article: The motion of ions, molecules or proteins in dendrites is restricted by cytoplasmic obstacles such as organelles, microtubules and actin network. To account for molecular crowding, we study the effect of diffusion barriers on local calcium spread in a dendrite. We first present a model based on a dimension reduction approach to approximate a three dimensional diffusion in a cylindrical dendrite by a one-dimensional effective diffusion process. By comparing uncaging experiments of an inert dye in a spiny dendrite and in a thin glass tube, we quantify the change in diffusion constants due to molecular crowding as Dcyto/Dwater = 1/20. We validate our approach by reconstructing the uncaging experiments using Brownian simulations in a realistic 3D model dendrite. Finally, we construct a reduced reaction-diffusion equation to model calcium spread in a dendrite under the presence of additional buffers, pumps and synaptic input. We find that for moderate crowding, calcium dynamics is mainly regulated by the buffer concentration, but not by the cytoplasmic crowding, dendritic spines or synaptic inputs. Following high frequency stimulations, we predict that calcium spread in dendrites is limited to small microdomains of the order of a few microns (<5 μm).
  Dendrites of neurons contain a complex intracellular organization made of organelles, such as mitochondria, endoplasmic reticulum, ribosomes and cytoskeletal network generated by actin and microtubules [1]–[3]. The cell cytoplasm is thus a crowded rather than diluted medium in which diffusional mobility of small molecules is restricted [3]–[7]. Molecular crowding can affect many biochemical processes such as, protein folding [8]–[10], enzymatic reactions [11]–[13] and signal transduction [14]. Although electromicroscopy images [15] reveal the complexity of dendritic organization, there are no direct methods to estimate the functional consequence on diffusion. Modeling in combination with Monte-Carlo methods [16]–[19] allowed to study diffusion in crowded media. Depending on the size of the diffusing molecule and the interactions with the heterogeneous media, crowding can lead to anomalous or normal diffusion [4], [20]–[24]. Neuronal calcium is an fundamental and ubiquitous messenger [25], [26]. It is regulated by cytoplasmic crowding, mobile and immobile calcium buffers [27]–[30], pumps and dendritic spines, which cannot be easily dissociated experimentally. It was already noticed and quantified [31] that cellular calcium buffers can determine amplitude and diffusional spread of neuronal calcium signaling. Precisely, fixed calcium buffers tend to retard the signal and to lower the apparent diffusion coefficient, whereas mobile buffers can contribute to calcium redistribution. To study calcium dynamics, we develop in the first part, a model of diffusion in a crowded three-dimensional dendrite, that we reduce to a one-dimensional effective diffusion process. The model is general and can be applied to protein diffusion in membranes or in endoplasmic reticulum-like networks [16], [32]. In a second part, we use uncaging experimental data of an inert dye (fluorescein) in a spiny dendrite and in a glass tube of similar size filled with aqueous solution to estimate the reduction of the diffusion constant in a dendrite. These experiments are repeated by Brownian simulations in a 3D model dendrite in order to validate our one-dimensional model. In the last part, we use the previously derived effective diffusion constant and simulate a system of reaction-diffusion equations in one dimension to study calcium dynamics in a dendrite. We accounted for calcium buffers, pumps, dendritic spines and synaptic inputs. We show that for moderate organelle crowding, calcium spread is mainly restricted by the buffer and the pump concentration and not by obstacles or dendritic spines. Although crowding restricts dendritic diffusion by a factor 20, it is not responsible for the high calcium compartmentalization () in dendrites [33], [34]. We further show that following high frequency stimulations, calcium spread does not exceed. In summary, calcium microdomains are highly regulated by various active processes such as calcium buffers, pumps and stores.
