Editor's evaluation: Patterning precision under non-linear morphogen decay and molecular noise
Year: 2023
Type: peer-review
Abstract: Full text Figures and data Side by side Abstract Editor's evaluation Introduction Results Discussion Methods Appendix 1 Data availability References Decision letter Author response Article and author information Metrics Abstract Morphogen gradients can instruct cells about their position in a patterned tissue. Non-linear morphogen decay has been suggested to increase gradient precision by reducing the sensitivity to variability in the morphogen source. Here, we use cell-based simulations to quantitatively compare the positional error of gradients for linear and non-linear morphogen decay. While we confirm that non-linear decay reduces the positional error close to the source, the reduction is very small for physiological noise levels. Far from the source, the positional error is much larger for non-linear decay in tissues that pose a flux barrier to the morphogen at the boundary. In light of this new data, a physiological role of morphogen decay dynamics in patterning precision appears unlikely. Editor's evaluation The authors use analytic calculations and numerical simulations to convincingly show that the purported benefits of nonlinear decay in morphogen gradients may be marginal in some cases and completely reversed in others (far from the concentration source). This is a valuable contribution to the field, as it questions common assumptions about the biological function of non-linear morphogen decays during development. https://doi.org/10.7554/eLife.84757.sa0 Decision letter Reviews on Sciety eLife's review process Introduction According to Wolpert’s famous French flag model (Wolpert, 1969), morphogen gradients encode readout positions xθ via concentration thresholds Cθ=C(xθ), and differentiating cells base their fate decisions on whether the local morphogen concentration lies above or below such thresholds (Figure 1A). Thus, these readout positions mark the boundary locations between domains of different cell fates. Variations in the morphogen profile result in variations in the readout positions. The accuracy of the spatial information carried by morphogen gradients can be quantified with the positional error, which is defined as the standard deviation of the readout positions over different gradient realizations (Vetter and Iber, 2022): (1) σx=stddev[xθ]. Figure 1 Download asset Open asset Comparison of linear and non-linear morphogen gradients. (A) According to the French flag model, morphogen gradients provide the spatial information required for tissue patterning via concentration thresholds Cθ, numbered by θ=1,2,3 etc. If a cell lies above or below a certain threshold Cθ, it switches fate, resulting in domain boundaries forming at the respective cell borders at x=xθ (blue and red lines). The morphogen source is located at x=x0=0. (B) Linear decay leads to exponential gradients. Changes in the gradient amplitude C0 (different colours) lead to a shift Δx that is independent of the amplitude. (C) Non-linear decay (n=2) leads to power-law gradients. The shift Δx due to a change of C0 is amplitude-dependent. (D, E) Noisy example gradients obtained numerically. Cell boundaries are denoted by black ticks along the patterning axis. Molecular kinetic noise and cell area variability leads to noisy gradients. For a fixed readout threshold Cθ, variable gradients result in different readout positions xθ,j (inset plots). Non-linear decay (E) leads to shallower gradients further in the patterning domain compared to linear decay (D). How the observed precision of tissue patterns arising from this principle is achieved, in spite of natural molecular noise in morphogen production, transport, decay, internalization, turnover and other sources of variability, is a key question in developmental biology (Lander, 2011; Vetter and Iber, 2022; Iber and Vetter, 2022). Morphogen dynamics are often described by reaction-diffusion equations of the form (Lander et al., 2009). (2) ∂C∂t=DΔC−dCn/Crefn−1 with morphogen concentration C, diffusion coefficient D, and decay rate d. Cref is a constant reference concentration that we introduce to make all units independent of n. The exponent n models linear (n=1) or non-linear (n>1) decay of the morphogen. Linear decay leads to exponential gradient profiles (Figure 1B) of the form (Lander et al., 2002). (3) C(x)=C0e−x/λ,λ=Dd with an amplitude C0 at the source at x=0, and a characteristic decay length λ, set by the diffusion coefficient D and the degradation rate d. Non-linear decay, on the other hand, results in shifted power-law gradients (Figure 1C; Eldar et al., 2003) (4) C(x)=C0(1+xmλm)−m,m=2n−1 where λm is a gradient length scale that depends on D/d, n and C0/Cref (see Appendix 1). Non-linear decay would for instance arise in case of cell lineage transport, when ligands interact with receptor clusters, or if ligand binding results in receptor upregulation, as is the case for several morphogens, most prominently for Hedgehog (Hh) (Eldar et al., 2003; Nahmad and Stathopoulos, 2009; Wartlick et al., 2009; Balaskas et al., 2012). Most reported morphogen gradient profiles have been fitted assuming linear decay (n=1) (Gregor et al., 2007; Gregor et al., 2008; Kicheva et al., 2007; Wartlick et al., 2011; Wartlick et al., 2014; Cohen et al., 2015; Zagorski et al., 2017; Mateus et al., 2020). For the FGF8 gradient in the developing