Understanding the principles underlying the design of robust, yet flexible patterning systems is a key problem in developmental biology. In the Drosophila wing, Hedgehog (Hh) signaling determines patterning outputs using dynamical properties of the Hh gradient. In particular, the pattern of collier (col) is established by the steady-state Hh gradient, whereas the pattern of decapentaplegic (dpp), is established by a transient gradient of Hh known as the Hh overshoot. Here, we use mathematical modeling to suggest that this dynamical interpretation of the Hh gradient results in specific robustness and precision properties. For instance, the location of the anterior border of col, which is subject to self-enhanced ligand degradation is more robustly specified than that of dpp to changes in morphogen dosage, and we provide experimental evidence of this prediction. However, the anterior border of dpp expression pattern, which is established by the overshoot gradient is much more precise to what would be expected by the steady-state gradient. Therefore, the dynamical interpretation of Hh signaling offers tradeoffs between robustness and precision to establish tunable patterning properties in a target-specific manner.
morphogen gradientHedgehog signalingrobustnessprecisionpatterningDrosophila wing discResearch organismD. melanogasterhttp://dx.doi.org/10.13039/501100003069Centre for Research and Advanced Studies of the National Polytechnic Institute, MexicoInstitutional SupportNahmadMarcosConsejo Nacional de Humanidades, Ciencias y TecnologíasGraduate FellowshipReyesRosalíohttp://dx.doi.org/10.13039/100000002National Institutes of HealthGM076516LanderArthur DThe funders had no role in study design, data collection, and interpretation, or the decision to submit the work for publication.Author impact statementTheoretical and experimental evidence that Hh signaling dynamics balances robust positioning and sharpness of different target genes in the Drosophila wing disc.Introduction
Developmental patterning must be robust to variety of genetic and environmental perturbations in order to ensure a reproducible and functional body plan. Since patterns of gene expression are often specified by morphogen gradients, there has been considerable interest in understanding how these gradients reliably establish positional boundaries (Neumann and Cohen, 1997; Gurdon and Bourillot, 2001; Lander, 2007; Claret et al., 2007; Ibañes and Izpisúa Belmonte, 2008; Rogers and Schier, 2011; Li et al., 2018; Stapornwongkul et al., 2018). This reliability depends on the robustness of pattern specification with respect to different perturbations, as well as the precision or sharpness of pattern boundaries. Several theoretical studies have investigated the properties in which patterning robustness is ensured (Eldar et al., 2003; Bergmann et al., 2007; Lander et al., 2009; Adelmann et al., 2023). These studies are generally based solely on steady-state morphogen profiles and therefore, robustness applies equally to all patterning targets. As a result, steady-state morphogen gradients cannot tune these patterning properties in a target-specific manner. The Drosophila wing imaginal disc has become a useful system to study the mechanisms of morphogen formation and interpretation and offers testable patterning outputs in terms of both robustness and precision in the adult wing (Hartl and Scott, 2014; Restrepo et al., 2014; Chen and Zou, 2019). Along the anterior–posterior (AP) axis, the Drosophila wing is patterned by the Hedgehog (Hh) and Decapentaplegic (Dpp) morphogen gradients that determine the position of the longitudinal veins L2–L5 (Blair, 2007). Hh is produced in cells of the posterior compartment during the third larval instar and forms a short-range signaling gradient into the anterior compartment (Tabata and Kornberg, 1994). The Hh gradient organizes AP patterning of the wing both directly and indirectly; it defines adult patterning outcomes, such as the expression of the transcription factor knot or collier (col) which sets the distance between the longitudinal veins L3 and L4 (Vervoort et al., 1999; Anonymous, 2000); and the expression of decapentaplegic (dpp) in a domain broader than col (Basler and Struhl, 1994; Vervoort, 2000). While dpp does not have a direct patterning output in the adult wing, Dpp then acts as a long-range morphogen to globally coordinate patterning and growth along the AP axis (Affolter and Basler, 2007).
Contrary to other signaling pathways in which a ligand activates a signaling cascade by binding to its receptor, Hh signaling is activated by removing the receptor Patched (Ptc) from the plasma membrane, a process that is promoted by Hh binding and endocytosis (Torroja et al., 2005). This suggests that Hh signaling activity solely depends on the number of unbound Ptc receptors. However, a study suggested that the levels of Hh-bound Ptc can titrate the inhibitory effects of unbound Ptc and proposed that Hh signaling activity is more accurately represented by the ratio of bound to unbound Ptc receptor (Casali and Struhl, 2004). Importantly, an evolutionary conserved feature of the Hh signaling pathway is that ptc is itself a target of the signal. Since Ptc expression attenuates the dispersion and strength of signaling activity, Hh-dependent Ptc upregulation acts as a negative feedback that self-limits the range of the gradient (Chen and Struhl, 1996; Briscoe et al., 2001). This feedback property of Hh signaling results in self-enhanced ligand degradation which makes a narrower, but more robust gradient to perturbations in ligand dosage (Eldar et al., 2003; Lander et al., 2009).
Hh-dependent Ptc upregulation also provides an alternative interpretation of positional information, in which instead of using multiple concentration thresholds of the Hh steady state as in the classical morphogen model, patterning is established by interpreting positional information in a temporal manner using a single-threshold signaling range defined by a transient and the steady-state gradients (Nahmad and Stathopoulos, 2009). In particular, the boundary of dpp is established by an extended pre-steady-state gradient, known as the overshoot, while the anterior border of col is established by the steady-state gradient. Since the overshoot occurs prior to Hh-dependent Ptc upregulation, dpp should not exhibit the robustness property offered by the self-enhanced ligand degradation mechanism, but this has not yet been documented experimentally.
A study by Irons et al. compared the width of col expression in the wing disc as well as the L3–L4 intervein distance in adult wings of hh heterozygous and wild-type animals and found that they are not statistically different, supporting that some robustness to Hh dosage is exhibited by the system (Irons et al., 2010). Furthermore, Hatori et al. showed that the widths of col or ptc patterns do not significantly change in discs with 1, 2, 3, or 4 hh gene copies (Hatori et al., 2021). However, it remains unclear if the same robustness is exhibited by dpp which depends on the dynamics of the Hh gradient. By using mathematical modeling, here we show that when patterns are established by steady-state models of patterning all target genes exhibit the same robustness with respect to changes in morphogen production, in agreement with prior theoretical work (Eldar et al., 2003). However, when the Hh gradient is interpreted dynamically through the overshoot model (Nahmad and Stathopoulos, 2009), robustness to hh dosage becomes target specific. In particular, the specification of the anterior border of col is more robust than that of dpp, since the latter is independent of Hh-dependent Ptc upregulation. In contrast, we show that the anterior border of dpp model under the overshoot model offers increased precision, relative to what would be expected in the steady state only patterning model. Taken together, our work shows that the overshoot model of Hh signaling enables tunable robustness and precision properties in a target-specific manner. We discuss implications of this dynamic patterning model in the context of balancing reliability and flexibility during developmental patterning.
ResultsSteady-state interpretation of morphogen gradients predicts identical robustness to morphogen dosage for all targets
Prior work on morphogen robustness has relied on quantifying displacements of the overall gradient shape (Gurdon and Bourillot, 2001; Tabata and Takei, 2004) or a single-threshold location of a gradient (Eldar et al., 2003). Robustness can be measured by computing the displacement (Δx) of the pattern boundary defined by a given morphogen threshold concentration, T, as result of a specific perturbation:Δx=|x(T)−x~(T)|,
where x(T) and x~(T) are the positions defined by the concentration threshold T of the unperturbed and perturbed morphogen gradients, respectively. Since Equation 1 is an absolute measure of robustness, in practice, perfect robustness occurs when Δx is less than the diameter of a single cell.
To investigate robustness of different target genes, we first analyze robustness predicted by classical morphogen models, that is, in which territories are defined by different thresholds of the steady-state gradient. As a starting model, we consider a free-diffusion, linear-degradation model at the steady state:d2Mdx2−1λ2M=0,
where M is the concentration of the morphogen and λ2 is the square of the characteristic gradient length, defined by the ratio between the diffusion coefficient and the degradation rate of the ligand M, subject to the following boundary conditions:B.C.1.M(0)=M0,B.C.2.limx→∞M(x)=0.
In this case, a perturbation in the morphogen source, M0→M~0, results in a uniform displacement of the gradient which is given by Δx=λln(M~0/M0) (Eldar et al., 2003), showing that patterns established by different thresholds exhibit the same response to this perturbation. This occurs because the solution of the perturbed problem is just a constant shift of the morphogen profile (Figure 1b–d).
