Modeling the role of lateral membrane diffusion in AMPA receptor trafficking along a spiny dendrite

Abstract

AMPA receptor trafficking in dendritic spines is emerging as a major postsynaptic mechanism for the expression of plasticity at glutamatergic synapses. AMPA receptors within a spine are in a continuous state of flux, being exchanged with local intracellular pools via exo/endocytosis and with the surrounding dendrite via lateral membrane diffusion. This suggests that one cannot treat a single spine in isolation. Here we present a model of AMPA receptor trafficking between multiple dendritic spines distributed along the surface of a dendrite. Receptors undergo lateral diffusion within the dendritic membrane, with each spine acting as a spatially localized trap where receptors can bind to scaffolding proteins or be internalized through endocytosis. Exocytosis of receptors occurs either at the soma or at sites local to dendritic spines via constitutive recycling from intracellular pools. We derive a reaction–diffusion equation for receptor trafficking that takes into account these various processes. Solutions of this equation allow us to calculate the distribution of synaptic receptor numbers across the population of spines, and hence determine how lateral diffusion contributes to the strength of a synapse. A number of specific results follow from our modeling and analysis. (1) Lateral membrane diffusion alone is insufficient as a mechanism for delivering AMPA receptors from the soma to distal dendrites. (2) A source of surface receptors at the soma tends to generate an exponential-like distribution of receptors along the dendrite, which has implications for synaptic democracy. (3) Diffusion mediates a heterosynaptic interaction between spines so that local changes in the constitutive recycling of AMPA receptors induce nonlocal changes in synaptic strength. On the other hand, structural changes in a spine following long term potentiation or depression have a purely local effect on synaptic strength. (4) A global change in the rates of AMPA receptor exo/endocytosis is unlikely to be the sole mechanism for homeostatic synaptic scaling. (5) The dynamics of AMPA receptor trafficking occurs on multiple timescales and varies according to spatial location along the dendrite. Understanding such dynamics is important when interpreting data from inactivation experiments that are used to infer the rate of relaxation to steady-state.

Appendices

Appendix 1

The diffusion model given by Eq. (1) can be viewed as a continuum approximation of a previous model of protein receptor trafficking along a dendrite, in which lateral diffusion is coupled to a discrete population of spines (Bressloff and Earnshaw 2007). The latter is obtained by taking the spine density to have the explicit form

$$ \label{rho} \rho(x)=\frac{1}{l}\sum_{j=1}^N \delta\left(x-x_j\right), $$

(28)

where δ(x) is the Dirac delta function and x _j is the distance of the jth spine from the soma. Substitution into Eq. (1) gives

$$ \frac{\partial U}{\partial t} = D\frac{\partial ^2 U}{\partial x^2} -\sum_{j=1}^N \frac{\Omega_j}{l}\left[U_j(t)-R_j(t)\right]\delta\left(x-x_j\right), \label{eq:U2} $$

(29)

where Ω _j  = Ω(x _j ), U _j (t) = U(x _j ,t) and R _j  = R(x _j ,t). Note that this discrete spine model ignores the spatial extent of each spine so that the domain over which free diffusion occurs is the whole cylindrical surface of the dendrite. This is motivated by the observation that the spine neck, which forms the junction between a synapse and its parent dendrite, varies in radius from ~0.02 − $0.2~\upmu$m, which is typically an order of magnitude smaller than the spacing between spines (~0.1–1 $\upmu$m) and the circumference of the dendritic cable (~1 $\upmu$m), see Sorra and Harris (2000). In other words, the disc-like region or hole forming the junction between a spine and the dendritic cable is relatively small, and can therefore be neglected in a one-dimensional cable model. As noted in the discussion, In the case of a full two-dimensional model of diffusion along the cylindrical surface of a dendritic cable, one can no longer ignore the effects of these holes due to the fact that the Green’s function associated with two-dimensional diffusion has a logarithmic singularity (Bressloff et al. 2007).