mouse brain, n≈4 has been reported (Chan et al., 2017). Embryos are subject to molecular noise, which can cause fluctuations in morphogen production and transport rates, and consequently, in the gradient amplitudes and morphogen fluxes from the source to the patterned cells. This results in shifts Δx between different gradient realisations (Figure 1B and C). In the case of linear decay, the shift is only related to the relative morphogen influx or amplitude and not to the absolute morphogen levels (Appendix 1, Equation 11). However, with non-linear decay, the shift depends on the absolute levels, with higher influxes resulting in smaller shifts (Appendix 1, Equation 12). Previous research (Eldar et al., 2003) suggested that the circumstance that this shift vanishes for power-law gradients at sufficiently large morphogen influx values leads to more robust patterning, because the readout position becomes independent of the influx in this limit when the morphogen decay is non-linear (Appendix 1). In other words, if two tissues are patterned by two different noise-free power-law gradients, both with high (but different) morphogen influxes from the source, the resulting gradients will be nearly identical, resulting in a reproducible pattern. For exponential gradients, the shift will not disappear, and the gradients will thus differ. However, the gradients that result from non-linear decay also possess significantly shallower tails, relative to the higher concentration (Figure 1C). Their usefulness for patterning has therefore been questioned (Lander et al., 2009), and it has remained unclear whether nonlinearity in the morphogen decay would in fact help achieving higher positional accuracy. Indeed, to first order, the positional error of variable gradients is inversely proportional to the magnitude of their slope (Gregor et al., 2007; Vetter and Iber, 2022) according to (5) σx≈|∂C∂x|−1σC, where σC is the standard deviation of local morphogen concentration and x denotes the patterning axis. This suggests that for patterning precision, the benefit of a smaller positional shift of gradients with n>1 might be offset or even overcompensated by their flatter shape further from the source. In the previous analysis (Eldar et al., 2003), molecular noise was considered only in the form of a fold-change in the morphogen amplitude or influx between different gradient realisations, resulting in shifted deterministic gradients, as shown in Figure 1B and C. To account for the intrinsic stochasticity of biological systems, we now extend this deterministic view by incorporating kinetic variability into the model, as depicted in Figure 1D and E. This is achieved by introducing randomness into all kinetic parameters of the reaction-diffusion equation. Our model is cell-based, meaning that each cell in the tissue is assigned its own specific variable kinetic parameters, emulating inter-cellular variability (Methods). With this quantitative statistical tool, we demonstrate numerically that the positional error of noisy morphogen gradients does not significantly improve with non-linear decay. In the contrary, if the morphogen cannot leave the patterned tissue opposite of the source, the power-law gradients become shallow in a substantial part of the domain, leading to reduced positional accuracy with non-linear decay. Results Noisy gradient model We simulated steady-state diffusion to study the impact of non-linear decay on the precision of noisy morphogen gradients. Our model uses a one-dimensional cellular domain composed of a source of length Ls and a patterning region of length Lp (Figure 1D and E). To represent morphogen-secreting source cells explicitly, the diffusion equation (Equation 2) was extended by a morphogen production term, resulting in (6) 0=D∂2C∂x2−dCn/Crefn−1+pH(−x). Here, H(x) is the Heaviside function, ensuring that production at rate p only occurs in the source (x<0). Zero-flux boundary conditions were used at both outer ends of the tissue, mimicking a situation in which morphogen molecules are restricted to the patterned tissue by an impermeable boundary: ∂C∂x(−Ls)=0=∂C∂x(Lp). We generated variable morphogen gradients by numerically solving Equation 6 with kinetic parameters pi, di and Di, and cell areas Ai independently drawn from log-normal distributions for each cell i=1,…,N in the domain (Vetter and Iber, 2022; Adelmann et al., 2022; Figure 2, for details see Methods). The individual gradient realisations Cj(x) can be thought of as representing different embryos, denoted by the index j. They exhibit inter- and intra-tissue variability due to the stochastic nature of the three kinetic parameters that vary from cell to cell. Cells have to convert the spatial morphogen distribution they are exposed to into a single concentration value, which determines their fate in the tissue according to the French flag model. There are several ways cells may achieve this, such as averaging the morphogen signal over their entire cell surface, beyond their cell surface via a cilium, or reading out the signal at a single point. In a recent study, we found that the readout mechanism has little impact on the gradient precision perceived by the cells (Adelmann et al., 2022). We therefore only analysed the case where cells average the morphogen signal over their cell surface, or over their diameter in the 1D model here, respectively. The concentration in each cell is then compared to the threshold