Dynamical interpretation model predicts differential robustness when morphogen dosage is reduced to half.
(a) Simple model of Hh signaling using a time-dependent step-wise degradation function. Diagrams displays a pre steady-state gradient that then retracts upon Hh-dependent ptc upregulation, resulting in a narrower gradient. (b, c) Plots of the analytical solution for the model in a using full (Hh(x=0)=1); (b, b’) or half (Hh(x=0)=0.5); (c, c’) Hh dosage. (d, d’) Displacements upon the above perturbation for the steady-state model with two thresholds (dotted horizontal lines corresponding to the locations of col and dpp) d; and for the dynamical interpretation model with a single-threshold readout (single dotted horizontal line) using the overshoot vs. the steady-state gradient predicts different shifts d’. The parameter values used for these plots are: λover=21μm, λSS=12μm which approximately correspond to the anterior border positions of col and dpp, respectively. The color coding of dpp in red and col in green, will be used in the rest of the article.
Code to generate <xref ref-type="fig" rid="fig1">Figure 1</xref>.
We then considered a very simple model of Hh signaling in the Drosophila wing. Since the expression of Ptc, the Hh receptor, is upregulated by Hh signaling and contributes to Hh degradation by binding the Hh ligand, we considered a model in which ligand degradation has different values within and beyond a presumptive Ptc expression domain:∂Hh∂t=D∂2Hh∂x2+θ(x)αHh−β(x,t)Hh
where θ(x)αHh represents the source of Hh in the posterior compartment of the wing disc (i.e., θ(x) is equal to 1 or 0, depending on whether x is a location in the posterior [x<0] or anterior compartment [x>0], respectively), andβ(x,t)=γHh_PtcPtc+βHh,
where γHh_Ptc is the mass action constant for Hh_Ptc binding. At the steady state, we expect that Ptc forms a uniform expression pattern over a stripe of anterior cells abutting the AP border (referred as Ptcss) and away from the stripe, Ptc is expressed at basal levels, Ptc0. Then, at the steady state βsteady-state(x) can be modeled as the step functionβsteady-state(x)={γHh_PtcPtcss+βHh,0<x<b,γHh_PtcPtc0+βHh,otherwise,
where b is the width of the Ptc stripe. For x>0, the steady-state solution of Equation 4 is given byHh(x)={Hhstripe(x)=Aex/λ2+Be−x/λ2,0<x<bHhbeyondPtc(x)=Ce−x/λ1,x≥b.
where λ1 and λ2 are the morphogens characteristic lengths within and beyond the Ptc stripe, and A, B, and C are constants determined by the boundary conditions. Upon a perturbation αHh→α~Hh, perturbed Hh concentrations are given by:Hh~stripe(x)=Ae[x+1λ2ln⟮α~HhαHh⟯]/λ2+Be−[x−1λ2ln⟮α~HhαHh⟯]/λ2,
andHh~beyondPtc(x)=Ce−[x−1λ1ln⟮α~HhαHh⟯]/λ1.
Note that once again, all territories defined by HhbeyondPtc are shifted by the same amount, λ1ln(α~HhαHh), upon variations in αHh. Therefore, any two target genes whose borders are defined by different concentration thresholds will exhibit the same robustness response.
Dynamic models of Hh signaling using a single threshold for different targets predict differential robustness
Previous work showed that Hh signaling in the Drosophila wing disc the anterior border of the Hh targets dpp and col are established by a single threshold at two time points during the formation of the Hh gradient; namely, at the overshoot and the steady state, respectively (Nahmad and Stathopoulos, 2009). To consider this dynamical patterning mechanism, we analyzed a simplified model which takes into accountx the temporal upregulation of ptc as a time-dependent switch function (Figure 1a). Following the overshoot model in Nahmad and Stathopoulos, 2009, we defined the overshoot gradient as the transient profile of maximum range. Since the timescale of Hh diffusion is much faster than the timescale of Ptc upregulation, we will assume that the Hh gradient reaches a pre-steady state with the first degradation rate, βearly, where the anterior border of dpp is approximately defined and then the real steady state with the second degradation rate, βlate (Figure 1b’). Under this simple model of Hh signaling, the shift in patterning borders defined by the overshoot (i.e., dpp) and the displacement at the steady state are related by the following simple equation:ΔxSS=λSSλoverΔxover,
where λover and λSS are the morphogen characteristic lengths before and after ptc upregulation, respectively (see Figure 1a). Since βearly<βlate, then λSS<λover and ΔxSS<Δxover, that is, overshoot-dependent targets are less robust than those established by the steady-state gradient. Then, in contrast to the steady-state model (Figure 1b–d), the overshoot model predicts differences in target gene displacement upon perturbation of morphogen dosages (Figure 1b’–d’), that is, robustness is target dependent, with higher robustness predicted for col patterning due to self-enhanced ligand degradation, than for dpp patterning (Figure 1d, d’). The ratio λSS/λover in Equation 10 may be written in terms of the kinetic parameters of Hh signaling (see Equation 6):ΔxSS=βoverβSSΔover=γHh_PtcPtcSS+βHhγHh_PtcPtcover+βHhΔxover≈PtcSSPtcoverΔxover.
The last approximation, which assumes that Ptc-dependent Hh degradation is much faster than other means of Hh degradation, provides an estimate of the difference in robustess for overshoot and steady-state targets as a function of Ptc levels. Note that in Equation 11, the difference in Δx between the steady state and overshoot model is independent of the specific threshold at which the Hh gradient establishes positional information. Thus, this equation provides a way to experimentally relate pattern robustness to actual patterning outputs in the system, such as Ptc expression levels (see Discusion).
We then asked if these results also hold in a more explicit model of the Hh pathway (Nahmad and Stathopoulos, 2009):∂Hh∂t=D∂2Hh∂x2+S+(x)αHh−γHh_PtcHh×Ptc−βHhHh,∂ptc∂t=S−(x)αptc0+αptcSignalmkptcm+Signalm−βptcptc,∂Ptc∂t=μPtcptc−γHh_PtcHh×Ptc−βPtcPtc,∂Hh_Ptc∂t=γHh_PtcHh×Ptc−βHh_PtcHh_Ptc,∂Signal∂t=S−(x)αSignal(Hh_PtcPtc)nkSignaln+(Hh_PtcPtc)n−βSignalSignal,
where Hh, ptc, Ptc, and Hh_Ptc are the concentrations of Hh, ptc (mRNA), Ptc (protein), and the Hh-Ptc complex, respectively. The coefficients α, β, γ, and μ represent the rates of synthesis, degradation, complex formation, and translation, respectively (see Figure 2-source data 6). We used a system of coordinates centered on the AP boundary with the anterior compartment on the negative side. S+(x) [alternatively, S−(x)] is a step function of the form S+(x)=1 if x>0 (alternatively, S−(x)=1 if x<0) and zero otherwise. Signal represents the intracellular response of Hh signaling activity that activates target gene expression. The system of Equations 12–16 is subject to the following boundary and initial conditions:I.C. 1ptc(x,0)=S−(x)αptc0βptc,I.C. 2Ptc(x,0)=μPtcβPtcptc(x,0)=S−(x)αptc0μPtcβptcβPtc,B. C.∂Hh∂x|x=−100=∂Hh∂x|x=100=0.
We solved Equations 12–16 numerically and computed Δx (as in Equation 1) for the overshoot and steady-state Signal gradients upon a range of perturbations of the wild-type Hh production rate, αHh0 (Figure 2a). In agreement with our previous result (Figure 1), we found that the steady-state outputs are more robust than the overshoot outputs (Figure 2a). Moreover, this result holds independently of the specific choice of model parameters (Figure 2b). We conclude that higher robustness is predicted for targets specified by the steady-state gradient (col), with respect to those specified by the overshoot profile (dpp).
Target-specific robustness still holds in an explicit model of Hh signaling and it is dependent on Hh-dependent Ptc upregulation.
(a) Δx (defined as in Equation 1, but for the Signal function, see Materials and Methods) for overshoot (red) vs. steady-state (green) outputs upon different perturbations in αHh using the values of the parameters reported in Nahmad and Stathopoulos, 2009 (Figure 2—source code 2 and 3 and Figure 2—source data 1, 2, and 6). (a’) Δx defined and color coded as in a, for different combinations of parameter runs, when all parameters (other than αHh) are varied through a random normal distribution around the mean value with a standard deviation of 10% of the mean value (Figure 2—source data 2). (b) Same as a, but for perturbations in αptc (Figure 2—source data 3). (b’) Comparison of Δx for different parameters runs as in a’ for steady-state outputs (light green dots) and when αptc=0 (dark green empty circles; Figure 2—source data 4). (c) Δx defined as in a, computed for the Signal gradient over time (Figure 2—source data 5). Red and green vertical lines indicate the overshoot and steady-state values corresponding to the anterior borders of dpp and col, respectively (Figure 2—source code 1).