  In addition to cytoplasmic crowding, calcium dynamics is regulated by many factors such as binding to buffer molecules (e. g. , calmodulin and calcineurin), dendritic spines and various types of pumps located on the dendritic surface (PMCA, NCX) and on the surface of internal organelles such as the endoplasmic reticulum (SERCA). It is usually not possible to dissect experimentally the contribution of each process, and we shall apply our previous result to study calcium spread in dendrites. We present a reaction-diffusion equation (Materials and Methods) to simulate calcium dynamics in both spiny and aspiny dendrites. At this stage, we do not take into account the intracellular calcium stores, and thus, we exclude the generation of calcium waves through CICR, nor do we model spontaneous dendritic calcium spikes or calcium transients associated with back-propagating action-potentials. We shall focus here on the local spread of calcium transients and we ignore global calcium events. We include in our simulations the effect of buffers, pumps, spines and synaptic input. The contribution to calcium dynamics for each active component is provided in the Materials and Methods. We first simulated calcium diffusion in an aqueous solution (contained in a glass pipette) by initiating a calcium transient and solving the one dimensional diffusion equation (41) – (45) with a diffusion constant of (Figure 4A). The effect of crowding alone on calcium diffusion in a dendrite was simulated by reducing the free diffusion constant to (Figure 4B). We assume here that the effects of crowding on motion are the same for fluorescein molecules and calcium ions attached to a dye molecules. As expected, crowding leads to a more localized and persistent calcium transient compared to free diffusion in an aqueous solution. We next added two types of imobile buffers, calmodulin (CaM) and calcineurine (CN), as well as pumps (NCX and PCMA) to the simulation. The buffer concentration was varied between low ([CaM]: , [CN]:), medium ([CaM]: , [CN]:) and high ([CaM]: , [CN]:) levels. Figure 4C and D show the effect of fast buffering on calcium dynamics in aqueous solution and in a crowded dendrite, respectively, for medium buffer concentration. The differences are small. The calcium signal in the crowded medium is more localized in space and slightly longer lasting than in aqueous solution. From these simulation results, we conclude that the spatiotemporal extent of the calcium signal is highly restricted by the stationary buffer activity. These results agree qualitatively with other uncaging experiments of calcium in glass tubes and dendrites [47]. We next analyze calcium spread originating from localized inputs such as synapses. At dendritic synapses calcium can enter through NMDA-receptors. To estimate calcium spread as a function of the synaptic input frequency, we simulated -influx in the middle of a dendritic segment (Figure 4E). Buffers and pumps were set to their default values (Table 1). We initiated calcium transients in the crowded model dendrite for different input frequencies (). The spatiotemporal extent of the calcium signal for different input frequencies is given in the intensity plots Figure 4F. Calcium spread is measured by the full width at half maximum (FWHM) of the calcium signal. Interestingly, for input frequencies larger than 20 Hz, the calcium signal in the dendrite reaches a stationary value. For high input frequencies (20 Hz) calcium spread does not exceed (= 0. 5FWHM) as measured from the input source. This is in agreement with the experimental data where calcium spread was contained within a domain of about. We conclude that buffer and pumps limit calcium spread to few micrometers.
  We have shown here that dendritic crowding reduces the diffusion constant of inert Brownian molecules by a factor of 20 when compared to diffusion in an aqueous solution. We have used this result to estimate calcium spread in dendrites. We found that in the absence of regenerative mechanisms (VSCC, calcium stores), the spread of calcium largely depends on the buffer concentration and moderate molecular crowding does not play a significant role in shaping calcium dynamics. Thus, crowding has only a minor effect compared to the cumulative effect of pumps and buffers. In addition, the presence of a single (passive) spine at the location of calcium release did not influence calcium diffusion in the dendrite. In this study, we have analyzed the effect of molecular crowding on calcium spread under the presence of stationary buffers. Assuming that the diffusion constant of calcium and fluorescein are reduced by the same factor due to the effect of molecular crowding, our results confirm previous studies that calcium spread is largely restricted by the effect of stationary buffers [31], [48]–[50]. Our analysis showed only a small effect of molecular crowding on calcium spread (Figure 4C and 4D): slightly more calcium molecules were bound to buffers in the crowded condition. These results are qualitatively consistent with stochastic simulations in a cubic cell model under different crowding and buffer mobility conditions [19], where it has been shown