Conceived and designed the experiments: EK. Performed the experiments: EK. Analyzed the data: AB. Contributed reagents/materials/analysis tools: AB EK DH. Wrote the paper: AB EK DH.
The authors have declared that no competing interests exist.
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 onedimensional 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
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 onedimensional 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
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
Neuronal calcium is an fundamental and ubiquitous messenger
In the last part, we use the previously derived effective diffusion constant and simulate a system of reactiondiffusion 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 (
Our results are divided into three sections. In the first section, we present the diffusion model for an inert dye in a crowded dendritic medium. The model is derived from a periodic compartmentalization of the dendritic domain. It is followed by an extension of the model to almost periodic compartments and the analysis of the mean time a particle takes to travel across the dendrite. In the second part, we present the outcome of the uncaging experiments of fluorescein to probe the dendritic medium and to estimate the model parameters. It is followed by a comparison to Brownian simulations, which repeat these experiments on a computer. Finally, we provide meanfield simulation results for calcium spread in a dendrite under the additional presence of stationary buffers, pumps and synaptic input.
To characterize diffusion in a heterogeneous dendrite, containing various organelles such as mitochondria, spine apparatus, endoplasmic reticulum and other structures, we propose to derive from a three dimensional analysis a onedimensional effective diffusion equation. In the limit where the space in between organelles is small, particles can still move inside a dendritic domain
The model dendrite is organized as a sequence of periodic compartments of length
Introducing the concentration
The compartment parameter is
The previous analysis can be applied to the motion of receptors on the surface of neurons, which contains impenetrable micro domains
To further analyze the effect of diffusion barriers, we investigate how our previous analysis is affected by an almost periodic distribution of barriers, where a random jitter is modelled as white noise. We will see that diffusion in such medium is characterized by a fourth order diffusion equation. This analysis shows that approximating diffusion by dimensional reduction can lead to a noneclassical diffusion description. We start with a compartment position
The effective diffusion equation (8) is recovered in the limit
A possible application of the previous theory and equation (6) is to estimate the mean time
The probability density function to find a molecule at position
To obtain the MFPT,
The solution is
We conclude that in a dendrite with an effective diffusion constant of
To study crowding inside a dendrite, we use a set of experiments in which we measure the diffusion time course of a caged inert dye molecule fluorescein (
(A) Images of the dendritic segments and the glass pipette used in the experiments. The sites of the uncaging spots are indicated. (B) Fluorescein transients in the pipette (black) and in the dendritic medium far away from any attached spine (green) at different distances from the uncaging spot. (C) Fluorescein transients in the dendrite near and far away of any attached dendritic spine are shown in blue and green, respectively. Fluorescein was uncaged at the base of the spine at location
For the pipette data, where fluorescein diffuses freely, this method led to a diffusion constant of
We further investigated the influence of spine on dendritic diffusion: we initiated a dye transient in the dendritic shaft at the base of a spine by uncaging fluorescein.
To support our modeling approach we use Brownian simulations (
(A) Model glass pipette (radius
We first calibrate the parameter of the model: according to equation (9), a reduction of diffusion constants by a factor of
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 reactiondiffusion equation (
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
(A) Calcium diffusion in an aqueous solution contained in a pipette of length
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]
We next analyze calcium spread originating from localized inputs such as synapses. At dendritic synapses calcium can enter through NMDAreceptors. To estimate calcium spread as a function of the synaptic input frequency, we simulated
Parameter  Value  Reference  
Glass tube geometry  
length of glass tube 

adjusted  
glass tube diameter 

adjusted  
Dendrite geometry  
length of dendritic segment 

adjusted  
dendrite diameter 

(Koch, 1999)  
dendritic cross section 
0.785 
adjusted  
Crowding  
compartment length 

adjusted  
opening size 

adjusted  
compartment parameter 
0.05  adjusted  
diffusion constant of free 

(Korkotian et al., 2004)  

(Korkotian et al., 2004)  
pump rate for PMCA 
0.27 
(Erler et al., 2004)  
pump density for PMCA 
9200/ 
(Erler et al., 2004)  
halfsaturation constant for PMCA 

