Figure 1

Continuum model of protein receptor trafficking along a dendrite. (a) Single dendritic cable. The concentration $U\left(x,t\right)$ of surface receptors at time $t$ varies with distance $x$ from the soma. A surface receptor current $I_{\text{soma}}$ is injected at the somatic end of the cable. The current $J\left(x,t\right)$ determines the rate at which receptors flow between the dendrite and an individual spine at $x$. (b) Simplified model of a spine showing constitutive recycling with an intracellular pool. Surface receptors of concentration $R\left(x,t\right)$ are internalized at a rate $k$ and are either reinserted into the spine surface from an intracellular pool of $C\left(x,t\right)$ receptors at a rate ${\sigma }^{\text{rec}}$ or sorted for degradation at a rate ${\sigma }^{\text{deg}}$. There is also a local production of intracellular receptors at a rate $\sigma$.Reuse & Permissions

Figure 2

(a) Steady-state concentration of surface receptors in the dendritic membrane as a function of distance $x$ from the soma. Solid curves show concentration profiles for diffusivity $D=0.1\mu {\text{m}}^2{\text{s}}^{-1}$ and dashed curves are for $D=0.45\mu {\text{m}}^2{\text{s}}^{-1}$. Lower two curves are for a zero background concentration, whereas upper two curves are for a nonzero background concentration. The length and circumference of the cable are $L=1\text{mm}$ and $l=1\mu \text{m}$. $N=1000$ identical spines are distributed uniformly along the cable with density ${\rho }_0=1\mu {\text{m}}^{-2}$. The spine parameters are as follows: surface area $A=1\mu {\text{m}}^2$, rate of endocytosis $k={10}^{-3}\mu {\text{m}}^2{\text{s}}^{-1}$, rate of recycling ${\sigma }^{\text{rec}}={10}^{-3}{\text{s}}^{-1}$, rate of degradation ${\sigma }^{\text{deg}}={10}^{-5}{\text{s}}^{-1}$, and hopping rates ${\Omega }_{\pm }=\Omega ={10}^{-3}\mu {\text{m}}^2{\text{s}}^{-1}$. The values of most parameters are based on experimental measurements, including the diffusivity [3, 4], the rates of constitutive recycling [7, 39, 40, 41], and the size and density of spines [13], whereas the hopping rate $\Omega$ is estimated using a simple model of diffusion within the spine neck [14]. The somatic current is taken to be $I_{\text{soma}}=0.1{\text{s}}^{-1}$ so that the maximum number of synaptic receptors per spine is consistent with experimental observations [42, 43]. (b) Space constant $\xi$ as a function of the diffusivity $D$, the diffusive coupling $\Omega$, and the rate of degradation ${\sigma }^{\text{deg}}$.Reuse & Permissions

Figure 3

Schematic diagram of a branched cable consisting of a primary branch with length $L_0$ and circumference $l_0$, and two secondary branches with lengths $L_1,L_2$ and circumferences $l_1,l_2$. The surface receptor concentration $U_B$ at the branch point is continuous, and the diffusive current $I_0$ flowing through the junction in the primary branch splits into two currents $I_1,I_2$.Reuse & Permissions

Figure 4

Time-dependent Green’s functions determining response to a surface current injection at the center of a dendritic cable of length $L=200\mu \text{m}$ and circumference $l=1\mu \text{m}$. All other parameters are as in Fig. 2. (a) Green’s function $\mathcal{G}\left(x,L/2,t\right)$ for surface receptors in dendrite plotted as a function of $x$ and various times $t$. (b) Green’s function $\mathcal{H}\left(x,L/2,t\right)$ for surface receptors in spines. (c) Green’s function $\mathcal{K}\left(x,L/2,t\right)$ for receptors in intracellular pools.Reuse & Permissions

Figure 5

Plots of time-dependent Green’s function $\mathcal{G}\left(x,L/2,t\right)$ for different values of the diffusive coupling $\Omega$. (a) Same as Fig. 4a with $\Omega ={10}^{-3}\mu {\text{m}}^2{\text{s}}^{-1}$. (b) $\Omega ={10}^{-4}\mu {\text{m}}^2{\text{s}}^{-1}$. (c) Zero diffusive coupling between dendrite and spines (pure diffusion).Reuse & Permissions

Figure 6

Same as Figs. 4a, 4b, 4c except that ${\Omega }_{-}=0.1{\Omega }_{+}={10}^{-4}\mu {\text{m}}^2{\text{s}}^{-1}$.Reuse & Permissions

Figure 7

Time courses of probabilities ${\mathcal{P}}_U\left(t\right),{\mathcal{P}}_R\left(t\right),{\mathcal{P}}_C\left(t\right)$ associated with dendritic surface, spines, and intracellular stores, respectively. (a) Same parameters as in Fig. 4 with ${\sigma }^{\text{deg}}={10}^{-5}{\text{s}}^{-1}$. (b) Same as (a) except for a faster rate of degradation ${\sigma }^{\text{deg}}={10}^{-3}{\text{s}}^{-1}$. Insets show same plots over a longer time interval.Reuse & Permissions

Figure 8

(Color online) Plot of asymptotic relaxation rate ${\lambda }_1$ (filled circles) and scaling factor $H\left(0\right)D$ (filled squares) as a function of the rate of degradation ${\sigma }^{\text{deg}}$. All other parameter values are as in Fig. 2.Reuse & Permissions

Figure 9

Schematic diagram of a branched dendritic tree showing a branch node $B$, a terminal node $T$, and the special terminal node $S$ adjoining the soma.Reuse & Permissions

Figure 10

Schematic diagram showing amplitude factors picked up by a trip on reaching (a) a branch node and (b) a terminal node.Reuse & Permissions

Figure 11

Simple configuration consisting of one main branch and two semi-infinite secondary branches. Two shortest trips starting at $X$ and ending at $Y$ are shown.Reuse & Permissions