Diffusion in a dendritic spine: The role of geometry

Dendritic spines, the sites where excitatory synapses are made in most neurons, can dynamically regulate diffusing molecules by changing their shape. We present here a combination of theory, simulations, and experiments to quantify the diffusion time course in dendritic spines. We derive analytical formulas and compared them to Brownian simulations for the mean sojourn time a diffusing molecule stays inside a dendritic spine when either the molecule can reenter the spine head or not, once it is located in the spine neck. We show that the spine length is the fundamental regulatory geometrical parameter for the diffusion decay rate in the neck only. By changing the spine length, dendritic spines can be dynamically coupled or uncoupled to their parent dendrites, which regulates diffusion, and this property makes them unique structures, different from static dendrites.

Figure 1

Comparisons of fluorescence kinetics in thin and thick dendrites following flash photolysis of caged fluorescein, detected with a fast line scan. (a) Cultured hippocampal cell, transfected at $1\text{week}$ in culture with DsRed plasmid to visualize the dendrites and spines using a lipofectamine 2000™ (Invitrogen) method (for details see Ref. 7). Image was taken $3\text{days}$ after transfection using a PASCAL confocal microscope (Zeiss). The helium neon $633\mathrm{nm}$ laser spot in the middle of box 1 represents the location of the uv $\left(355\mathrm{nm}\right)$ flash. White boxes, containing thinner (1) and thicker (2) dendritic segments are shown at higher magnification in (b) 1 and 2. The neuron was patch clamped at the soma with a glass pipette (arrow at the bottom of a) containing $100\mu \mathrm{M}$ of caged fluorescein. (b) 1 and 2: thin (1, less than $1\mu \mathrm{m}$ in diameter) and thick (2, about $2.5\mu \mathrm{m}$ in diameter) dendritic segments, line scanned at a rate of $0.7\mathrm{ms}$ per line in the middle and along the segment. Measurements were performed at the focus of uncaging (flashes, arrow at $0\mu \mathrm{m}$) and then 0.6, 1.2, 1.8, 2.4, and $3.6\mu \mathrm{m}$ apart, as shown in (b). Responses on the left and the right sides of the focus were found symmetric (data not shown). (c) Graphic representations of line scan recordings at the distances, shown in (b).

Figure 2

(Color online) Molecular simulations in a three-dimensional (3D) model of a dendritic spine. The spine geometry is approximated by a spherical head of radius $R=0.5\mu \mathrm{m}$ and a cylindrical neck with radius $\epsilon$ and length $L$. All particles were initially released from the center of the spine head. (a), (b) Mean sojourn time (± standard) of a diffusing molecule in the spine neck and in the spine as a function of the spine neck length (a) and the spine neck radius (b) assuming that no reentries into the spine neck occur after the molecule entered the neck. The theoretical results as predicted by formula (1) are shown as solid lines. (c) and (d) Same settings as in (a) and (b), except that multiple reentries into the spine head were allowed. The theoretical prediction according to formula (2) are shown as solid lines for $\alpha =0.84$ (e) Snapshots of one realization of the Brownian trajectory as traced by one diffusing molecule in a 3D model of a dendritic spine with possible reentries into the spine head ($R=0.5\mu \mathrm{m}$, $L=1.0\mu \mathrm{m}$, and $\epsilon =0.1\mu \mathrm{m}$). The snapshots are taken at different times $t=0$, 1, 5, 10, and $57.4\mathrm{ms}$, where the particle reached the dendritic shaft. Simulation parameters in (a)–(e). Diffusion constant $D=400\mu {\mathrm{m}}^2∕\mathrm{s}$ (free calcium), time $\text{step}=1\mathrm{e}\text{−}7\mathrm{s}$, total simulation time: $250\mathrm{ms}$.