Shh.

A. Probability distribution of the change in FP size (Δl_FP) for successful solutions with production of Shh by FP switched off (κ_F→S = 0) compared to the default model (κ_F→S > 0). Rectangular frames indicate solutions with FP formation dependent on Shh^FP (Δl_FP = l_FP; sensitive, red) and independent of Shh^FP (Δl_FP = 0; insensitive, grey). Colors correspond to log-scaled conditional probability of observing Δl_FP for a given l_FP (legend), n = 168 074 (all solutions except 1905 with FP extending across the whole tissue at t = 60 h). B. Distribution of model parameters for Shh^FP-sensitive (red) and insensitive (grey) classes of solutions, sensitive (n = 45 336), insensitive (n = 66 827).

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To understand what distinguishes these classes of networks, we compared the parameter distributions within each class (Figs 3B, S3A and S3B). The most notable difference was that Shh^FP-sensitive solutions required smaller basal activation κ_F and at the same time had overall higher strength of activation of F by Shh (c_S→F), consistent with their strong dependence on Shh. By contrast, solutions that are insensitive to Shh^FP require strong basal activation and have lower c_S→F. Furthermore, insensitive networks have higher repression of F by N. This suggests that in the insensitive class, FP formation results from strong basal uniform activation, strong repression by Shh-dependent repressors, weak initial activation by Shh and no activation by FP-derived Shh.

Rapid floor plate formation is followed by expansion via tissue growth

To investigate how FP formation can occur without being affected by Shh production in the FP, we analysed the time course of FP formation in the insensitive class of solutions compared to the sensitive class. We observed that as the overall tissue length increases over time, the FP size also increases (Fig 4A and 4B). Notably, the FP size occupies a constant fraction of the overall tissue length from 10 h onwards (Fig 4A’ and 4B’). This indicates that after this time point, the FP size scales with tissue growth (Fig 4A’ and 4B’). These dynamics were unchanged in the absence of Shh^FP in the insensitive class of solutions (Fig 4C and 4C’), while in the sensitive class, the FP domain was lost after 10 h (Fig 4D and 4D’).

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Fig 4. Two time scales of FP formation.

A, B. Absolute FP size as a function of the increasing tissue size L over time for Shh^FP-insensitive (A) and sensitive (B) solutions for the default model with Shh^FP. The tissue growth rate is varied from k_p = 0 μm/h to 10 μm/h for the linear growth condition (color code in legend), and follows L = L₀ exp(t/τ) with τ = 43.3 h for the exponential growth condition (purple dashed curve). The inset shows a magnified view for small tissue sizes. A’, B’. Relative FP size as a function of time for insensitive (A’) and sensitive (B’) solutions with Shh^FP and varied growth rate. Growth rates are k_p = 0, 5, 10 μm/h for linear growth (color-coded as indicated), and exponential growth condition (purple dashed curve) is as in A and B. C, D. The same as A and B for linear growth, respectively, but in the absence of Shh^FP. C’, D’. The same as A’ and B’ for linear growth, respectively, but in the absence of Shh^FP. A-D’, each curve is the mean of n = 10 solutions. E. Average FP formation time T_FP (defined as the time at which FP reaches its final relative size) as a function of the degradation rate of N (γ_N), see Eq 1. Pairwise comparisons two-tailed t-test: * 0.05 ≥ P > 0.01; *** 0.001 ≥ P > 0.0001. n = 10 per condition, error bars SEM. F. Relative FP size as a function of time for different γ_N. The default condition is γ_N = 0.72 h^-1, dashed line indicates the position of the average T_FP = 10 h for that condition.

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The scaling of the FP with tissue size from 10 h onwards implies that growth passively expands the FP domain after this time. To test this, we varied the growth rate k_p from 0.5 μm/h to 10 μm/h, resulting in tissues with different final sizes at 60 h, from 100 μm to 700 μm, respectively. In addition, we also simulated an exponentially, rather than linearly growing tissue. Strikingly, the FP scaled with tissue size in these different growth regimes, indicating that the relative FP size was growth rate-invariant (Fig 4A’ and 4B’). Weak deviations from perfect scaling were only observed for large FPs from the sensitive class of networks (Fig 4B’).

