Morphogen transport in epithelia

We present a general theoretical framework to discuss mechanisms of morphogen transport and gradient formation in a cell layer. Trafficking events on the cellular scale lead to transport on larger scales. We discuss in particular the case of transcytosis where morphogens undergo repeated rounds of internalization into cells and recycling. Based on a description on the cellular scale, we derive effective nonlinear transport equations in one and two dimensions which are valid on larger scales. We derive analytic expressions for the concentration dependence of the effective diffusion coefficient and the effective degradation rate. We discuss the effects of a directional bias on morphogen transport and those of the coupling of the morphogen and receptor kinetics. Furthermore, we discuss general properties of cellular transport processes such as the robustness of gradients and relate our results to recent experiments on the morphogen Decapentaplegic (Dpp) that acts in the wing disk of the fruit fly Drosophila.

Figure 1

Schematic representation of ligand transport by transcytosis in a chain of cells of diameter $a$ indexed by $n$. The rates of ligand-receptor binding and un-binding, internalization and externalization of ligand-receptor pairs are denoted $k_{\text{on}}$, $k_{\text{off}}$, $b_{\mathrm{int}}$, and $b_{\mathrm{ext}}$. Degradation of ligand occurs inside the cells with rate $b_{\mathrm{deg}}$ and in the extracellular space with rate $e_{\mathrm{deg}}$. Ligands can also hop directly between neighboring extracellular spaces at a rate $2D_0∕a^2$ which describes their movement in the extracellular space around the cells by passive diffusion with diffusion coefficient $D_0$ (not shown). Figure modified from [20].

Figure 2

(Color online) Time development of gradient formation in our description of ligand transport. Ligand densities in the presence of a source at $x=0$ at different times $t{b}_{\mathrm{deg}}=0.72,2.16,3.6$ during gradient formation (black lines) and in steady state (red lines). Lines indicate solutions to Eq. (6), while symbols indicate solutions to Eq. (1) for comparison. (a–c) Time development of the profiles of the total ligand density $\lambda (x,t)$ (a), the total receptor density $\rho (x,t)$ (b), and the receptor bound ligand density $s_i(x,t)+s_{+}(x,t)$ (c) in the absence of extracellular diffusion, i.e., for $D_0=0$. (d) Like (a) but with $2D_0∕a^2{b}_{\mathrm{deg}}=10∕3$, i.e., in the presence of extracellular diffusion. All concentrations are normalized to the steady state value of the surface receptor concentration in the absence of ligands $r_0$. Initial conditions at $t=0$: $\lambda \left(x\right)=0$ and $\rho \left(x\right)=(1+f_{\mathrm{int}}∕f_{\mathrm{ext}})r_0$. Parameters are $k_{\text{off}}∕b_{\mathrm{deg}}=b_{\mathrm{int}}∕b_{\mathrm{deg}}=f_{\mathrm{int}}∕b_{\mathrm{deg}}=1000∕3$, $k_{\text{on}}a{r}_{\max }∕b_{\mathrm{deg}}=8000∕3$, $b_{\mathrm{ext}}∕b_{\mathrm{deg}}=f_{\mathrm{ext}}∕b_{\mathrm{deg}}=2000∕3$, $e_{\mathrm{deg}}∕b_{\mathrm{deg}}=2∕3$, $f_{\mathrm{deg}}∕b_{\mathrm{deg}}=1$, $f_{\mathrm{syn}}^0∕a{r}_{\max }{b}_{\mathrm{deg}}=1∕12$, $\psi =2$, $j_0∕b_{\mathrm{deg}}a{r}_{\max }=25∕6$, $r_0∕r_{\max }=1∕7$, and $j=0$ at $x∕a=50$.