that molecular crowding affects the calcium signaling system mainly through crowding-induced binding of calcium to buffer molecules and less through the direct hindrance of calcium diffusion. This study showed further that these effects are not additive. Interestingly, the reduction in diffusion constant due to molecular crowding was found to be for moderately crowded environments with excluded volume fraction. In our study, the reduction of the calcium diffusion constant was extrapolated from fluorescein uncaging experiments in the dendritic medium, which resulted in a much higher value. This difference might result from additional crowding effects such as cavities that were not modelled in the stochastic simulations. Calcium microdomains have been observed during spontaneous and electrically evoked activation of synapses on dendritic shafts in aspiny neurons [34]. Compartmentalization into domains of about resulted from fast kinetics of calcium permeable AMPA receptors and fast local extrusion via the exchanger [34]. In general, as observed in Figure 4, calcium spread is robustly confined in a domain of less than from the input source and this seems to be independent of the synaptic firing frequency. Thus, calcium dynamics seems to be well regulated by buffers, stores and extrusion mechanisms. It is certainly a requirement for dendrites to prevent calcium spread over large distances because it is not only the primary messenger in the induction of synaptic plasticity, such as long term potentiation (LTP) [51], but it is also involved in morphological changes and in the regulation of receptor trafficking such as AMPA [52]. While organelle localization might depend on the dendritic local needs (protein syntheses, energy supply and local calcium stores), calcium pump densities and calcium buffer concentrations might be regulated independently to maintain calcium homeostasis. It remains an unsolved question to determine how pumps and calcium buffer molecules are regulated along a dendrite. Using our previous computations, we found that (passive) dendritic spines in this mean-field approach do not contribute much in dendritic calcium regulation (data not shown). In general, our result suggests that spines should not significantly affect the movement of diffusing particles along the dendrite. However, in the case of calcium, we have not taken into account a possible calcium propagation through the endoplasmic reticulum network, which may lead to a very different type of propagation. Dendritic spines can be seen as the ultimate place of confinement in dendrites: indeed, calcium exchangers located on the endoplasmic reticulum surface or on the spine neck membrane can prevent calcium from diffusing into the spine head [53], [54]. In addition, large crowding observed at the spine base due to various types of organelles such as the endoplasmic reticulum or the spine apparatus [2], [15] can prevent diffusing molecules from entering the spine neck. However, it is not clear whether mRNA or transcription factors can enter dendritic spines by passive diffusion or whether active processes are required.
  Cultures were prepared as detailed in [47]: we use wistar rat pups at P1. Hippocampal tissue was mechanically dissociated and plated on 12 mm glass coverslips at 3–4105 cells per well in a 24 well plate. Cells were left to grow in the incubator at C, 5% CO for 4 days, at which time the medium was changed to 10% HS in enriched MEM. The medium was changed four days later to 10% HS in enriched MEM. Cells were transfected at 1 wk in culture with DsRed plasmid to visualize the dendrites and spines using a lipofectamine 2000 (Invitrogen) method. On the day of imaging, the glass was transferred to the recording medium containing (in mM): NaCl 129, KCl 4, MgCl 1, CaCl 2, glucose 10, HEPES 10, and TTX. pH was adjusted to 7. 4 with NaOH, and osmolarity to 315 mOsm with sucrose. Ten-fourteen day old cultured cells were patch clamped at the soma and recorded with a glass pipette containing (in mM): K-gluconate 140, NaCl 2, HEPES 10, EGTA 0. 2, Na-GTP 0. 3, Mg-ATP 2, phosphocreatine 10, and 100 of caged fluorescein (Molecular Probes) at pH 7. 4 having a resistance of 6–12 M. Signals were amplified with Axopatch 200 (Axon Instruments Inc. Foster City, CA). Cells were imaged with a 63× water immersion objective (NA = 0. 9). UV laser was aimed at a spot of 1 m in the center of the field of view. A line scan mode (0. 7 msec/line) was used along an imaged dendrite to measure fast changes in fluorescence following flash photolysis of caged fluorescein. In the second stage of the experiment, the content of patch pipettes, containing caged fluorescein, was sucked out and introduced into additionally prepared pipettes with long and sharp tips, having tens of microns in length and about 1–2 in diameter making their geometry similar to a “typical” dendrite. Same line scan mode was used to compare changes in fluorescence in a dendrite and in a glass tube, containing similar concentrations of caged fluorescein. Data were analyzed using custom made MATLAB-based programs. Steps of 0. 