The time course analysis suggests that there are two time scales of FP formation (Fig 4A’ and 4B’). Initially, the FP rapidly, within 10 h, reaches a size that is defined by the model parameters. Subsequently, the FP scales with tissue size, hence the absolute FP size is defined by the amount of growth. Consistent with this, in the absence of growth, FP formation is completed within 10 h. We then asked which parameters control this initial rapid phase of FP formation. We found that there is no obvious correlation between the time it takes for the relative FP size to become constant (T_FP) and most individual parameters in the computational screen (S4A Fig), consistent with observations that in non-linear systems, time scales may be determined by a set of parameters [37,38]. Nevertheless, we found that the time it takes for the relative FP size to become constant (T_FP) was most prominently influenced by the degradation rate of the reacting species, notably γ_N (Fig 4E), and to a lesser extent γ_F (S4B and S4C Fig). A 2-fold increase in the degradation rate of N leads to an approximately 2-fold decrease in T_FP (Fig 4E and 4F).

The scaling of FP with tissue size after 10 h suggests that the FP is established in response to Shh initially present in the system and after 10 h is not sensitive to changes in Shh concentration. To test whether this is the case, we investigated how the FP size changes when the Shh input is configured differently. In our model so far, Shh is initially provided as a fixed pulse of 100 a. u. in the first spatial bin at t = 0. In the absence of this initial pulse, FP formation does not occur. The initially provided amount of Shh rapidly diffuses through the tissue and forms an exponential gradient which transiently declines in amplitude and extends in decay length until Shh production starts in the FP (S5A Fig). The values of D_S and γ_S determine the speed of this transient behaviour. Thus, increasing D_S leads to extended decay lengths and correspondingly increased F domain sizes, while lowering D_S has the opposite effect (S5B Fig). Moreover, for fixed values of D_S and γ_S, increasing values of S_init correlate with increased Shh gradient amplitudes and also with an increased size of the F domain (S5C Fig). The magnitude of the Shh pulse S_init correlated with the size of the F domain in a logarithmic manner (Fig 5A), which is on average consistent with the positioning of the F boundary position at a threshold concentration of Shh during this early phase of the dynamics [2].

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Fig 5. FP formation does not require continuous Shh flux.

A. Mean FP size for varied magnitude of initial Shh and no flux for insensitive solutions resulting in large FPs (l_FP/L_end = 20%; dark grey) or small FPs (l_FP/L_end = 5%; light grey). FP size relative to tissue size at t = 60 h, left y-axis; absolute FP size, right y-axis. Inset, subregion on linear x-axis. B, C. Mean FP size for insensitive (B) and sensitive (C) solutions for varied Shh flux and without initial Shh. Solutions resulting in large FPs (l_FP/L_end = 20%), dark grey/red, small FPs (l_FP/L_end = 5%), light grey/red. The magnitude of flux that yields approximately the same FP size as the default condition (no flux, with initial Shh) is indicated with dashed lines. Insets, linear scale for j_Shh ≤ 10 a. u./s. Y-axes as in A. D. Relative FP size as a function of time for the default condition (no flux, with initial Shh, dashed) vs analogous flux condition (solid). j_Shh = 0.1 a. u./s for large FPs (top) and j_Shh = 1 a. u./s for small FPs (bottom). E, F. Relative FP size with flux abruptly removed at the indicated time (colored) for insensitive (E) and sensitive (F) solutions for large FPs. With Shh^FP (top), without Shh^FP (κ_F→S = 0; bottom). A-F, the average relative FP size is shown, n = 10 per point. A-C, at 60 h, error bars SEM.

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While these observations demonstrate the importance of the initial Shh signalling levels for setting the FP size, in the embryo Shh is continuously produced by the notochord, rather than as a short burst, which raises the question of how continuous production of Shh affects the dynamics. To simulate the effect of the notochord, we compared our default model with an initial pulse of Shh to a model with a constant flux of Shh j_Shh, representing the notochord. We found that for a specific magnitude of flux, the model produced a similar relative FP size as in the model with a pulse (Fig 5B). Similar to the magnitude of S_init (Fig 5A), increasing flux led to a logarithmic increase in the relative FP size (Fig 5B). This was the case for both sensitive and insensitive classes of solutions (Fig 5B and 5C). Crucially, the presence of flux within a test range spanning >4 orders of magnitude did not alter the temporal dynamics of FP formation–FP size still scaled with tissue size after 10 h (Fig 5D). This supports the idea that the Shh flux from the notochord contributes to FP formation before 10 h, but that the FP does not respond to the notochord Shh flux after this time point.