Figure 3

Effective transport coefficients and degradation rates for transcytosis. (a,b) Coefficients $D_{\lambda }(\lambda ,\rho )$, $D_{\rho }(\lambda ,\rho )$, and $k_{\lambda }(\lambda ,\rho )$ in the transport equation (6) as a function of the dimensionless ratio $\lambda ∕\rho$ of the total ligand concentration $\lambda$ and the total receptor concentration $\rho$. The solid lines show the coefficients $D_{\lambda }$ and $D_{\rho }$ in presence of extracellular diffusion with $2D_0∕a^2{b}_{\mathrm{deg}}=10∕3$. Broken lines show these coefficients in the absence of extracellular diffusion, i.e., with $D_0=0$. Inset in (a): the solid line shows $k_{\lambda }$ with $e_{\mathrm{deg}}∕b_{\mathrm{deg}}=2∕3$ and the broken line with $e_{\mathrm{deg}}=0$. ${\rho }_0$ denotes the steady state total receptor concentration in the absence of ligands. (c) Effective diffusion coefficient $D\left(\lambda \right)$ in the transport equation for the constant surface receptor approximation (11) as a function of the ligand concentration $\lambda ∕r$ for $2D_0∕a^2{b}_{\mathrm{deg}}=10∕3$ (solid line) and $D_0=0$ (dashed line). Inset: effective degradation rate $k\left(\lambda \right)$ as a function of $\lambda ∕r$ for $e_{\mathrm{deg}}=0$ (dashed line) and $e_{\mathrm{deg}}∕b_{\mathrm{deg}}=2∕3$ (solid line). The constant total surface receptor concentration is denoted $r$. Parameters as in Fig. 2 with $k_{\text{on}}a\rho ∕b_{\mathrm{deg}}=8000∕3$ in (a) and (b) and $k_{\text{on}}ar∕b_{\mathrm{deg}}=8000∕3$ in (c).

Figure 4

(Color online) Ligand transport by transcytosis with a directional bias. (a) Drift velocity $V_{\beta }$ from Eq. (B3) as a function of $\lambda ∕\rho$. (b, c) Time development of gradient formation with directional bias. Profiles of the total ligand concentration $\lambda (x,t)$ (b) and the total receptor concentration $\rho (x,t)$ (c) in the presence of a source at $x=0$ at different times $t{b}_{\mathrm{deg}}=0.72,2.16,3.6$ during gradient formation (black lines) and in steady state (red lines). Lines indicate solutions to Eq. (9), while symbols indicate solutions to Eq. (B1) for comparison. All concentrations are normalized to the steady state value of the surface receptor concentration in the absence of ligands $r_0$. Initial condition: $\lambda \left(x\right)=0$ and $\rho \left(x\right)=(1+f_{\mathrm{int}}∕f_{\mathrm{ext}})r_0$. Parameters as in Fig. 2 with $\beta =0.1$, $D_0=0$, and $j=0$ at $x∕a=100$.

Figure 5

Nonrobust and robust steady state gradients in our description of morphogen transport with constant surface receptor concentration given by Eq. (11). (a) Ligand profiles in the steady state for $D_0=0$ and $j_0∕b_{\mathrm{deg}}R=7$ where robustness is small, $\mathcal{R}\simeq 0.1$. The profile is strongly affected by halving (dotted line) or doubling (dashed line) the ligand current of the reference state (solid line). The positions of an arbitrarily chosen concentration threshold are indicated. (b) Ligand profiles in the steady state for $D_0=0$ and $j_0∕b_{\mathrm{deg}}R=70$ where robustness is large, $\mathcal{R}\simeq 470$. The reference profile (solid line) is almost unaffected by halving (dotted line) or doubling (dashed line, covered by the solid line) the ligand current of the reference state. (c) Like b but with extracellular diffusion $(D_0∕a^2{b}_{\mathrm{deg}}=50)$ which reduces robustness to $\mathcal{R}\simeq 0.32$. The insets in a–c show the respective profiles of the receptor-bound ligand concentration $s_{+}+s_i$ which is the biologically relevant quantity that triggers signal transduction in the cells. (d) Robustness $\mathcal{R}$ of steady state ligand profiles as a function of the ligand current $j_0$ from the source for different values of the ratio of the extracellular diffusion length ${\xi }_d=\sqrt{D_0∕e_{\mathrm{deg}}}$ and the cell size $a$. The description of morphogen transport with constant surface receptor approximation given by Eq. (11) was used to calculate $\mathcal{R}$. Figure modified from [20]. Parameters are $b_{\mathrm{int}}∕b_{\mathrm{deg}}=b_{\mathrm{ext}}∕b_{\mathrm{deg}}=3\times {10}^3$, $k_{\text{on}}ar∕b_{\mathrm{deg}}=1.1\times {10}^4$, $e_{\mathrm{deg}}∕b_{\mathrm{deg}}=5$, and $k_{\text{off}}∕b_{\mathrm{deg}}=7\times {10}^2$.