Code to generate <xref ref-type="fig" rid="fig2">Figure 2</xref>.
Code to solve steady-state solution of <xref ref-type="disp-formula" rid="equ18">Equation 18</xref>.
Code to solve transient solution of <xref ref-type="disp-formula" rid="equ12 equ13 equ14 equ15 equ16 equ17">Equations 12–17</xref>.
Raw data to generate <xref ref-type="fig" rid="fig2">Figure 2a</xref>.
Raw data to generate <xref ref-type="fig" rid="fig2">Figure 2a’</xref>.
Raw data to generate <xref ref-type="fig" rid="fig2">Figure 2b</xref>.
Raw data to generate <xref ref-type="fig" rid="fig2">Figure 2b’</xref>.
Raw data to generate <xref ref-type="fig" rid="fig2">Figure 2c</xref>.
Parameters used to solve <xref ref-type="disp-formula" rid="equ12 equ13 equ14 equ15 equ16 equ17">Equations 12–17</xref> (same values as in <xref ref-type="bibr" rid="bib28">Nahmad and Stathopoulos, 2009</xref>).
Robustness of steady-state outputs depends on Hh-dependent Ptc regulation
Since previous work suggests that Hh-dependent ptc upregulation determines the range of the signal (Chen and Struhl, 1996), we wanted to confirm that Hh-dependent ptc regulation is responsible for the difference in robustness of Hh outputs. We perturbed the ptc production rate, αptc, and noticed that Δx computed using the steady-state Signal profile is clearly reduced, but has little effect when computed with the overshoot Signal function (green vs. red dots in Figure 2b). Once again, this result is largely independent of the choice of parameters since robustness always improves compared to the case when αptc=0 (Figure 2b’). Therefore, we suggest that Hh-dependent Ptc upregulation is responsible for differential robustness in this system by making steady-state outputs more robust with respect to overshoot-defined outputs.
Prior theoretical work suggests that when positional information is established before the steady state, it enhances robustness (Bergmann et al., 2007). This idea appears to contradicts our finding that overshoot-dependent patterning (which occurs prior to steady state) is less robust than steady-state-dependent patterning (Figure 2a, b). In order to understand the relative robustness of pre-steady-state gradients, we computed Δx, upon perturbations of αHh as a function of time in our model of Hh signaling. We found that early transient states exhibit the smallest Δx and therefore are the gradients that drive the more robust outputs, although they have a very limited range (Figure 2c), in agreement with the study of Bergmann et al., 2007. Then, Δx increases as the gradient approaches the overshoot when it reaches a maximum, before it starts to decrease again toward the steady state (Figure 2c).
<italic>col</italic> expression is more robust than <italic>dpp</italic> expression in the <italic>Drosophila</italic> wing disc
We then proceeded to test experimentally whether Hh targets are diferentially robust to changes in Hh dosage as predicted by the overshoot model. Previous studies showed that the width of the col domain is largely unaffected in hh heterozygous wing discs (Irons et al., 2010; Hatori et al., 2021). To investigate if this robustness property also holds for dpp, which is established by the overshoot (Nahmad and Stathopoulos, 2009), we examined the patterns of col (using a Col antibody) and dpp (using a dpplacZ reporter) in discs carrying 1 or 2 copies of hh (referred as hh(+/−) and hh(+/+), respectively). We found that the width of the Col pattern in hh(+/−) mutant discs is reduced by 1.66 μm relative to hh(+/+) wild-type discs (Figure 3a, b, e). Although this difference is statistically significant, it is less than the average diameter of a single cell (about 2.5 μm) and therefore, it confirms previous experimental findings (Irons et al., 2010; Hatori et al., 2021). However, the pattern of dppLacZ is reduced by 4.44 μm in hh(+/−) discs relative to hh(+/+) controls (Figure 3c–e). This result does not depend on the size of the wing disc, since the pouch area in both, hh(+/−) and hh(+/+) discs are approximately the same (Figure 3—figure supplement 1), nor on the threshold used to measure the width of the patterns (see Figure 3—figure supplement 2). We conclude that, in agreement with the overshoot model of Hh signaling, but not with any of the steady-state models, the pattern width of Col is more robust than the pattern width of anterior dppLacZ.
Differential robustness of Hh targets to <italic>hh</italic> dosage.
(a-d) Representative third-instar wild-type, hh(+/+) (a, c), and hh heterozygous hh(+/−) (b, d) wing discs immunostained with Col (a, b) and β-galactosidase (c, d) antibodies. Both hh(+/+) and hh(+/−) flies carry a transgene with a dppLacZ enhancer trap, so β-galactosidase marks the pattern of dpp expression. The scale bars in a, a’ apply to b, b’; c, c’; and d, d’ panels, respectively. (a’-d’) Enlarged areas of the white boxes shown in (a-d). (e) Widths of the col and dppLacZ patterns (color coded as in a–d) measured in the region marked by the white rectangle (see Figure 3—source data 1 and Figure 3—source code 1). The brackets on the right represent the difference between the medians of both groups. A non-parametric Mann–Whitney U test was applied in both cases (Figure 3—source data 1). Statistical p-values are 3.0×10−4 for Col (**) and 6.0×10−3 for dppLacZ (***). hh(+/−) discs (n = 14). hh(+/+) discs (n = 23). See Figure 3—source code 1.
Code to generate <xref ref-type="fig" rid="fig3">Figure 3</xref>.
Raw data represented in <xref ref-type="fig" rid="fig3">Figure 3e</xref> .
Raw data represented in <xref ref-type="fig" rid="fig3s1">Figure 3—figure supplement 1</xref>.
The wing disc pouch area does not change in mutant discs.
The pouch area was calculated as the area enclosed by the hinge-pouch folds. Statistical p-value after a Student t-test.
Differences in the width of col and <italic>dpp</italic>LacZ patterns in <italic>hh</italic>(+/+) discs at different threshold values.
Width of the gene patterns measured at different threshold concentrations measured as in Figure 3e (color coding is as in Figure 3). Numbers at the top are the differences between medians of both groups. **** indicates that p-values after a non-parametric Mann-Whitney U test are less than 0.00005.
The overshoot model predicts higher precision in the establishment of the <italic>dpp</italic> border than would be expected from the classical steady-state model
Our findings that the width of dpp is less robust than the width of col in agreement with the overshoot model is puzzling. Why would Hh patterning uses a dynamic mechanism that patterns dpp at the time of least robustness (Figure 2c)? Why would col and dpp have different robustness properties (Figures 1d, 2a, and 3)? We wondered if this dynamical model trades off one patterning advantage over another in a target-specific manner. Morphogen concentrations are naturally noisy, which may cause territories to have a diffuse border especially when the morphogen narrowly declines due to self-dependent ligand degradation (Lander et al., 2009). In particular, we noticed that if dpp had to be specified by the steady-state gradient subject to Ptc-dependent degradation, instead that with the overshoot gradient, it would have to be specified at a location where the Hh gradient is nearly flat (Figure 4a). But at this same location, the Hh gradient is not as flat (Figure 4b). Therefore, we predicted that the overshoot model would establish a more precise dpp anterior boundary compared to a steady-state model, suggesting that the dynamic interpretation of Hh signaling would trade off robustness for precision. Therefore, we analyzed the performance of the overshoot and steady-state models at specifying the sharpness of a pattern boundary. We defined a measure of precision, σx, for an experimental or simulated pattern boundary as the standard deviation of different measurements along the extension of the pattern (Figure 4c). Evidently, perfect precision occurs for σx=0, when the pattern would be completely sharp. In contrast, as σx increases, the less precise the pattern boundary is.
The overshoot model predicts more precision of the anterior border of <italic>dpp</italic> than the steady-state model.