To test this directly, we performed a simulation in which there was no initial pulse of Shh and instead, Shh flux was provided for 10 or 20 hours and subsequently removed. This showed that both in the sensitive and insensitive classes of networks, external (notochord) Shh input is required for only 10 h, and subsequently the FP is maintained without the need for continuous Shh flux (Fig 5E and 5F). Notably, in the absence of Shh production in the FP, the FP was maintained after flux removal in the insensitive class of networks, but was lost in the sensitive class (Fig 5E and 5F). This indicates that in the sensitive class, FP maintenance requires either continuous flux or Shh^FP. By contrast, in the insensitive class neither of these sources of Shh production is required for FP maintenance, indicating that networks in the insensitive class are bistable with respect to Shh.

Altogether, these results show that in Shh^FP-insensitive networks, the FP forms independent of Shh^FP because of the fast time scale of initial FP formation, which is set by the degradation rate of N. The FP is maintained independent of Shh due to the hysteresis inherent to this dynamical system with respect to the Shh input.

The floor plate amplifies the Shh gradient amplitude at late stages

Our model indicates that the initial FP formation depends on the Shh gradient, however, the contribution of different factors to Shh gradient formation is unclear. To address this, we analyzed the Shh gradient shape and asked how it depends on different conditions and model parameters. The decay length λ of steady state exponential morphogen gradients formed by diffusion with diffusion coefficient D, uniform degradation with rate γ and localized production is equal to [39]. Thus, because the Shh diffusion coefficient D_S and degradation rate γ_S in our model are fixed, the decay length of the Shh gradient λ_Shh in the receiving tissue is expected to reach a constant value, corresponding to 23.45 μm. Consistent with this, we find that at t = 60 h, λ_Shh estimated from a fit to the Shh profile in the N domain in the set of successful solutions has a narrow distribution centred at 23.32 ± 0.09 μm (mean ± SE) (S6A Fig). By contrast, the Shh amplitudes of successful solutions exhibited a broad ordinary normal distribution in the range between 100 and 500 a. u. with 〈A_Shh〉 = 293 ± 75 a. u. (mean ± SE) (S6B Fig). Therefore, the Shh gradient changes mainly through its amplitude.

To understand what factors influence the Shh amplitude, we first asked how it depends on changes in the model parameters. We found that the Shh amplitude and the FP size varied with respect to most parameters in a qualitatively similar manner (Fig 6A compare to Fig 2). There was one notable exception–while the FP size did not depend strongly on the strength of Shh production by the FP, κ_F→S, the Shh amplitude increased with increasing κ_F→S. Thus, while Shh^FP is dispensable for regulating the FP size, it influences the Shh gradient amplitude. This suggests that changes in the amplitude of Shh over time results from changes in the net production of Shh^FP over time.

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Fig 6. The increase in Shh amplitude over time depends on floor plate growth.

A. Sensitivity of the Shh amplitude at the end of simulations to parameter perturbations. The parameters are changed in 40 equally distributed logarithmic steps from 0.1 to 10-fold of their value. Networks with l_FP/L_end = 20% were randomly selected (n = 214, including n = 100 sensitive, n = 100 insensitive and n = 14 partially sensitive). The shaded regions are SE. B. Shh amplitude as a function of tissue size for insensitive solutions with Shh^FP, resulting in small FPs (l_FP/L_end = 5%). Growth rates from k_p = 0 μm/h to 50 μm/h are color-coded, n = 10 per condition, sampled every 10 min. Note that the saturation of A_Shh at large growth rates will be reached faster for solutions resulting in large FPs. C. Shh amplitude as a function of time in the absence of Shh^FP. The curves are identical for all n = 10. D. Shh amplitude for large FPs (l_FP/L_end = 20%) with varied flux of Shh and no initial pulse of Shh. The notochord flux is present at all times (black solid) or is removed at t_off = 10 h (white dashed), error bars SEM. E. Shh amplitude over time for large FPs with flux removed at specific times. The flux j_Shh = 1 a. u./s is present (solid) or removed at 10 h (dashed), or at 20 h (dot-dashed). The condition with initial pulse of Shh S_init = 100 a. u. is shown for comparison (solid grey). F. The same as E, but for 10-fold lower flux j_Shh = 0.1 a. u./s. D-F, n = 10 per condition.