Figure 6

(Color online) Ligand transport in two dimensions. (a) Tissue in the region of the wing disk where the Dpp gradient forms. Cell membranes are labeled in red, the morphogen Dpp is shown in green. (b) Schematic of this tissue. Source cells which produce ligands are shown in darker gray, the cells in the target tissue in light gray. (c) The rates for the various cellular processes are denoted as in one dimension, see Fig. 1. For the triangular lattice with hexagonal cells which we use here, receptor bound ligands can be present on the six edges of each cell and inside the cells. The concentration of the receptor-bound ligands on edge $j$ of cell $n$ is denoted $S_{n,j}$, that inside cell $n$ is termed $S_n^{\left(\mathrm{i}\right)}$. Free ligands exist in the gaps between two cell surfaces. Their concentration in gap $⟨n,j⟩$ which is located adjacent to edge $j$ of cell $n$ is denoted $L_{⟨n,j⟩}$. (d) Triangular lattice structure with hexagonal cells used in our discrete theoretical description of ligand transport in two dimensions. This lattice structure approximates the situation shown in b. The source cells which are shown in darker gray secrete ligands into the extracellular spaces surrounding them. The cell diameter is $a$.

Figure 7

(Color online) Time development of ligand densities $\lambda (\vec{x},t)$ in two dimensions. The solution to the continuum transport equation (22) is compared to that of the discrete description given by Eq. (21) in the presence of a region which the ligand cannot enter located at $6⩽x∕a⩽11$ and $-4⩽y∕a⩽4$. In the discrete description, this is realized by setting $b_{\mathrm{int}}$ to zero in the region. In the continuum description, zero flux boundary conditions are imposed on the outlining edge of the region. (a–d) Two-dimensional ligand profiles at $t{b}_{\mathrm{deg}}=3.6$ (a,b) and at $t{b}_{\mathrm{deg}}=17.3$ which is close to the steady state (c,d). These profiles were obtained by solving the discrete (a,c) and continuum description (b,d), respectively. The rectangular region which the ligand cannot enter appears in white. It appears smaller in a and c than in b and d because in the discrete description, ligands can still bind to surface receptors on the cells located at the edge of the region so that $\lambda >0$ for these cells. (e–g) Profiles of $\lambda (\vec{x},t)$ along the slices indicated in b at $t{b}_{\mathrm{deg}}=0.72,2.16,3.6$ (black lines) and in steady state (red lines). Lines indicate solutions to Eq. (22), while symbols indicate solutions to Eq. (21) for comparison. (h) Contrast of the depletion of $\lambda (\vec{x},t)$ shown in g. The contrast is defined as $c\left(t\right)=1-{\lambda }_b\left(t\right)∕{\lambda }_o\left(t\right)$ where ${\lambda }_b\left(t\right)$ and ${\lambda }_o\left(t\right)$ are the total ligand concentration at the locations shown by the crosses in a, i.e., directly behind the clone and far away from it, respectively. Initial condition: $\lambda \left(\vec{x}\right)=0$. Parameters as in Fig. 2 with $D_0=0$, $j_0∕b_{\mathrm{deg}}ar=25∕3$ at $x=0$, $j=0$ at $y∕a=\pm 25$ and at $x∕a=50$.