(a, b). Representation of the steady-state (a) and overshoot (b) Hh gradients. At the location of the dpp anterior border, the slope of the gradient is steeper for the overshoot gradient than for the steady-state gradient. (c) Schematic representation of how we define our measure of precision for a patterning border (both in experimental and in simulated patterns). First, a box defines the region of interest (ROI) in the pattern. Then, this ROI is subdivided in n boxes, each of which define a position xi. The measure of precision is the standard deviation of all the xi values. (d, e) Representative Col (d) and dppLacZ (e) patterns in which the xi for each ROI as defined in c is measured and marked with an asterisk along the anterior border. (f) Quantification of σx in several experimental (exp) and simulated (sim) patterns of col (green) and dpp (red). In the simulated patterns, noise levels are adjusted so that the distributions of col are not statistically significant and these noise levels are used to computed the simulated σx of dpp as determined by the steady state (ss) or overshot (over) models. exp sample sizes as in Figure 3. sim sample sizes is n = 50 in all cases. For the statistical comparison, a Mann–Whitney U tests were applied in all cases. Statistical p-value for col was p=0.42. For experimental vs. overshoot dpp: p=1.0×10−3 (**), and for experimental vs. simulated steady-state dpp: p=9.0×10−3 (**).
Code to generate <xref ref-type="fig" rid="fig4">Figure 4</xref>.
Raw data represented in <xref ref-type="fig" rid="fig4">Figure 4f</xref>.
We first measured σx at the anterior border of col and dpp in hh(+/+) wing discs reported in Figure 3. We found that col is about twice more precise than dpp (Figure 4d–f). Then, we compared the precision of the anterior border in simulated patterns of col and dpp (as defined both by the overshoot and steady-state gradients). To do so, we introduced Gaussian noise in the threshold T at which the Signal function establishes a patterning position (see Materials and methods). Since the mechanism that sets the anterior border of the col pattern is the same in both the overshoot and steady-state interpretations, we fitted the extent of noise in the threshold T such that the precision of the simulated border of col is the same as the one we measured in the experimental pattern (σx = 1.23 μm). At this extent of noise in T, we compared the simulated border of dpp defined by the overshoot (dppover) and steady-state models (dppSS). We found that under the overshoot model, the anterior border of dpp is predicted to be more precise than under the steady-state model (Figure 4f). Indeed, the overshoot model predicts a sharper border to what is observed experimentally, but this is not biologically significant since σx is less than one cell diameter in both cases. However, the mean of σx for the simulated dpp border under the steady-state model is 4.36 μm, suggesting that if the anterior border of dpp was established by a steady-state gradient, it would have an imprecision of approximately two cell diameters, which could have some patterning impact in the adult wing (see Discussion).
Discussion
The robust architecture of body plans to genetic and environmental perturbations is a general feature of developmental systems (Waddington, 1942; Csete and Doyle, 2002; Kitano, 2004). At the same time, this robust design should also admit some flexibility in order to allow the system to evolve and adapt under certain genetic or environmental challenges (Barkai and Shilo, 2007). While much work has been dedicated to the understanding of network features that confer robustness in developmental patterning, it is unclear how a robust, yet flexible architecture could be encoded in the interpretation of morphogen gradients (Lander et al., 2009; Lo et al., 2015). In particular, despite much prior theoretical work, the ability of a single morphogen to produce different patterning outputs with target-specific properties has not been studied in detail.
Relative to the classical view of morphogen interpretation, in which different morphogen concentration thresholds at the steady state define different borders of gene expression patterns, two strategies have been proposed to increase robustness to changes in the rates of morphogen production. First, morphogen gradients that promote their own degradation and sharply decay near the source of ligand production (Eldar et al., 2003). And second, gradients that specify patterns prior to steady state (Bergmann et al., 2007). When implementing either of these strategies, increased robustness is achieved for all gene expression patterns, regardless of the concentration thresholds at which they are established. However, both of these strategies have a clear inconvenience; they significantly narrow the patterning domain, and therefore, morphogen readout occurs where the gradient is essentially flat (Adelmann et al., 2023). Thus, these strategies provide robustness at the expense of a narrower gradient which may result in an imprecise border of gene expression. In agreement with this idea, Adelmann et al., 2023 recently showed that a linearly decaying gradient establishes more precise patterning boundaries with respect to a gradient established by a self-enhanced ligand degradation mechanism when interpreted several cells away from the morphogen source. The dynamic interpretation of Hh patterning in the Drosophila wing disc (Nahmad and Stathopoulos, 2009) offers a mechanistic implementation of this idea. First, a linearly decaying Hh gradient (the overshoot gradient) establishes the anterior border of dpp prior to upregulation of the Hh receptor, Ptc; once Ptc is upregulated, self-enhanced ligand degradation narrows the gradient and the anterior border of col is established (Figure 1). Under this model, the col border exhibits higher robustness than the dpp border to hh dosage (Figure 2), and our experimental data supports this prediction (Figure 3). This reduced robustness of dpp patterning occurs as a trade off for increased precision, relative to what would be expected by the steady-state interpretation model (Figure 4). Therefore, the dynamical interpretation of Hh signaling offers a target-specific, robust-yet-flexible architecture of patterning in this system (Figure 5).
The dynamical interpretation of the Hh gradient trades off robustness for higher precision in a target-specific manner.
In the steady-state interpretation, all the target genes are established with the same robustness (Δx) upon perturbations in the amount of ligand. In the overshoot model interpretation one of the target genes (red) is established with less robustness than the other (green). However, it allows the less robust gene to be defined with greater precision than the steady state would define it (compare the sharpness of the boundaries of these patterns).
The finding that the displacement of the anterior borders of Hh targets is more than twice for dpp than for col (≈2.65, from their median values; Figure 3e) provides a interesting prediction about the overshoot gradient. From Equation 11, it can be inferred that the overshoot occurs when Ptc expressions is about twice its basal levels in the anterior compartment, but estimates suggest that Ptc reaches about seven times its basal levels in Ptc domain (Casali and Struhl, 2004). This suggests that the overshoot occurs significantly earlier than Ptc reaches its steady-state levels and that Ptc is produced at much larger amounts than what actually is needed to control the range of the Hh gradient. But since unbound Ptc represses Hh signaling, perhaps the purpose of building very high levels of Ptc is to desensitize Hh signaling over time as has been proposed for the vertebrate neural tube (Dessaud et al., 2008).
Why does this patterning system is wired to ensure robustness for the col border, but favors precision over robustness for dpp? In the Drosophila wing, the expression of col defines directly a specific feature in the adult wing, the L3–L4 intervein area (Vervoort et al., 1999), which corresponds to the more central area of the wing, whereas the dpp pattern does not have a direct positional role in the adult wing, but it acts as the source of another morphogen. As suggested by prior theoretical work, the source where a morphogen is produced does not have a significant impact on patterning (Mizutani et al., 2006), so the robustness of the dpp pattern may not subject to strong selection pressure during evolution, or perhaps other mechanisms downstream of Hh signaling exist to provide robustness at the level of Dpp signaling (Aguilar-Hidalgo et al., 2018; Romanova-Michaelides et al., 2022). In contrast, in the adult wing of Drosophila, precision could have a direct role on the sharpness of vein patterning. Thus, robustness ensures the correct positioning of veins whereas precision may be related to ensure straight veins. While it is unclear if a more imprecise dpp pattern would impact the straightness of veins 2 and 5 which are positioned by Dpp signaling, it suggests that in general, the overshoot model ensures robust positioning close to the morphogen source, but prioritize straightness of stripe-like patterns over positioning in more distant locations. Given that Ptc-dependent Hh degradation is evolutionary conserved (Chen and Struhl, 1996), our findings could have implications for robust and precise patterning in other systems as well.
Customized source codes (available from this paper)
Fly stocks and crosses
Fly crosses were conducted at 25°C. For experiments using one copy of hh [hh(+/-)] (Figure 3), ry[506],hh[AC]/TM3,Sb[1] flies (Bloomington Drosophila Stock Center, BDSC, #1749) were crossed to cn[1],dpp10638/CyO (BDSC # 12379) flies at 25°C to obtain cn[1],dpp[10638]/+; ry[506],hh[AC]/ry[506] discs. hh[AC] is a lost of function hh allele and dpp10638 is a transgene containing a LacZ reporter that drives nuclear β-galactosidase in the location of the dpp gene. Control discs with two copies of hh [hh(+/+)] are obtained from crossing the dppLacZ reporter stock to wild-type flies.
Wing imaginal disc dissection and immunostaining
Wing imaginal discs were dissected from third-instar larvae. Third-instar larvae were dissected under a stereoscopic microscope and fixed in PEM-T (PEM with 0.1% of Triton X-100) with 4% paraformaldehyde, washed three times, and blocked in PEM-T with 0.5% of bovine serum albumin for 2 hr at room temperature. Then, samples were stained with primary antibodies at 4°C overnight at the following dilutions: monoclonal mouse anti-Col (a gift from M. Crozatier, 1:250), rabbit anti-β-gal (MP Biomedicals, Cat. # 55976, 1:250). Primary antibodies were detected with Alexa Fluor 488 anti-mouse and Alexa Fluor 555 anti-rabbit secondary antibodies (1:1000). Imaging was done in a Leica TC5 SP8 confocal microscope using a 40× oil-immersion objective.