(Korkotian et al., 2004)  
hill coefficient for PMCA 
1.0  (Stauffer et al., 1995)  
pump rate for NCX 
0.48 
(Erler et al., 2004)  
pump density for NCX 
300/ 
(Erler et al., 2004)  
halfsaturation constant for NCX 

(Fujioka et al., 2000)  
hill coefficient for NCX 
1.7  (Fujioka et al., 2000)  
Calmodulin  
total concentration 
10, 25 (default), 100 
(Volfovsky et al., 1999)  
forward binding rate for 1st binding 
160 
(Johnson et al., 1996)  
backward binding rate for 1st binding 
405 
(Johnson et al., 1996)  
forward binding rate for 2st binding 
160 
(Johnson et al., 1996)  
backward binding rate for 2st binding 
405 
(Johnson et al., 1996)  
forward binding rate for 3st binding 
2.3 
(Johnson et al., 1996)  
backward binding rate for 3st binding 
2.4 
(Johnson et al., 1996)  
forward binding rate for 4st binding 
2.3 
(Johnson et al., 1996)  
backward binding rate for 4st binding 
2.4 
(Johnson et al., 1996)  
Calcineurine  
total concentration 
5, 10 (default), 25 
(Volfovsky et al., 1999)  
forward binding rate 
50 
(Volfovsky et al., 1999)  
backward binding rate 
25 
(Volfovsky et al., 1999)  
Calcium dye (Fluo4)  
total concentration 
2 
(Korkotion et al., 2004)  
forward binding rate 
60 
(Korkotion et al., 2004)  
backward binding rate 
170 
(Korkotion et al., 2004)  
NMDAR  
current through a single NMDAR 
9 pA  (PinaCrespo and Gibb, 2002)  
fraction of current carried by 

(Burnashev, 1995)  
time constant (decay) 
80 ms  (Zador and Koch, 1994)  
time constant (rise) 
3 ms  (Zador and Koch, 1994)  
radius of receptor 
0.025 
adjusted  
Spines  
spine radius 
0.05–0.16 
(Koch, 1999) 
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
These results are qualitatively consistent with stochastic simulations in a cubic cell model under different crowding and buffer mobility conditions
Calcium microdomains have been observed during spontaneous and electrically evoked activation of synapses on dendritic shafts in aspiny neurons
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)
Using our previous computations, we found that (passive) dendritic spines in this meanfield 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
Cultures were prepared as detailed in
We implemented the Brownian simulations in MATLAB using a raytracing algorithm. To overcome the huge computational burden that Brownian simulations in complex domains impose, we made heavily use of MATLAB's objectoriented programming and vectorization features as well as of the external C/C++ interface functions capabilities (MEXfiles). We first constructed a triangular mesh of the simulation domain (e.g., cylinder, cylinder with spine, see
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 precomputed and provided to the algorithm during execution.
The Brownian simulation was implemented using an Eulerscheme with adaptivestep 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
The spatiotemporal calcium signal in the dendrite is regulated by several active and passive components that are described next.
Dendrites contain a large number of different buffers. The reactions of a buffer
Two basic mechanisms are responsible for the removal of calcium ions across the neuron membrane: the ATPdriven plasma membrane
Dendritic spines are modeled as passive calcium absorbers. In our model, calcium ions entering a dendritic spine are totally absorbed. The flux of calcium ions into the dendritic spine depends on the spine neck radius. It can be computed in the configuration where the dendrite is compartmentalized and the compartments are connected through small openings (
The effect of the calcium dye is modeled as buffer by the reaction
Calcium influx on dendritic spines is mediated primarily by slow NMDA currents
The total effect of buffers, pumps and spines on the cytosolic calcium concentration can be summarized in form of a reactiondiffusion equation:
We included in the above equations the effect of mobile buffers. However, in the following, we assume that the buffers are fixed and set the buffer diffusion constants,
The reactiondiffusion equations (42)–(45) were solved numerically using MATLAB. The partial differential equations were solved using the numerical method of lines which is implemented in the MATLAB solver. Space and time discretizations were set to
Movie of a stochastic simulation of
(AVI)
Movie of a diffusion experiment of
(AVI)
Validation study for testing the algorithm implemented in the stochastic simulation tool.
(PDF)
(A) Global particle concentration
(EPS)