Figure 1

Diffusion-trapping model of receptor trafficking along a dendritic cable (diagrams not to scale). (A) A population of dendritic spines are distributed on the surface of a dendritic cable of length $L$. Each receptor diffuses freely until it encounters a spine where it may become trapped. Within a spine receptors may be internalized via endocytosis (END) and then either recycled to the surface via exocytosis (EXO) or degraded (DEG), see inset. Synthesis of new receptors at the soma and insertion into the plasma membrane generates a surface flux $\sigma$ at one end of the cable. (B) Reduced one-dimensional model showing a set of discrete trapping sites (spines) and their associated state transition diagrams. Here $R_j$ denotes the concentration of surface receptors inside the $j\text{th}$ spine, and $S_j$ denotes the number of receptors within the corresponding intracellular pool. Freely diffusing surface receptors can enter/exit the spine at a hopping rate ${\Omega }_j$, be endocytosed at a rate $k_j$, exocytosed at a rate ${\sigma }_j^{\mathit{rec}}$, and degraded at a rate ${\sigma }_j^{\mathit{deg}}$. The spatial extent of each trapping region is neglected so that free diffusion occurs over the whole length of the cable.Reuse & Permissions

Figure 2

Dendritic receptor concentration plotted as a function of distance from the soma for a uniform distribution of identical spines. The dendritic cable has length $L=200\mu \mathrm{m}$ and circumference $l=1\mu \mathrm{m}$. We consider $N=200$ spines having spacing $d=1\mu \mathrm{m}$ and surface area $A=1\mu {\mathrm{m}}^2$. (A)–(D) The black curve is the receptor profile for the following baseline parameter values: $D=0.1\mu {\mathrm{m}}^2{\mathrm{s}}^{-1}$, $k={10}^{-3}{\mathrm{s}}^{-1}$, $\Omega ={10}^{-3}\mu {\mathrm{m}}^2{\mathrm{s}}^{-1}$, ${\sigma }^{\mathit{rec}}={10}^{-3}{\mathrm{s}}^{-1}$ and ${\sigma }^{\mathit{deg}}={10}^{-4}{\mathrm{s}}^{-1}$ (so that $\lambda =0.909$), and $\sigma =1{\mathrm{s}}^{-1}$. Gray curves correspond to receptor profiles when one or more parameters are changed from the baseline as indicated.Reuse & Permissions

Figure 3

(Color online) Inhomogeneities in spine density and surface area. (A) Dendritic receptor concentration plotted as a function of distance from the soma for various distributions of identical spines: spines are spaced $2\mu \mathrm{m}$ apart for the first $134\mu \mathrm{m}$ of the cable, then $0.5\mu \mathrm{m}$ thereafter (solid gray line), or spaced $4.88\mu \mathrm{m}$ for the first $166\mu \mathrm{m}$ of the cable, then $0.205\mu \mathrm{m}$ thereafter (dashed gray line); all other parameters are taken at baseline as defined in Fig. 2. Note that for both distributions the total number of spines is still $N=200$. In both cases there are more receptors at each location of the cable than in the uniform distribution (black). (B) Same as (A), except parameters are chosen as in Fig. 2D. Again, there are more receptors everywhere. (C) Dendritic receptor concentration (solid gray line) and receptor numbers in spines (dashed blue line) plotted as functions of distance from the soma for a uniform distribution of nonidentical spines. (The numerical range of both quantities are taken to be the same.) Spine surface area is $1\mu {\mathrm{m}}^2$ for the first $100\mu \mathrm{m}$ of the cable, then $2\mu {\mathrm{m}}^2$ thereafter; all other parameters are taken at baseline as defined in Fig. 2. There is relatively little change from baseline (black). (D) Same as (C), except parameters are chosen as in Fig. 2D. Receptor numbers at spines increase dramatically beginning $100\mu \mathrm{m}$ from the soma.Reuse & Permissions