Numerical simulations
For computations in Figure 2, a Forward-in-Time-Centered-in-Space (FTCS) algorithm (using space and time steps of 1μm and time steps of 0.5s, respectively) was implemented to solve Equations 12–16 in Python, using the parameters reported by Nahmad and Stathopoulos, 2009. At the steady state, the equations can be reduced to a single equation in each compartment (Nahmad and Stathopoulos, 2009):Dd2HhSSdx2+S+(x)αHh−χS−(x)HhSSγHh_PtcHh+βPtc[αptc0+αptcHhSSnmηm(kn+HhSSnm)+S−(x)HhSSnm]−βHhHhSS=0,
The steady-state Equation 18 was solved using solve_bvp and solve_ivp from scipy.integrate Python package. Plots were made with matplotlib and seaborn libraries of Python (see Figure 3—source code 1 and Figure 4—source code 1). To compute Δx as defined in Equation 1 in Figure 2, we used 0.2 of the maximum value of the Signal function and numerically solved for corresponding location x.
For simulations of col and dppSS patterns in Figure 4, we considered an exponential decay gradient of Hh, like obtained with the simple model in Figure 1, evaluated on a matrix of 80×50. Patterns were determined by the position defined by the threshold T of 20% of the maximum Signal value (with a Gaussian noise with mean T and standard deviation determined in such a way that noise of simulated col coincides width background distribution noise of experimental col pattern, i.e., 1.23 μm). For dppover, we used numerical solution of Signal overshoot (i.e., the Signal function at the time of maximum range) to fit a Hill function, AHhnkn+Hhn, using the function fit_curve of scipy.optimize. We found A=0.2372, k=0.0483, and n=4.6212. Then, we used the approximation function of Signal overshoot to evaluate an exponential decay gradient of Hh overshoot and made an analysis analogous to what was done at the steady state. We measure the width of the pattern at 0.2 of the profile maximun obtained through a vertical projection of the simulated pattern.
Image analysis
For image analysis, we took the Z projection of the confocal images using ImageJ. 16-bit resolution images were saved in TIF format and then processed to measure the width of the fluorescence patterns using OpenCv library of Python. We normalized the intensity values after dividing them by the maximum intensity value and then we measured the width of each pattern domain at 0.2 of relative intensity (in Figure 3—figure supplement 2 we varied this threshold value from 0.1 to 0.6). Graphs were plotted with matplotlib and seaborn libraries of Python (see Source code for each panel of Figure 2). The same images were used to measure robustness (Figure 3) and precision (Figure 4).
All data generated or analyzed during this study (including the source code) are included in this submission.
Acknowledgements
We thank M Crozatier for kindly providing us with an aliquot of Collier antibody. We also thank Fanis Misirlis, José Antonio Arias, and members of the Nahmad laboratory for useful discussions, and Rafael Rodríguez-Muñoz and José Luis Fernández for technical assistance.
ReferencesAdelmannJAVetterRIberD2023Patterning precision under non-linear morphogen decay and molecular noiseeLife12e8475710.7554/eLife.8475737102505AffolterMBaslerK2007The Decapentaplegic morphogen gradient: from pattern formation to growth regulationNature Reviews. Genetics866367410.1038/nrg216617703237Aguilar-HidalgoDWernerSWartlickOGonzález-GaitánMFriedrichBMJülicherF2018Critical point in self-organized tissue growthPhysical Review Letters12019810210.1103/PhysRevLett.120.19810229799239AnonymousJ2000Activation of knot (kn) specifies the 3–4 intervein region in the Drosophila wingDevelopment127556310.1242/dev.127.1.55BarkaiNShiloBZ2007Variability and robustness in biomolecular systemsMolecular Cell2875576010.1016/j.molcel.2007.11.01318082601BaslerKStruhlG1994Compartment boundaries and the control of Drosophila limb pattern by hedgehog proteinNature36820821410.1038/368208a08145818BergmannSSandlerOSberroHShniderSSchejterEShiloBZBarkaiN2007Pre-steady-state decoding of the Bicoid morphogen gradientPLOS Biology5e4610.1371/journal.pbio.005004617298180BlairSS2007Wing vein patterning in Drosophila and the analysis of intercellular signalingAnnual Review of Cell and Developmental Biology2329331910.1146/annurev.cellbio.23.090506.12360617506700BriscoeJChenYJessellTMStruhlG2001A hedgehog-insensitive form of patched provides evidence for direct long-range morphogen activity of sonic hedgehog in the neural tubeMolecular Cell71279129110.1016/s1097-2765(01)00271-411430830CasaliAStruhlG2004Reading the Hedgehog morphogen gradient by measuring the ratio of bound to unbound Patched proteinNature431768010.1038/nature0283515300262ChenYStruhlG1996Dual roles for patched in sequestering and transducing HedgehogCell8755356310.1016/s0092-8674(00)81374-48898207ChenZZouY2019Anterior-posterior patterning of Drosophila wing discs I: a baseline mathematical modelMathematical Biosciences314132710.1016/j.mbs.2019.05.00131085191ClaretSSanialMPlessisA2007Evidence for a novel feedback loop in the Hedgehog pathway involving Smoothened and FusedCurrent Biology171326133310.1016/j.cub.2007.06.05917658259CseteMEDoyleJC2002Reverse engineering of biological complexityScience2951664166910.1126/science.106998111872830DessaudEMcMahonAPBriscoeJ2008Pattern formation in the vertebrate neural tube: a sonic hedgehog morphogen-regulated transcriptional networkDevelopment1352489250310.1242/dev.00932418621990EldarARosinDShiloBZBarkaiN2003Self-enhanced ligand degradation underlies robustness of morphogen gradientsDevelopmental Cell563564610.1016/s1534-5807(03)00292-214536064GurdonJBBourillotPY2001Morphogen gradient interpretationNature41379780310.1038/3510150011677596HartlTAScottMP2014Wing tips: The wing disc as a platform for studying Hedgehog signalingMethods6819920610.1016/j.ymeth.2014.02.00224556557HatoriRWoodBMOliveira BarbosaGKornbergTB2021Regulated delivery controls Drosophila hedgehog, wingless, and decapentaplegic signalingeLife10e7174410.7554/eLife.7174434292155IbañesMIzpisúa BelmonteJC2008Theoretical and experimental approaches to understand morphogen gradientsMolecular Systems Biology417610.1038/msb.2008.1418364710IronsDJWojcinskiAGliseBMonkNAM2010Robustness of positional specification by the Hedgehog morphogen gradientDevelopmental Biology34218019310.1016/j.ydbio.2010.03.02220363217KitanoH2004Biological robustnessNature Reviews. Genetics582683710.1038/nrg147115520792LanderAD2007Morpheus unbound: reimagining the morphogen gradientCell12824525610.1016/j.cell.2007.01.00417254964LanderADLoW-CNieQWanFYM2009The measure of success: constraints, objectives, and tradeoffs in morphogen-mediated patterningCold Spring Harbor Perspectives in Biology1a00202210.1101/cshperspect.a00202220066078LiPMarksonJSWangSChenSVachharajaniVElowitzMB2018Morphogen gradient reconstitution reveals Hedgehog pathway design principlesScience36054354810.1126/science.aao064529622726LoW-CZhouSWanFY-MLanderADNieQ2015Robust and precise morphogen-mediated patterning: trade-offs, constraints and mechanismsJournal of the Royal Society, Interface122014104110.1098/rsif.2014.104125551154MizutaniCMMeyerNRoelinkHBierE2006Threshold-dependent BMP-mediated repression: a model for a conserved mechanism that patterns the neuroectodermPLOS Biology4e31310.1371/journal.pbio.004031316968133NahmadMStathopoulosA2009Dynamic interpretation of hedgehog signaling in the Drosophila wing discPLOS Biology7e100020210.1371/journal.pbio.100020219787036NeumannCJCohenSM1997Long-range action of Wingless organizes the dorsal-ventral axis of the Drosophila wingDevelopment12487188010.1242/dev.124.4.8719043068RestrepoSZartmanJJBaslerK2014Coordination of patterning and growth by the morphogen DPPCurrent Biology24R245R25510.1016/j.cub.2014.01.05524650915RogersKWSchierAF2011Morphogen gradients: from generation to interpretationAnnual Review of Cell and Developmental Biology2737740710.1146/annurev-cellbio-092910-15414821801015Romanova-MichaelidesMHadjivasiliouZAguilar-HidalgoDBasagiannisDSeumCDuboisMJülicherFGonzalez-GaitanM2022Morphogen gradient scaling by recycling of intracellular DppNature60228729310.1038/s41586-021-04346-w34937053StapornwongkulKSSalbreuxGVincentJP2018Developmental biology: morphogen in a dishCurrent Biology28R755R75710.1016/j.cub.2018.05.04729990461TabataTKornbergTB1994Hedgehog is a signaling protein with a key role in patterning Drosophila imaginal discsCell768910210.1016/0092-8674(94)90175-98287482TabataTTakeiY2004Morphogens, their identification and regulationDevelopment13170371210.1242/dev.0104314757636TorrojaCGorfinkielNGuerreroI2005Mechanisms of Hedgehog gradient formation and interpretationJournal of Neurobiology6433435610.1002/neu.2016816041759VervoortMCrozatierMValleDVincentA1999The COE transcription factor Collier is a mediator of short-range Hedgehog-induced patterning of the Drosophila wingCurrent Biology963263910.1016/s0960-9822(99)80285-110375526VervoortM2000hedgehog and wing development in Drosophila: a morphogen at work?BioEssays: News and Reviews in Molecular, Cellular and Developmental Biology2246046810.1002/(SICI)1521-1878(200005)22:5<460::AID-BIES8>3.0.CO;2-GWaddingtonCH1942Canalization of development and the inheritance of acquired charactersNature15056356510.1038/150563a010.7554/eLife.85755.sa0Editor's evaluationJamesDavid EReviewing Editorhttps://ror.org/0384j8v12University of SydneyAustralia
This study presents a valuable finding on the precision conferred by dynamical interpretation of morphogen gradients. The evidence supporting the claims of the authors is convincing, with compelling theoretical analysis and solid experimental data. The authors have adequately addressed most concerns raised and so the work will be of considerable interest to the developmental biology and developmental systems biology communities.