concentration Cθ and the location xθ,j of the first cell whose sensed concentration subceeds this threshold is recorded. This process is repeated for all gradients j, allowing to compute the positional error according to its definition (Equation 1), σx=stddevj{xθ,j} to quantify the precision of the positional information conveyed by the morphogen gradients. Figure 2 Download asset Open asset Numerical model to simulate noisy gradients. A 1D cellular domain is constructed by drawing cell areas from log-normal distributions with mean cell area μA and standard deviation σA. Cell areas are then converted to diameters (δi). This procedure is repeated N times until source and patterning domains of length Ls and Lp are filled with cells. Kinetic parameters k=p,d,D are drawn independently from log-normal distributions with a mean μk and standard deviation σk for each cell. Production only takes place in the source (blue cells). Then, the reaction-diffusion equation (Equation 6) is solved on the cellular domain, generating one noisy gradient Cj(x). To determine a unique readout concentration of a cell, the average concentration along the cell boundary is computed for each cell in the patterning domain. Based on these concentrations the readout position xθ,j where Cj(xθ,j)=Cθ is recorded for each gradient. This step is repeated 1000 times. Lastly, the average readout position μx and the positional error σx is calculated based on the 1000 noisy gradients. PDF denotes the probability density function. Model parameters To define the stochastic nature of the morphogen kinetics involved in the formation of the gradients, we express the mean value and standard deviation of a parameter q by μq and σq, respectively (Figure 2). Based on measurements of the Hedgehog morphogen gradient in the Drosophila wing disc and mouse neural tube (Kicheva et al., 2007; Cohen et al., 2015; Vetter and Iber, 2022), we used a mean diffusivity of μD=0.033 µm2/s and a mean gradient length μλ=20 µm. We furthermore set the average degradation rate to μd=μD/μλ2, and the average production rate to μp=μdCref, where Cref=1 arb. units to normalise the concentrations. Other specific values of gradient parameters would not change the results reported here, which are for the steady state, but would only alter the timescale it takes for the steady state to be reached. The noise-to-signal ratio in each quantity q is given by the corresponding coefficient of variation, CVq=σq/μq. Reported physiological noise levels in morphogen production, decay, and transport differ between morphogens and tissues, but are around CVp,d,D≈0.3 (Vetter and Iber, 2022), which we use to define the distribution widths of the kinetic parameters. In addition to the morphogen kinetics, our simulations also include cell-to-cell variability in the cell areas. The widths and cross-sectional areas of cells vary in all layers along the apical-basal axis (Gómez et al., 2021). Most quantifications have been carried out on the apical surface. One of the highest reported values for the apical area CV is found in the vertebrate neural tube (CVA≈0.9) (Escudero et al., 2011; Guerrero et al., 2019; Bocanegra-Moreno et al., 2022), but most values are considerably lower (Kokic et al., 2019). We therefore used CVA=0.5 in all simulations unless specified otherwise. The diffusion coefficient, D, and the degradation rate, d, set the steady-state patterning length scale, λ=D/d. Thus, our results are independent of the absolute values chosen for D and d, and only depend on their ratio. Positional quantities, such as the positional error, are reported relative to the average cell diameter, which in turn was chosen to be a fixed multiple of the average gradient decay length. We fixed the average cell diameter at a fourth of the exponential gradient length, μδ/μλ=1/4, as found in the developing mouse neural tube (Kicheva et al., 2014; Cohen et al., 2015). Impact of non-linear decay on gradient precision We previously showed that in case of linear decay, there is a negligible impact of cell area variability as long as CVA<1 (Adelmann et al., 2022). We now find that this holds similarly for non-linear decay (Figure 3A), justifying the use of a fixed CVA=0.5 in the remainder of our analysis. Figure 3 Download asset Open asset Impact of non-linear decay on gradient precision. (A) Physiological variability in the cross-sectional cell areas has no significant impact on gradient precision. The positional error σx is plotted in units of the mean cell diameter μδ at different readout positions in the patterning domain (symbols), and for different degrees of non-linearity (colours). (B) The positional error increases with the square root of the cell diameter, irrespective of n. Dotted lines show σx=αμδ for α=2.6,5.2 for reference, with lengths in units of µm. Lp=100μδ. (C) Non-linear decay leads to a marginally lower positional error close to the morphogen source. Inset plot shows σx/μδ at a distance of two cells from the source as a function of decay non-linearity. With a no-flux boundary at x=Lp, the shallowness of gradients from non-linear decay lets the positional error increase strongly far from the source. Colours correspond to different decay exponents n, as specified in panel D. (D) Variability in the production rate alone has no effect on the positional error along the domain for linear decay (blue). The stronger the non-linearity, the smaller the positional error close to the source (inset). Far from the source, the positional error increases rapidly with non-linear decay. (E) Difference between the positional error for n>1 and for n=1 relative to the mean cell diameter, at fixed readout positions (colours). (F) Effect of finite patterning domain size. The positional error increases close to the distant zero-flux boundary in case of non-linear decay (shades of blue, n=2). Patterning remains precise across a larger distance in the case of linear decay (black, n=1). In all panels, each data point represents the mean from 103 independent simulations. Error bars represent standard errors. Much as for linear decay (Adelmann et al., 2022), the positional error scales with the square root of the mean cell diameter also for non-linear decay (Figure 3B). Small cell diameters, as observed in all known tissues that employ gradient-based patterning (Adelmann et al., 2022), are therefore important for high spatial precision also in case of non-linear decay. The positional error increases from less than one cell diameter close to the source to about two cell diameters at a distance of 75 cell diameters away from the source (Figure 3C). Close to the distant domain boundary opposite of the source, where a no-flux condition was imposed, the positional error rapidly increases for non-linear decay, while remaining relatively low for linear decay. If only the production in the source is varied (CVp=0.3, CVd,D=0), the positional error remains constant as the readout distance from the source increases, but increases again sharply close to the distant end in case of non-linear decay (Figure 3D). But even for strong non-linearity (n=4), the positional error remains in the sub-cellular range when only production noise is considered, as long as the readout position is further than about λ away from the distal end. Independent of whether all parameters are varied or only the production rate, the positional error drops in close vicinity to the source with stronger non-linearity in the decay (insets of Figure 3C and D). However, with less than 20% of a single cell diameter from n=1 to n=4, the effect is likely too small to be physiologically relevant. Further away from the source, linear decay yields a smaller positional error than non-linear decay (Figure 3D–E). No matter how long the patterning domain is, non-linearity always increases the positional error as the distal tissue boundary is approached (Figure 3F). What then causes the increased positional errors with non-linear decay near the distal domain boundary? A zero-flux boundary condition there implies shallower gradients than on infinite domains: C′(x)→0 as x→Lp. This effect occurs irrespective of n, but the spatial range over which the gradient flattens (and thus deviates from the pure exponential and shifted power-law forms for infinite domains, Equations 3; 4) increases with n. By virtue of Equation 5, non-linear decay thus leads to greater positional errors at readout positions in the vicinity of the distal boundary compared to linear decay. In summary, our computer simulations of noisy morphogen gradients suggest that it is insufficient to quantify gradient robustness and patterning precision by considering variability in the morphogen production alone. Moreover, if the morphogen cannot exit the patterning domain opposite of the source, shifted power-law gradients that result from non-linear morphogen decay flatten over a significantly larger range than exponential gradients, leading to increased positional errors. The gain in positional accuracy close to the source for non-linear decay is negligible and therefore barely physiologically relevant. Overall, exponential gradients lead to more robust patterning. Impact of boundary condition at the source Given the impact of the distal domain boundary, we whether the of the morphogen source by a spatial production domain (Figure by a flux boundary condition (Figure as used by Eldar et al., or by a fixed gradient amplitude (Figure would the positional error by the model. While there are small quantitative the gradient (Figure and positional errors (Figure are very Figure Download asset Open asset Impact of the boundary condition at the source. gradient when the morphogen is in a source domain at rate p (Equation 6) with flux or were at at the far end of the tissue Positional error as a function of morphogen variability, at different readout positions and degrees of non-linearity (colours). variability in the morphogen production rate influx and gradient amplitude (F) leads to a larger positional error above a certain threshold variability Kinetic variability was fixed at for in D). Further In each data point represents the mean from 103 independent simulations. Error bars represent standard errors. we increase the variability in the production rate via (Figure in the influx from the source via (Figure or in the gradient amplitude at the source boundary via (Figure we find the increase in the positional error for the production rate and the increase for the gradient amplitude. or boundary conditions thus the positional error when the variability in the source is of such boundary a spatial source domain be where With the physiological values and (Vetter and Iber, 2022), variability in the morphogen production a to role in the gradient Molecular noise in morphogen degradation and diffusivity the patterning precision. Impact of the morphogen source the gradient amplitude determines the