6 from the center of the uncaging sphere were defined through the line scans and pixels inside every step were horizontally averaged. Every line scan trial was repeated 7–14 times. Statistical comparisons were made with t-tests. We implemented the Brownian simulations in MATLAB using a ray-tracing algorithm. To overcome the huge computational burden that Brownian simulations in complex domains impose, we made heavily use of MATLAB' s object-oriented programming and vectorization features as well as of the external C/C++ interface functions capabilities (MEX-files). We first constructed a triangular mesh of the simulation domain (e. g. , cylinder, cylinder with spine, see Figures 3 A, B) using a simple mesh generator based on distance function (DistMesh package, [55]). The (meshed) simulation domain was then equipped with user-defined sampling boxes, an initial distribution of particles and diffusion barriers (e. g. , disks with small holes, see Figures 3 B, C). We predefined a sampling interval (ms) at which the particle concentrations in the sampling boxes were measured. Surface mesh elements were defined to be either reflective or absorbing. The top and the bottom of the cylindrical domain was set to be absorbing while all other surface elements were defined to be reflective. Particle rays crossing reflecting boundaries or obstacles were reflected according to the law of light reflection. To speed up the code we divided the simulation domain into partition voxels. For each partition voxel a list of contained objects (mesh elements, obstacles) was pre-computed and provided to the algorithm during execution. The Brownian simulation was implemented using an Euler-scheme with adaptive-step size. Steps were defined by the distance to mesh elements and obstacles. The closer the particles were to objects the smaller the step size was chosen. As a rule of thumb, the minimal step size was determined by 0. 3–0. 5 of the smallest length scale that had to be resolved (e. g. , the radius of the hole of the disks, see Figure 3 B). The (vectorized) particle rays were traced in the voxels and tested for intersections with mesh elements or objects. If intersections occurred the particles were either reflected or absorbed. It is important to note that an adaptive-step size algorithm leads for each particle to a different progress in physical time. Hence, the measurement of particle concentrations at fixed sampling times, required the implementation of a scheduler that removed particles temporarily from the simulation and stored their positions. Our simulations lasted between several hours to several days on a cluster depending on the number of particles, number of objects and the minimal step size. We have made extensive use of MATLAB' s visualization tools to monitor the simulations and to generate visual outputs of the simulation results (see snapshots in Figures 3 A–C and a movie (Video S1) in the Text S1). We have included in the Text S1 a validation study of diffusion in a cylindrical domain with absorbing boundaries at the top and bottom. Different measures such as global and local particle concentrations as well as the mean first passage time to the absorbing boundaries are extracted from the simulations and compared with existing analytical results. The test-simulation is shown in Video S2. A good agreement between these results was obtained, and thus, evidence for the correctness of the implemented algorithm in the Monte-Carlo simulation tool is provided. The spatiotemporal calcium signal in the dendrite is regulated by several active and passive components that are described next.
summary: Diffusion is one of the main transport phenomena involved in signaling mechanisms of ions and molecules in living cells, such as neurons. As the cell cytoplasmic medium is highly heterogeneous and filled with many organelles, the motion of a diffusing particle is affected by many interactions with its environment. Interestingly, the functional consequences of these interactions cannot be directly quantified. Thus, in parallel with experimental methods, we have developed a computational approach to decipher the role of crowding from binding. We first study here the diffusion of a fluorescent marker in dendrites by a one-dimensional effective diffusion equation and obtained an effective diffusion constant that accounts for the presence heterogeneity in the medium. Furthermore, comparing our experimental data with simulations of diffusion in a crowded environment, we estimate the intracellular calcium spread in dendrites after injection of calcium transients. We confirm that calcium spread is mainly regulated by fixed buffer molecules, that bind temporarily to calcium, and less by the heterogeneous structure of the surrounding medium. Finally, we find that after synaptic inputs, calcium remains restricted to a domain of 2. 5 µm to each side of the input location independent of the input frequency.
keywords: physics
  computer science
  mathematics
  theoretical biology
  interdisciplinary physics
  applied mathematics
  biology
  computational biology
  computerized simulations
  biophysics
  neuroscience
title: Barriers to Diffusion in Dendrites and Estimation of Calcium Spread Following Synaptic Inputs