10.7554/eLife.85755.sa1Decision letterJamesDavid EReviewing Editorhttps://ror.org/0384j8v12University of SydneyAustralia
Our editorial process produces two outputs: i) public reviews designed to be posted alongside the preprint for the benefit of readers; ii) feedback on the manuscript for the authors, including requests for revisions, shown below. We also include an acceptance summary that explains what the editors found interesting or important about the work.
Decision letter after peer review:
Thank you for submitting your article "Dynamic readout of the Hh gradient in the Drosophila wing disc reveals pattern-specific tradeoffs between robustness and precision" for consideration by eLife. Your article has been reviewed by 2 peer reviewers, and the evaluation has been overseen by David James as the Senior and Reviewing Editor.
The reviewers have discussed their reviews with one another, and the Reviewing Editor has drafted this to help you prepare a revised submission.
Essential revisions:
While your manuscript was deemed of interest there were significant shortcomings identified that need to be addressed. Most notably both referees felt the experimental part was less compelling than the modelling part and in fact, one referee indicated that they felt this was at best an incremental advance over previous findings. We would like to provide you with an opportunity to address these serious concerns as both reviewers did see positive aspects of the study. However, it is critical that you address the issue concerning the nature of the advance compared to previous studies before we can proceed.
The reviewer found this study presents at best incremental advances to the field. It doesn't provide substantial progress conceptually or experimentally from Eldar et al., 2003, Adleman et al., 2022 and particularly Nahmad and Stathopoulos, 2009. The experimental data and interpretation appear to lack the rigor needed to challenge the model predictions.
Reviewer #1 (Recommendations for the authors):
The manuscript presents an elegant theoretical analysis of robustness and precision in morphogen ingredients, focusing on hedgehog signaling. I have found the proposal made by the authors interesting and convincing. However, I have found that some parts of the manuscript are not very clear. In addition, I believe the experimental results need to be improved in their presentation and to be broadened in scope if possible. Here below I detail my comments:
1) In the Introduction, paragraph starting at 75 indicates the properties of Hh signaling as if they were disconnected to the features described in the previous paragraph. Please, rewrite it to make all appropriate connections with the previous paragraph.
2) Clarify how robustness is exactly defined. The displacement of the boundary of the pattern upon perturbation of Hh level is used in Figure 1 to say whether a target is more robust. However, the coefficient of robustness is not defined as such displacement. These different definitions should be related and preferably refer to them with different names. In addition, the meaning of m in the definition of the coefficient of robustness is not totally clear to me. A plot depicting it would help. Is m the slope of the non-perturbed gradient at the threshold?
3) The coefficient of robustness used is a different measure of the Robustness introduced by Eldar et al.2003. The latter one considered the displacement upon perturbation relative to the extent of the unperturbed gradient. Why the authors do not use the definition of robustness introduced by Eldar et al? Why the definition of robustness in this manuscript does not take into account whether the gradient spans over a larger or a smaller spatial region? The overshoot gradient produces larger displacements yet it is a gradient spanning a larger domain than the steady-state gradient. I am not sure whether the over-shoot gradient is less robust than the steady gradient if the definition of robustness introduced by Eldar et al. 2003 is used. Please justify and clarify all this.
4) These differences in definitions (point 3) make the comparison of the analysis in Box2 with the results from Eldar et al.2003, described in lines 168-169, awkward. Box 2 analyses exponential gradients. It compares the robustness of two exponential gradients with different spatial characteristic lengths (λ). Based on the definition of the coefficient of robustness of this manuscript, these two exponential gradients have a different robustness. However, if we use the definition of robustness by Eldar et al. 2003, all exponential gradients have the same robustness, R=1, independently of their characteristic length λ. Please clarify.
5) In the text, at the beginning of section 2.3, state more explicitly the concept of precision.
6) Define mathematically how precision is measured. The text refers to Box2 (line 187) but there is no definition of coefficient of precision in that Box (nowhere else either).
7) As far as I understand, precision is related to how fluctuations (noise) on the amount of morphogen impact on the position of the boundary. These fluctuations can be from cell to cell and over time within the same cell. The current manuscript does not model fluctuations or noise. Instead, it uses the slope of the deterministic gradient to define the precision (lines 188-190, using Figure 2A to visualize this idea). The manuscript would benefit from indicating the assumptions behind this claim :
A) It assumes uniform noise, i.e. that noise/fluctuations are independent of the slope of the gradient, in other words, are of the same amplitude at any spatial position. Indeed, what we may expect is not this, since intrinsic noise is proportional to the square root of the number of molecules. Hence, the fluctuations will be larger where the morphogen is in high amounts than where it is in low amounts.
B) It also assumes that the range of Hh concentrations that are not discernible/distinguishable under fluctuations (i.e the widths of the red and green bands in the Hh axis) is independent of the Hh concentration (i.e the width of the red band is located around Hh=0.1 and has the same width as that of the green band which is located at Hh=0.77), and that this range does not change over time (it is the same for the steady and the overshoot gradients).
8) The "Dynamical interpretation" model is used with two (related) different meanings, in my opinion, and this drives confusion. On the one hand, according to Figure 1A',B',C', the Dynamical interpretation model corresponds to a single threshold used by different targets: one uses it in the steady gradient and the other target uses it in the overshoot gradient. On the other hand, in the text, in line 198, the dynamic interpretation is used only to refer to the overshoot gradient. I suggest revising how "dynamical interpretation" is used: whether it applies only to the overshoot gradient and then whether a different name must be used to the whole framework of single-threshold interpretation.
9) The results assume that Dpp and col use the same threshold. This is supported by Nahmad and Stathopoulos 2009. Which threshold value is used? Which value is used for the simulations with different sets of the parameter values?
10) Why Robustness is not analysed for the Signal (x)? I would expect that the target is activated by the Signal and not directly by the morphogen gradient. Hence it is valuable to analyse the robustness in the signal and to add these results. Perhaps Figure 3A-C already compute the magnitudes from the signal profile (and not from the morphogen Hh(x) profile), but it is unclear from the main text and figure caption.
11) In Figure 3 precision is much less analyzed than robustness. I suggest that the type of analysis already done in Figure 3B and C for robustness is also done for precision. These analyses will show whether the conclusions on precision are maintained for different parameter values. By the way, "parameters are varied between 0,5 and 2 of the reported values" means that they are varied between 0,5 and 2 TIMES the reported values? Perhaps is standard but the meaning of the sentence was unclear to me.
