Protein concentration gradients and switching diffusions

Morphogen gradients play a vital role in developmental biology by enabling embryonic cells to infer their spatial location and determine their developmental fate accordingly. The standard mechanism for generating a morphogen gradient involves a morphogen being produced from a localized source and subsequently degrading. While this mechanism is effective over the length and time scales of tissue development, it fails over typical subcellular length scales due to the rapid dissipation of spatial asymmetries. In a recent theoretical work, we found an alternative mechanism for generating concentration gradients of diffusing molecules, in which the molecules switch between spatially constant diffusivities at switching rates that depend on the spatial location of a molecule. Independently, an experimental and computational study later found that Caenorhabditis elegans zygotes rely on this mechanism for cell polarization. In this paper, we extend our analysis of switching diffusivities to determine its role in protein concentration gradient formation. In particular, we determine how switching diffusivities modifies the standard theory and show how space-dependent switching diffusivities can yield a gradient in the absence of a localized source. Our mathematical analysis yields explicit formulas for the intracellular concentration gradient which closely match the results of previous experiments and numerical simulations.

Figure 1

Schematic illustration of heterogeneous diffusion due to temporal disorder. A Brownian particle randomly switches between two conformational states $n=0$ or 1 having different diffusivities such that $D_0<D_1$. The switching rates $\alpha \left(x\right)$ and $\beta \left(x\right)$ are taken to be dependent on the spatial position $x$, $0\le x\le L$. In this particular example, the relative rate of switching $\beta \left(x\right)/\alpha \left(x\right)$ increases to the right as indicated by the color gradient. Hence, the particle spends more time in the slow-diffusing state at the end $x=0$ and more time in the fast-diffusing state at the end $x=L$.

Figure 2

Comparison of the fast-switching limit to the steady-state solution for various values of $\epsilon$. (a) $\eta =0.1$ and (b) $\eta ={10}^{-3}$. Other parameters are $L=100$ $\mu \mathrm{m}$ and $Q_0=5\times {10}^{-3}$ $\left[C\right]$ $\mu \mathrm{m}$/s.

Figure 3

The fast-switching limit as a function of $\eta$. Parameters are $L=100$ $\mu \mathrm{m}$ and $Q_0=5\times {10}^{-3}$ $\left[C\right]$ $\mu \mathrm{m}$/s.

Figure 4

Plot of numerical solution and first-order asymptotics for MEX-5 and PIE-1 spatial switching rates given by Eqs. (4.11) and (4.12), respectively, with the values of the coefficients listed in the tables. Parameter values are taken from Ref. [12], which were chosen to match their experimental measurements of protein concentrations.

Figure 5

Equilibrium density with switching rates given by $\alpha \left(x\right)=2$ and $\beta \left(x\right)={(x+0.65)}^4$. The diffusion coefficients were set to $D_0=0.5$ and $D_1=5$, with $\epsilon =5\times {10}^{-4}$. The red and black dashed lines show the increased accuracy achieved by the $O\left(\sqrt{\epsilon }\right)$ terms.

Figure 6

(a) MEX-5 concentration over space and time. (b) MEX-5 concentration plotted versus time $t$ at $x=16$ $\mu \mathrm{m}$ showing slight overshooting. Initial state concentrations are $C_0^{*}=0.75$ and $C_1^{*}=0.25$.