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To test this, we analysed the temporal profiles of Shh gradient formation. Similar to the FP size, which continuously increases over time, the Shh amplitude also increases (Fig 6B). Furthermore, we observed two clear time scales of Shh amplitude change. Up to ~10 h after simulation onset, the Shh amplitude increased rapidly. This was followed by a second phase, in which the increase was slower (Fig 6B). To test whether the increase in Shh amplitude over time is due to Shh^FP, we compared the temporal dynamics in the default case to a situation in which the floor plate does not produce Shh (κ_F→S = 0). We found that for κ_F→S = 0, the Shh amplitude did not increase, but continuously declined and reached negligibly small values (< 10⁻⁴ a.u.) after 10 h (Fig 6C). This supports the conclusion that the increase in Shh amplitude over time depends on Shh produced by the FP.

Dependence of the Shh amplitude on Shh^FP implies that the Shh amplitude should depend on the tissue growth rate in a similar manner to the FP. To test this prediction, we analysed the Shh amplitude and decay length over time and at different tissue growth rates. In the first 10 h, the Shh decay length exhibited transient dynamics corresponding to a shift from the initial Shh present in the system to the decay length predicted by the fixed Shh diffusion and degradation rate = 23.45 μm (S6C Fig). After 10 h, the decay length remained approximately constant for a wide range of growth rates (S6C Fig). In contrast to the decay length, the Shh amplitude increased with increasing growth rates (Fig 6B). Furthermore, the temporal changes in amplitude depended on the growth rate. Crucially, in the absence of growth, the amplitude reached saturation levels in less than 10 h (Fig 6B), indicating that growth is essential for the Shh amplitude to increase after this time point. For low and intermediate growth rates, the Shh amplitude increased linearly over time between 10 h and 60 h (Fig 6B), as well as with respect to the growing tissue length L (S7A Fig), indicating that the amplitude of Shh scales with the tissue size after 10 h.

This suggests that tissue growth leads to an expansion of the FP size, which in turn leads to an increase in the overall Shh production, the net Shh flux into the target tissue and ultimately to an increase in the Shh amplitude over time (Fig 6B). The size of the morphogen source, however, is expected to be linearly related to net morphogen flux through the source boundary and to the gradient amplitude only within a certain range. At large source sizes, newly produced molecules are degraded before they spread to the source boundary, hence further increase in the source size will not increase the flux [40,41]. Consistent with this, our simulations indicate that at an unrealistically high growth rate of k_p = 50 μm/h where the absolute FP size rapidly becomes large, the amplitude of Shh saturates over time (Fig 6B). Similarly, in the complete set of successful solutions in the screen, we found that the Shh amplitude is on average linearly related to FP size for small FP sizes (< ~60 μm), and saturates for large FP sizes (S7B Fig).

Altogether, our analysis suggests that the floor plate and the Shh it produces have an essential contribution to the Shh amplitude dynamics. Nevertheless, it is possible that Shh derived from the notochord also contributes to the Shh amplitude. To investigate this, we first asked how the Shh gradient amplitude behaves in a model where Shh is continuously supplied by the notochord (j_Shh) but is not produced in the FP. In this case, the Shh amplitude was linearly related to j_Shh, A_Shh~j_Shh, as expected from a model of gradient formation by diffusion, degradation and localized production [39], and was lower compared to the condition with Shh^FP production (S7C Fig). This supports the conclusion that the presence of the FP amplifies the Shh amplitude. We then analysed the converse scenario, in which Shh is produced in the FP (κ_F→S > 0) but not in the notochord. To this end, we compared the model with continuous flux from the notochord j_Shh to the conditions with initial burst of Shh or with j_Shh abruptly removed at 10 h, which have no continuous notochord contribution. We found that if the Shh flux is abruptly stopped or Shh is provided only as an initial burst, A_Shh does not reach the same extent as with continuous j_Shh (Fig 6D–6F), however, the difference between the conditions depends on the magnitude of the flux. In a regime with very high flux (1 a. u./s, 10-fold higher than the default condition), the flux has a strong contribution to the overall Shh production until the end of the simulation (53% ± 4% of A_Shh, mean ± SEM, Fig 6E), while for smaller flux values–the effect of continuous flux on the Shh amplitude was minor (11% ± 9%, mean ± SEM, Fig 6F). Thus, our analysis indicates that besides the floor plate, continuous flux from the notochord can potentially contribute to increasing the Shh gradient amplitude over time.