12) How the overshoot gradient is identified for the different set of parameters to compute Figure 3B?
13) I suggest computing Figure 4B for the overshoot gradient and therefore show that the trend in Figure 4A is kept for different parameter values.
14) Figures 5-6 should be improved by adding: Scale bars, magnifications of images, and detail at cell resolution to observe the displacements in terms of cell length scales. What is exactly measured should be also depicted: How the width is measured and which width is measured for the blurry boundary of Dpp? Which is the number of samples?
15) The finding that the robustness of Col depends on Ptc regulation supports the results by Eldar et al. 2003 and that Col is a target of the steady gradient. Hence these new experimental results support proposals made in previous papers. In my opinion, this experimental result in this manuscript (section 2.7) is not very relevant since it validates previous proposals but not the new ones from this manuscript.
16) The manuscript indicates that Dpp is less robust but more precise than it would be if it was specified by the steady-state gradient. Since the authors have analysed the case of non-regulated patch, I suggest addressing how Dpp would change when patched is not regulated, and to address it both theoretically, and if possible, experimentally. If Patched is not regulated, then there will not be an overshoot gradient and Dpp should be as robust as col. Is this indeed the theoretical prediction? And experimentally: what is observed? In addition, will precision become worse or better? What is the prediction from the model when patch is not regulated?
Reviewer #2 (Recommendations for the authors):
Figure 5 – elaborate on how exactly the results are consistent with the model predictions? While the Dpp width changes more, the width is also larger to begin with- taking into account these rather small changes, can a much simpler model with noise explain the experimental results already (does one have to resort to overshoot and dynamic interpretation?)
Width panels: individual data points should be shown, with "n" defined in the legends
10.7554/eLife.85755.sa2Author response
Essential revisions:
While your manuscript was deemed of interest there were significant shortcomings identified that need to be addressed. Most notably both referees felt the experimental part was less compelling than the modelling part and in fact, one referee indicated that they felt this was at best an incremental advance over previous findings. We would like to provide you with an opportunity to address these serious concerns as both reviewers did see positive aspects of the study. However, it is critical that you address the issue concerning the nature of the advance compared to previous studies before we can proceed.
The reviewer found this study presents at best incremental advances to the field. It doesn't provide substantial progress conceptually or experimentally from Eldar et al., 2003, Adleman et al., 2022 and particularly Nahmad and Stathopoulos, 2009. The experimental data and interpretation appear to lack the rigor needed to challenge the model predictions.
We truly appreciate the summary and the valuable criticisms that the reviewers raised about of our work. We think that the reviewers’ comments have made the current, major-revised manuscript a much better paper, so we very grateful for their feedback.
First of all, we would like to admit that in the original manuscript our definitions of precision and robustness were confusing and we agree that there is not a consensus about these concepts in the literature. In the revised version of the manuscript, we have independently defined the positional buffering effects as a result of hh dosage, as robustness, and sharpness of the anterior borders of patterns, as precision. We also clarified our image analysis of the col and dpp patterns in agreement with the reviewers’ suggestions. When we used our new definitions to compute robustness and precision on both experimental and simulated patterns, we confirmed the findings of our original manuscript, namely, that the anterior border of col that depends on the steady-state Hh gradient subject to self-enhanced degradation gradient is more robust than that of dpp, which is set by the overshoot gradient. Nonetheless, the lack of robustness of the dpp anterior border with the dynamic gradient is compensated with more precision than would be expected from the steady-state gradient interpretation at this position. In our revised manuscript, we provide experimental evidence for differential robustness to hh dosage and use simulations to support our precision hypothesis. Taken together, our work stands out from previous work such as Eldar et al. 2003, Adelmann et al. 2023, and Nahmad and Stathopoulos 2009, to show for the first time that a dynamic interpretation through the overshoot model modulates positioning and sharpness in a target-specific manner during Hh patterning in the Drosophila wing disc.
Reviewer #1 (Recommendations for the authors):
The manuscript presents an elegant theoretical analysis of robustness and precision in morphogen ingredients, focusing on hedgehog signaling. I have found the proposal made by the authors interesting and convincing. However, I have found that some parts of the manuscript are not very clear. In addition, I believe the experimental results need to be improved in their presentation and to be broadened in scope if possible. Here below I detail my comments:
1) In the Introduction, paragraph starting at 75 indicates the properties of Hh signaling as if they were disconnected to the features described in the previous paragraph. Please, rewrite it to make all appropriate connections with the previous paragraph.
Thank you for this comment. We have re-written the introduction to integrate these paragraphs.
2) Clarify how robustness is exactly defined. The displacement of the boundary of the pattern upon perturbation of Hh level is used in Figure 1 to say whether a target is more robust. However, the coefficient of robustness is not defined as such displacement. These different definitions should be related and preferably refer to them with different names. In addition, the meaning of m in the definition of the coefficient of robustness is not totally clear to me. A plot depicting it would help. Is m the slope of the non-perturbed gradient at the threshold?
We appreciate this criticism. We agree that the use of the coefficient of robustness was not necessary and could lead to unnecessary confusion. Therefore, in the revised manuscript, we completely removed the use of the robustness coefficient and instead, we use the very intuitive notion of pattern displacement as a measure of robustness (equation 1 in the revised manuscript). Note that for simple models of Hh signaling such as the one now depicted in Figure 1, the displacement resulting from the steady state and the overshoot gradients can be compared and our claim that steady-state outputs are more robust than overshoot outputs can be directly demostrated (equation 2 in the revised manuscript).
3) The coefficient of robustness used is a different measure of the Robustness introduced by Eldar et al.2003. The latter one considered the displacement upon perturbation relative to the extent of the unperturbed gradient. Why the authors do not use the definition of robustness introduced by Eldar et al? Why the definition of robustness in this manuscript does not take into account whether the gradient spans over a larger or a smaller spatial region? The overshoot gradient produces larger displacements yet it is a gradient spanning a larger domain than the steady-state gradient. I am not sure whether the over-shoot gradient is less robust than the steady gradient if the definition of robustness introduced by Eldar et al. 2003 is used. Please justify and clarify all this.
As we mentioned in the previous point, we no longer use a definition of robustness coefficient, as this is not necessary to address our hypothesis. While the definition of robustness coefficient introduced by Eldar et al. and others allows generalizing the notion of robustness, we think that simply using the displacement between a perturbed and unperturbed location established by a morphogen (equation 1) is a more intuitive measure of robustness that allows to directly show that in fact the outputs of the overshoot gradient are less robust than those of the steady-state gradient (equation 10). Indeed, Eldar et al. 2003 also use this displacement to show the differences in robustness between linear and non-linear gradients (see Eldar et al. 2003, equations 2-5).
4) These differences in definitions (point 3) make the comparison of the analysis in Box2 with the results from Eldar et al.2003, described in lines 168-169, awkward. Box 2 analyses exponential gradients. It compares the robustness of two exponential gradients with different spatial characteristic lengths (λ). Based on the definition of the coefficient of robustness of this manuscript, these two exponential gradients have a different robustness. However, if we use the definition of robustness by Eldar et al. 2003, all exponential gradients have the same robustness, R=1, independently of their characteristic length λ. Please clarify.
We agree that the use of boxes was not a very convenient way of presenting the information. In the revised manuscript, we removed all boxes and put all relevant information directly in the text. Exponential gradients that are perturbed at the boundary conditions do display a shift that depends on the characteristic length λ (as shown also by Eldar et al. 2003, equation 2). Using this displacement as a direct measure of robustness, a larger displacement (i.e., less robustness) occurs in the overshoot gradient that has a larger λ.
5) In the text, at the beginning of section 2.3, state more explicitly the concept of precision.
We have introduced a more direct notion of precision in terms of sharpness of a 2D border (Figure 4c). We think that this new notion of precision is very easy to compute directly from experimental or simulated 2D patterns and effectively reflects the sharpness of a border (see last subsection of the Results).
6) Define mathematically how precision is measured. The text refers to Box2 (line 187) but there is no definition of coefficient of precision in that Box (nowhere else either).
Sorry for being unclear about this in the original manuscript. In the revised manuscript, we completely separated the notions of robustness and precision, so that they can be independently evaluated. Similarly as with robustness, we no longer rely on the coefficient of precision. As stated in our previous point, a mathematical definition of precision is provided in the last subsection of the Results and in Figure 4c.