sensitivity of the readout position to amplitude for non-linear decay (Appendix 1, Equation but not for linear decay (Appendix 1, Equation the average morphogen production rate is to the patterning accuracy in the case of non-linear decay, but not for linear decay. We this to the by the mean relative production rate, the mean influx from the source, and the mean morphogen amplitude in the three different simulated morphogen production Changes in these parameters have no effect on the positional error if morphogen degradation is which is with the (Figure lines). With non-linear decay, on the other hand, we the positional error to be on morphogen previously for deterministic morphogen gradients (Eldar et al., 2003) not to quantitatively to the positional error in where cell-to-cell variability is and where morphogen production remains at physiological levels. Figure Download asset Open asset Impact of the morphogen source obtained spatial patterning accuracy in units of average cell diameters μδ at different positions in the tissue and for different degrees of non-linearity (colours). close to the source, at far from the source, at Morphogen production are to Figure Production in a source domain with morphogen-secreting cells with a morphogen influx from the source at the source boundary and with a specified morphogen concentration at the source boundary low influxes or amplitudes lead to gradients at strong decay non-linearity, the parameter range over which the positional error can be for In all panels, each data point represents the mean from 103 independent error bars represent standard errors. For high morphogen levels, non-linear decay leads to a smaller positional error close to the source (Figure The effect is, less in the model that a spatial morphogen source domain (Figure than in that not (Figure and again the of the With an source domain, non-linear decay yields only marginally more spatial when production is high production levels increase the positional error close to the source in all three several cell diameters, for The gradients flatten out at low production, reducing their usefulness for spatial tissue patterning. The stronger the non-linearity in the the more this of patterning precision. Further away from the source, the benefit of non-linear decay is and exponential gradients more precise than shifted power-law gradients also at high morphogen levels (Figure In summary, models of morphogen-secreting cells the impact of non-linear decay on patterning precision. In all models considered here, the benefit of non-linear morphogen decay is restricted to a close vicinity of the morphogen source, where patterning precision is high (Vetter and Iber, 2022) and may thus not be as for robust and to a of very strong morphogen Further into the tissue, and at morphogen linear decay yields more patterning. Discussion Non-linear morphogen decay was as a mechanism for tissue patterning in the by Eldar et al., in a deterministic where morphogen gradients are of we have this with a stochastic model, noisy gradients into as they arise from cell-to-cell variability in morphogen The of our quantitative analysis is while a small effect near the morphogen source, this gain is by a substantial of precision in the spatial information that gradients provide to cells in the and distal of a patterned tissue when morphogen decay is In tissues that pose a diffusion barrier to the at their boundary, shifted power-law gradients that with flatten out over a substantial of the spatial domain, exponential gradients more (Figure 1). This leads to greater spatial precision with linear decay (Figure and is to the (Eldar et al., This boundary effect is not the only linear morphogen decay is for precise The positional error, which is the quantity that the spatial accuracy with which cells can determine their location in the and their fate in is to morphogen levels when morphogen decay is but when decay is linear (Figure This implies that patterning is more robust to variations in the and of the morphogen-secreting source, if decay is results the view that power-law gradients fluctuations in morphogen production (Eldar et al., We find that the positional error in the opposite production fluctuations only with but not with non-linear decay. an the linear case may be as patterning precision is by in the and kinetics of the morphogen-secreting source only if Our study that a stochastic is required to quantify patterning precision of noisy gradients. Moreover, we find the positional error to be in models that the morphogen-secreting cells by a or boundary condition (Figure Based on this, we to include an of the source in or numerical on the as we with Equation exponential gradients from shifted can be very in as they can over the over which they can be with The FGF8 gradient in the developing mouse is the only case we are of where n>1 has been reported (Chan et al., and whether this is to patterning precision in remains gradient data in other systems, such as Hedgehog and in the neural tube et al., is too variable to the that Most further of morphogen gradient (Gregor et al., 2007; Gregor et al., 2008; Kicheva et al., 2007; Wartlick et al., 2011; Wartlick et al., 2014; Cohen et al., 2015; Zagorski et
Source: eLife
Author: Gordon J Berman
Institution: Emory University
Language: English
Subfield: Molecular Biology
Domain: Life Sciences
Open Access status: gold
Estimated APC paid: $2,500
OpenAlex ID: W4367339646