Experimental validation of the model

Our model revealed that FP formation can in principle occur via different mechanisms, depending on whether FP-derived Shh contributes to FP formation. To determine which mechanism is relevant to the in vivo situation, we deleted Shh production from the floor plate, while leaving production in the notochord intact. To this end, we used embryos homozygous for a Shh^Flox allele and carrying one copy of the tamoxifen-inducible Sox2^CreERT2 [31] (Methods), which is expressed in floor plate cells, but not in the notochord. Endogenous Shh expression in the mouse floor plate is initiated around E9 [42], hence we injected mothers with tamoxifen at E7.5 of development to eliminate any Shh production in the floor plate (Fig 7A). In this condition, at E10.5 of development, embryos had a significant reduction of approximately 94% in their Shh gradient amplitude, while the floor plate size, assessed by Arx immunostaining, was unchanged compared to their control littermates (Fig 7B–7D). This indicates that FP-derived Shh is not required for the formation of the floor plate. These results are consistent with prior Shh^FP deletion experiments, in which no effect was observed on FoxA2 expression [14], as well as with observations that the FP becomes refractory to Shh signaling from E9.5 onwards [5]. Taken together, these results support a Shh^FP insensitive mechanism of FP formation in vivo.

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Fig 7. FP size is independent of FP-derived Shh, while the Shh gradient is maintained by the FP.

A. Schematic illustrating the experimental set-up for the tissue and time-specific deletion of Shh in the notochord and FP. Noto = notochord. The time (embryonic day) of tamoxifen injection is indicated with yellow lines and the resulting Shh protein expression with red bars. The genotypes in each condition are listed in parentheses. B. Representative mouse brachial sections of E10.5 control littermates and Sox2^CreERT2/+, Shh^Flox/Flox embryos injected with 3mg tamoxifen at E7.5 (condition i). Immunostaining as indicated. Scale bar = 30 μm. C. Quantification of the Arx-positive area in the experiment in B. Two-tailed t-test: not significant. D. Quantification of the Shh FI profile in the receiving tissue in B. Mean profiles normalized to the maximum FI in the control condition with 95% CI (shaded) are shown. Number of sections: n = 49 (control), n = 40 (mutant). E. Representative images of E10.5 control littermates and Shh^CreERT2/Flox embryos injected with 3mg tamoxifen at E5.5 and E6.5 (condition ii). Immunostaining as indicated. F. Quantification of the Arx-positive area in the experiment in E. Two-tailed t-test: P ≤ 0.0001. G. Quantification of the Shh gradient in E, as in D. Number of sections: n = 52 (control), n = 81 (mutant). H. Representative images of E10.5 control littermates and Shh^CreERT2/Flox embryos injected with 3mg tamoxifen at E8.5 (condition iii). I. Quantification of the Arx-positive area in the experiment in H. Two-tailed t-test: not significant. J. Quantification of the Shh gradient in H, as in D. Number of sections: n = 26 (control), n = 36 (mutant). Error bars in C, F, I; SEM.

https://doi.org/10.1371/journal.pcbi.1012508.g007

The insensitive class of networks is characterised by properties that can be experimentally tested. The first key prediction of the model is that Shh input is required for FP production only at early developmental times, but is dispensable later on. To test this prediction, we used the Shh^CreERT2 mouse line [32], in which a tamoxifen-inducible Cre is knocked into the Shh locus, creating a Shh null allele (Methods). We crossed Shh^CreERT2/+ mice to Shh^Flox mice [33] to generate Shh^CreERT2/Flox embryos in which all Shh production from both the FP and notochord is deleted upon tamoxifen injection (Fig 7A). We found that early deletion of Shh by injection at E5.5 and E6.5 resulted in a severe reduction of 65% of FP size compared to control wildtype littermates (Fig 7E–7G). By contrast, in embryos with late deletion by injection at E8.5, the FP size was not altered compared to controls (Fig 7H–7J). These results demonstrate that Shh input is dispensable for FP formation after E8.5 and confirms the model prediction.

