Morphogen profiles can be optimized to buffer against noise

Morphogen profiles play a vital role in biology by specifying position in embryonic development. However, the factors that influence the shape of a morphogen profile remain poorly understood. Since morphogens should provide precise positional information, one significant factor is the robustness of the profile to noise. We compare three experimentally relevant classes of morphogen profiles (linear, exponential, and algebraic) to see which is most precise when subject to both external embryo-to-embryo fluctuations and internal fluctuations due to intrinsically random processes such as diffusion. We find that both the kinetic parameters and the overall gradient shape (e.g., exponential versus algebraic) can be optimized to generate maximally precise positional information.

Figure 1

Numerical results: (a) ${\rho }_{quad}$ against $x$ averaged for $\tau =5000\text{s}$. Line is fit to Eq. (4). (b) Mean and standard deviation of $f\left(x\right)$ (error bars from ten simulations). (c) Comparing ${\omega }_{quad}$ from simulations with Eq. (7) for $J=1$ (●), $J=2$ (○), and $J=5$ (△) molecules $\mu {\text{m}}^{-2}{\text{s}}^{-1}$, with various $\tau$ and $x_T$. (d) ${\omega }_{quad}$ against nonlinear decay probability averaged for $\tau =1000\text{s}$. Line is prediction from Eq. (7), where $\alpha =P\gamma /\left(1+cP\gamma \right)$ [25] [$\gamma ={\left(\delta x\right)}^3/\delta t$, $c=12.5\mu {\text{m}}^{-3}\text{s}$]. Parameters unless stated otherwise: $J=1\text{molecule}\mu {\text{m}}^{-2}{\text{s}}^{-1}$; $x_T=3\mu \text{m}$; $D=D_0=0.67\mu {\text{m}}^2{\text{s}}^{-1}$; $P={10}^{-2}$; $L=50\mu \text{m}$; lattice spacing $\delta x=0.01\mu \text{m}$; time step $\delta t=2.5\times {10}^{-4}\text{s}$.

Figure 2

(Color online) Optimizing morphogen profiles. (a) ${ϵ}_{lin}$ versus $\lambda$ (solid line) and ${ϵ}_{quad}$ versus $x_0$ (dashed line) with ${\omega }_{lin}$ (○) and ${\omega }_{quad}$ (△). The contributions solely from external fluctuations, which are linear in $\lambda$ and $x_0$, respectively, are omitted for clarity. (b) Effective phase diagram for relative precision of the three models considered with varying internal and external fluctuations. (c) Relative precision of models with varying $x_T$ in presence of internal fluctuations with fixed $\delta J/J=0.1$. Solid and dashed lines in (b) and (c) correspond to ${ϵ}_{lin}={ϵ}_{s\text{−}s}$ (solid line) and ${ϵ}_{lin}={ϵ}_{quad}$ (dashed line) and color map defined in text. (d) Optimal $n$ for different averaging times $\tau$. Parameters unless stated otherwise: $J=2\text{molecules}\mu {\text{m}}^{-2}{\text{s}}^{-1}$, $\delta J/J=0.1$, $\Delta x=3\times {10}^{-3}\mu \text{m}$, $D=15\mu {\text{m}}^2{\text{s}}^{-1}$, $D_0=0.5\mu {\text{m}}^2{\text{s}}^{-1}$, $x_T=240\mu \text{m}$, $\tau =8\text{min}$, $N_{spat}=0.06\tau$, and $L=550\mu \text{m}$.