7) As far as I understand, precision is related to how fluctuations (noise) on the amount of morphogen impact on the position of the boundary. These fluctuations can be from cell to cell and over time within the same cell. The current manuscript does not model fluctuations or noise. Instead, it uses the slope of the deterministic gradient to define the precision (lines 188-190, using Figure 2A to visualize this idea). The manuscript would benefit from indicating the assumptions behind this claim :
A) It assumes uniform noise, i.e. that noise/fluctuations are independent of the slope of the gradient, in other words, are of the same amplitude at any spatial position. Indeed, what we may expect is not this, since intrinsic noise is proportional to the square root of the number of molecules. Hence, the fluctuations will be larger where the morphogen is in high amounts than where it is in low amounts.
B) It also assumes that the range of Hh concentrations that are not discernible/distinguishable under fluctuations (i.e the widths of the red and green bands in the Hh axis) is independent of the Hh concentration (i.e the width of the red band is located around Hh=0.1 and has the same width as that of the green band which is located at Hh=0.77), and that this range does not change over time (it is the same for the steady and the overshoot gradients).
We appreciate the reviewer pointed this out as it helped us to redefine our notion of precision. It is true that precision is ill-defined in the literature and therefore people refer to precision in different ways, so it is really important to set a clear definition of precision. In her/his comment, the reviewer refers to precision as a change in a patterning position due to fluctuations in morphogen amounts. This notion, applied to a 2D border results in local changes in position along the pattern (Figure 4c), so that effectively it is a measure of sharpness of the patterning border. We no longer need to clarify which of the assumptions referred by the reviewer hold because we are no longer defining precision in terms of the slope of a deterministic gradient. Thanks to this reviewer’s comment, we approached the notion of precision in a different way than in the original manuscript. Namely, we generated simulated data by directly introducing Gaussian noise to the morphogen threshold at which cell fate takes place; we then fixed these noise levels so that simulated data fits the experimental data for the anterior border of col (which was not the focus of the precision analysis), and finally, we tested our hypothesis for the anterior border of the dpp pattern (see Materials of Methods in the revised manuscript).
8) The "Dynamical interpretation" model is used with two (related) different meanings, in my opinion, and this drives confusion. On the one hand, according to Figure 1A',B',C', the Dynamical interpretation model corresponds to a single threshold used by different targets: one uses it in the steady gradient and the other target uses it in the overshoot gradient. On the other hand, in the text, in line 198, the dynamic interpretation is used only to refer to the overshoot gradient. I suggest revising how "dynamical interpretation" is used: whether it applies only to the overshoot gradient and then whether a different name must be used to the whole framework of single-threshold interpretation.
We understand and apologize for the confusion. In response to this comment, in the revised manuscript we use the term overshoot model every time we refer to the interpretation of Hh signaling (both with the overshoot and steady-state gradient), and we only use dynamical interpretation when referring more generally to the use of dynamic properties of a morphogen gradient.
9) The results assume that Dpp and col use the same threshold. This is supported by Nahmad and Stathopoulos 2009. Which threshold value is used? Which value is used for the simulations with different sets of the parameter values?
We always use 0.2 of the maximum value as the threshold value not only for the simulated data, but also as a threshold to determine pattern boundaries (see Materials and methods’ sections 4.3 and 4.4). It is important to state, however, that our main robustness result that steady-state outputs are more robust than overshoot outputs is independent of the threshold used.
10) Why Robustness is not analysed for the Signal (x)? I would expect that the target is activated by the Signal and not directly by the morphogen gradient. Hence it is valuable to analyse the robustness in the signal and to add these results. Perhaps Figure 3A-C already compute the magnitudes from the signal profile (and not from the morphogen Hh(x) profile), but it is unclear from the main text and figure caption.
We did use Signal when computed x in Figure 2 of the revised manuscript. Thank you for the observation. We are now stating this in the main text as well as in the legend of Figure 2.
11) In Figure 3 precision is much less analyzed than robustness. I suggest that the type of analysis already done in Figure 3B and C for robustness is also done for precision. These analyses will show whether the conclusions on precision are maintained for different parameter values. By the way, "parameters are varied between 0,5 and 2 of the reported values" means that they are varied between 0,5 and 2 TIMES the reported values? Perhaps is standard but the meaning of the sentence was unclear to me.
In the revised manuscript, robustness and precision are analyzed independently, so I don’t think that the first part of the comment applies. Please keep in mind, that our analysis of robustness is limited to changes in hh dosage, but for precision, we simply use Gaussian noise in the interpretation of the gradient. Thus, the analysis for robustness presented in Figure 2 of the revised manuscript does not apply to precision and the analysis of precision presented in Figure 4 of the revised manuscript does not apply to robustness. Regarding the second part of the comment; yes, it refers to 0.5 to 2 TIMES and this is now clearly specified in Figure 2a,b.
12) How the overshoot gradient is identified for the different set of parameters to compute Figure 3B?
The overshoot is defined in the main text as ‘the transient gradient of maximum range’. In our simulations of Figure 2 in the revised manuscript, this is how overshoot displacements are computed.
13) I suggest computing Figure 4B for the overshoot gradient and therefore show that the trend in Figure 4A is kept for different parameter values.
Done. This is shown in new Figure 2b’.
14) Figures 5-6 should be improved by adding: Scale bars, magnifications of images, and detail at cell resolution to observe the displacements in terms of cell length scales. What is exactly measured should be also depicted: How the width is measured and which width is measured for the blurry boundary of Dpp? Which is the number of samples?
Thank you for the suggestions. In the new manuscript, we have updated new Figure 3 (which corresponds to Figure 5 in the original manuscript; note that Figure 6 from the original manuscript is no longer in the revision) incorporating these suggestions. Information on the number of samples and how widths are measured is clearly explained in the legend of Figure 3 and in Materials and methods section.
15) The finding that the robustness of Col depends on Ptc regulation supports the results by Eldar et al. 2003 and that Col is a target of the steady gradient. Hence these new experimental results support proposals made in previous papers. In my opinion, this experimental result in this manuscript (section 2.7) is not very relevant since it validates previous proposals but not the new ones from this manuscript.
We agree with the reviewer that Figure 6 did not add anything new to the manuscript and it was difficult to fit in the story. In addition, it was difficult to interpret experimentally due to the effects in changes in temperature (a point that was also raised the other reviewer). Therefore, we decided to remove it from the revised manuscript. This decision of course does not affect the conclusions of the paper and makes it a more concise story. We thank the reviewer for this suggestion.
16) The manuscript indicates that Dpp is less robust but more precise than it would be if it was specified by the steady-state gradient. Since the authors have analysed the case of non-regulated patch, I suggest addressing how Dpp would change when patched is not regulated, and to address it both theoretically, and if possible, experimentally. If Patched is not regulated, then there will not be an overshoot gradient and Dpp should be as robust as col. Is this indeed the theoretical prediction? And experimentally: what is observed? In addition, will precision become worse or better? What is the prediction from the model when patch is not regulated?
As the reviewer rightly pointed out, when ptc is not upregulated, there is no overshoot and therefore, the anterior border of col and dpp overlap. Indeed, Nahmad and Stathopoulos (2009) already showed that this is the case experimentally. The prediction of the model, with no overshoot is that the patterns would be identical (simply, because by definition of overshoot as the gradient with maximum range, the overshoot and the steady-state gradients would be the same) and therefore their robustness and precision would be also identical. However, it is possible to compare, both in the simulations and experimentally, the precision of the anterior borders of col and dpp. In the revised manuscript, we computed the precision of col and dpp experimentally using the same protocol and showed that the anterior border of col is more precise than that of dpp which confirms the observation by eye that the dpp border is more fuzzy than the col border (Figure 3). Moreover, using this measure of precision, we are able to show that the anterior border of dpp in simulations is more precise when obtained it from the overshoot gradient than from the steady-state gradient.
Reviewer #2 (Recommendations for the authors):
Figure 5 – elaborate on how exactly the results are consistent with the model predictions? While the Dpp width changes more, the width is also larger to begin with- taking into account these rather small changes, can a much simpler model with noise explain the experimental results already (does one have to resort to overshoot and dynamic interpretation?)
We did not make measurements relative to the width of each pattern because robustness (as a displacement) and precision (sharpness) should be interpreted in absolute units. In our manuscript, we show that simply adding noise to a steady-state model could not explain differential robustness. We cannot rule out that other models could also explain differential robustness, but we are favoring our interpretation on a model that has been validated by prior experimental work (i.e., the overshoot model; Nahmad and Stathopoulos, 2009).
Width panels: individual data points should be shown, with "n" defined in the legends
We now include individual data points in Figure 3 of the revised manuscript and display the sample numbers ‘n’ in the figure legend.