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Figure 1

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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 $C_{θ}$ , numbered by $θ = 1, 2, 3$ etc. If a cell lies above or below a certain threshold $C_{θ}$ , it switches fate, resulting in domain boundaries forming at the respective cell borders at $x = x_{θ}$ (blue and red lines). The morphogen source is located at $x = x_{0} = 0$ . (B) Linear decay leads to exponential gradients. Changes in the gradient amplitude C₀ (different colours) lead to a shift $Δ x$ that is independent of the amplitude. (C) Non-linear decay ( $n = 2$ ) leads to power-law gradients. The shift $Δ x$ 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 $C_{θ}$ , variable gradients result in different readout positions $x_{θ, j}$ (inset plots). Non-linear decay (E) leads to shallower gradients further in the patterning domain compared to linear decay (D).

Figure 2

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Numerical model to simulate noisy gradients.

A 1D cellular domain is constructed by drawing cell areas from log-normal distributions with mean cell area $μ_{A}$ and standard deviation $σ_{A}$ . Cell areas are then converted to diameters ( $δ_{i}$ ). This procedure is repeated $N$ times until source and patterning domains of length $L_{s}$ and $L_{p}$ are filled with cells. Kinetic parameters $k = p, d, D$ are drawn independently from log-normal distributions with a mean $μ_{k}$ and standard deviation $σ_{k}$ 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 $C_{j} (x)$ . 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 $x_{θ, j}$ where $C_{j} (x_{θ, j}) = C_{θ}$ is recorded for each gradient. This step is repeated 1000 times. Lastly, the average readout position $μ_{x}$ and the positional error $σ_{x}$ is calculated based on the 1000 noisy gradients. PDF denotes the probability density function.

Figure 3

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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 $σ_{x}$ is plotted in units of the mean cell diameter $μ_{δ}$ 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 $σ_{x} = α \sqrt{μ_{δ}}$ for $α = 2.6, 5.2$ for reference, with lengths in units of µm. $L_{p} = 100 μ_{δ}$ . (C) Non-linear decay leads to a marginally lower positional error close to the morphogen source. Inset plot shows $σ_{x} / μ_{δ}$ at a distance of two cells from the source as a function of decay non-linearity. With a no-flux boundary at $x = L_{p}$ , 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.

Figure 4

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Impact of the boundary condition (BC) at the source.

(A–C) Noise-free gradient shapes when the morphogen is either secreted in a source domain at rate $p$ (Equation 6) (A), with flux BC, $- {D \partial C / \partial x |}_{x = 0} = j_{0}$ (B), or Dirichlet BC, $C (0) = C_{0}$ (C). No-flux BC were imposed at at the far end of the tissue (at $x = L_{p}$ ). (D–F) Positional error as a function of morphogen abundance variability, at different readout positions (symbols) and degrees of non-linearity (colours). Greater variability in the morphogen production rate (D), influx (E), and gradient amplitude (F) leads to a larger positional error above a certain threshold variability $C V ⪆ 0.1 - 0.3$ . Kinetic variability was fixed at ${C V}_{d, D} = 0.3$ (except for ${CV}_{p}$ in D). Further parameters: $μ_{j_{0}} = μ_{D} C_{ref} / μ_{λ}$ (E), $μ_{C_{0}} = C_{ref}$ (F). In panels D–F, each data point represents the mean from 10³ independent simulations. Error bars represent standard errors.

Figure 5

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Impact of the morphogen source strength.

Numerically obtained spatial patterning accuracy in units of average cell diameters $μ_{δ}$ at different positions in the tissue (symbols) and for different degrees of non-linearity (colours). (A–C) Readout close to the source, at $x_{θ} = 5 μ_{δ}$ ; (D–F) Readout far from the source, at $x_{θ} = 150 μ_{δ}$ . Morphogen production scenarios are identical to Figure 4: Production in a source domain with morphogen-secreting cells (A,D), with a morphogen influx from the source at the source boundary (B,E), and with a specified morphogen concentration at the source boundary (C,F). Very low (high) influxes or amplitudes lead to flat (steep) gradients at strong decay non-linearity, limiting the parameter range over which the positional error can be reliably determined for $n = 4$ (B,C,E,F). In all panels, each data point represents the mean from 10³ independent simulations, error bars represent standard errors.

Appendix 1—figure 1

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Shift in morphogen gradients due to changes in morphogen production.

(A) Comparison of noise-free gradients arising from linear (blue) and non-linear (green) decay. A fold-change in the influx j₀ from the source shifts the gradients by $Δ x$ . (B) Positional shift of the morphogen gradient as a function of the amplitude and degree of non-linearity, for a fold-change in the amplitude, $C_{0}^{*} / C_{0} = e$ . (C) Positional shift as a function of the influx and degree of non-linearity, for a fold-change in the influx, $j_{0}^{*} / j_{0} = e$ .

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Jan Andreas Adelmann
Roman Vetter
Dagmar Iber

(2023)

Patterning precision under non-linear morphogen decay and molecular noise

eLife 12:e84757.

https://doi.org/10.7554/eLife.84757

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