The second prediction of the model is that the floor plate size depends on the initial levels of Shh (Fig 5A). To test this, we took advantage of experimental conditions in which the production of Shh from the notochord and floor plate was affected to varying extents from the onset of neural tube formation. One such condition is Shh heterozygous embryos, which have only one functional copy of the Shh gene–we found that in these embryos the Shh levels in the notochord (at E8.5) and the gradient amplitude (at E10.5) were reduced compared to wildtype controls (Figs 8A, 8B and S8A, S8B). To reduce Shh production even further, we generated Shh^CreERT2/Flox embryos induced with a low dose of tamoxifen. In this condition, only a fraction of notochord and floor plate cells contain one functional Shh copy, whereas the other cells do not express any Shh. This reduces the mean Shh amplitude to levels between the heterozygous mutants and the high dose-injected mutants (Fig 8A and 8B). Together, the set of the wildtype, heterozygous and homozygous conditional mutants had a broad range of Shh amplitudes. By plotting the Shh amplitude and corresponding FP size in individual tissue sections across these conditions, we found that floor plate size was non-linearly related to the Shh amplitude at E10.5 (Fig 8C), reminiscent of the logarithmic dependence of l_FP on S_init and j_Shh (Fig 5A and 5B, respectively). Strikingly, simulations of the relationship between the final amplitude and final floor plate size for conditions with decreased Shh production (α_S and j_Shh) was in excellent agreement with the experimental data (Fig 8C). This result supports the validity of the model and shows that the floor plate is dependent on the levels of Shh production from the earliest developmental stages. Furthermore, this analysis shows that the relationship between amplitude and FP size in embryos with different levels of Shh production is non-linear: FP size sharply increases with A_Shh at low production values and is less sensitive to changes in the amplitude at high values. This non-linear relationship provides an explanation for the relatively unperturbed floor plate formation that is observed in Shh heterozygous embryos (Fig 8A and 8D).

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Fig 8. FP size depends non-linearly on changes in Shh production.

A. Representative mouse brachial sections at E10.5 from control, Shh^CreERT2/+ heterozygous knock-ins, Shh^CreERT2/Fl injected with 2mg tamoxifen at E6.5 (low injection), and Shh^CreERT2/Fl injected with 3mg tamoxifen at E5.5 and E6.5 (high injection). The data for the control and high injection conditions are the same as in Fig 7E and are repeated here to facilitate comparison. Immunostaining as indicated. Scale bar = 30 μm. B. Mean Shh FI profiles in the receiving tissue with 95% CI (shaded) for the conditions shown in A, normalized to the maximum FI in the control condition. Number of sections: n = 83 (control), n = 29 (Shh^CreERT2/+), n = 26 (low injection), n = 81 (high injection). C. FP size and Shh amplitude measured from individual sections at E10.5 for the indicated conditions with different levels of Shh production. Orange line and points show simulation data at t = 60 h for insensitive solutions resulting in large FPs (l_FP/L_end = 20%) for different values of j_Shh as indicated, and α_S, where α_S was varied in proportion to j_Shh. α_S = 0.1 h^-1 (default) for j_Shh = 0.1 a. u./s. Error bars, SEM. D. Quantification of the Arx-positive area in the experiment in A. Error bars, SEM. One-way ANOVA test: * 0.05 ≥ P > 0.01; **** P ≤ 0.0001.

https://doi.org/10.1371/journal.pcbi.1012508.g008

A third prediction of the model arises from the short (~1h) half-life of Shh, which has been experimentally measured [25] and is implemented in the model, which implies that the Shh gradient is continuously and rapidly turned over. This implies that Shh production is continuously required to increase the Shh gradient amplitude over time. To test this, we compared the Shh gradient shape in the Shh^CreERT2/Flox conditional mutants, in which all Shh production was eliminated at early or late time points (Fig 7A). As expected, deletion of Shh at both time points resulted in a severe reduction (by 91% and 88%, respectively) in the Shh gradient amplitude (Fig 7G and 7J). Crucially, this result confirms that new production of Shh is continuously required to maintain the gradient amplitude.

The model results further indicated that continuous flux of Shh from the notochord can contribute to the Shh gradient amplitude only if the magnitude of flux is high relative to the amount of Shh produced by the FP (Fig 6E and 6F). Our experimental data indicate that the effect of Shh deletion from both the notochord and FP with Shh^CreERT2 (Fig 7G) compared to the deletion from FP alone with Sox2^CreERT2 (Fig 7D) have a similar effect on the Shh gradient amplitude. This indicates that in the embryo, after the initial rapid FP establishment phase, Shh flux from the notochord does not have a major contribution to the Shh gradient profile. Instead, our results suggest that once the floor plate is formed, it becomes the main source of the Shh gradient observed in the neural tube.

Overall, the experimental results support a mechanism of FP formation that is independent of FP-derived Shh. Our results demonstrate that FP formation requires only initial Shh input from the notochord and is not sensitive to loss of Shh signalling at later stages. By contrast, FP-derived Shh is essential for the increasing dynamics of the Shh gradient amplitude.