For the given spine density (28), the steady-state diffusion equation (11) reduces to

$$ \label{eq:Udis} 0 = D\frac{d^2 U}{d x^2} -\sum_{j=1}^N \widehat{\Omega}_j\left[U_j -r_j\right]\delta\left(x-x_j\right) , $$

(30)

where r _j  = r(x _j ) etc. Integrating Eq. (30) over the interval 0 ≤ x ≤ L leads to the conservation condition

$$ l J_{\rm soma} = \sum_{j=1}^N \widehat{\Omega}_j\left[U_j -r_j\right]\ $$

(31)

Equation (30) can be solved in terms of the generalized one-dimensional Green’s function H(x,x ^′), defined according to the solution of the equation

$$ \label{eq:H} \frac{d ^2 H\left(x,x^\prime\right)}{d x^2} =-\delta\left(x-x^\prime\right) +L^{-1}, $$

(32)

with reflecting boundary conditions at the ends x = 0, L. A standard calculation shows that

$$ H(x,x') = \frac{L}{12} \left [g\left(\left[x+x'\right]/L\right)+ g\left(|x-x'|/L\right) \right],\quad $$

(33)

where g(x) = 3x ² − 6|x| + 2. Given the Green’s function H, the dendritic surface receptor concentration has an implicit solution of the form

$$ \label{eq:Uchi} U(x)= \chi -\sum_{j=1}^N \frac{\widehat{\Omega}_j\left[U_j -r_j\right]}{lD}H\left(x,x_j\right)+\frac{J_{\rm soma}}{D}H(x,0), $$

(34)

where the constant χ is determined from the conservation condition (31).

We can now generate a matrix equation for the concentration of dendritic receptors U _i at the i ^th spine, i = 1,...,N, by setting x = x _i in Eq. (34):

$$ \label{eq:Ui} U_i = \chi -\sum_{j=1}^N {\mathcal H}_{ij}\left[U_j-r_j\right]+ J_i, $$

(35)

where

$$ \label{eq:cBcC} {\mathcal H}_{ij}=\frac{\widehat{\Omega}_j}{lD} H\left(x_i,x_j\right),\quad J_i = \frac{J_{\rm soma}}{D}H(x_i,0). $$

(36)

If the matrix ${\mathcal H}=({\mathcal H}_{ij})$ does not have − 1 as an eigenvalue (which is the generic case), then the matrix $\ {\mathcal I}+{\mathcal H}$, where ${\mathcal I}$ is the N ×N identity matrix, is invertible and we can solve the system (35). That is, setting ${\mathcal M}= ({\mathcal I}+{\mathcal H})^{-1}$, we have

$$ U_i -r_i= \sum_j {\mathcal M}_{ij}\left[\chi+{J}_j-r_j\right]. $$

(37)

The conservation condition (31) then determines χ according to

$$\chi = {\left[ {\frac{{lJ_{{soma}} - {\sum\nolimits_{k,l} {\widehat{\Omega }} }_{k} {\mathcal M}_{{kl}} {\left[ {J_{l} - r_{l} } \right]}}}{{{\sum\nolimits_{k,l} {\widehat{\Omega }} }_{k} {\mathcal M}_{{kl}} }}} \right]}.$$

(38)

Equations (37) and (38) determine the dendritic receptor concentration U _j at the discrete site x _j of the jth dendritic spine. Substituting this solution into Eq. (34) then generates the full receptor concentration profile U(x). For a large number of spines distributed along a dendrite, the resulting solution matches that obtained from the continuum model of Section 2. Treating the spines as a continuous population makes the analysis more transparent than the matrix solution of the discrete model. For example, it generates a simple expression for the effective space constant of receptor diffusion.

Appendix 2

Here we provide a rough estimate for the hopping rate Ω₀ based on diffusion through the spine neck. For purposes of illustration, let us assume that the spine neck is a uniform cylinder of length L _n and radius r _n . Consider the steady state diffusion equation along the surface of the cylinder:

$$ D_n \frac{d^2 U_n}{d s^2}=0, \quad s \in (0,L_n). $$

(39)

with boundary conditions

$$ U_n(0)= U, \quad U_n(L_n)=R, $$

where U is the dendritic receptor concentration at the junction between the spine neck and cable and R is the receptor concentration within the ESM. Denoting the constant flux through the spine neck by J _n , we can solve Eq. (39) to obtain

$$ U-R = \frac{J_nL_n}{ D_n}. $$

(40)

Given that the total number of receptors per unit time flowing across either end of the spine neck is 2πr _n J _n , we deduce that

$$ \Omega_0 = \frac{2 \pi r_n D_n}{L_n}. $$

(41)

Using $L_n = 0.45~\upmu$m and $r_n = 0.075~\upmu$m (Sorra and Harris 2000) and $D_n = 6.7\times 10^{-3}~\upmu$m²s^− 1 (Ashby et al. 2006), we find that $\Omega \approx 7 \times 10^{-3}~\upmu$m²s^− 1, which is consistent with the baseline value shown in Table 1.