Stochastic modeling of calcium in 3D geometry

Abstract

Release of inflammatory mediators by mast cells in type 1 immediate-hypersensitivity allergic reactions relies on antigen-dependent increases in cytosolic calcium. Here, we used a series of electron microscopy images to build a 3D reconstruction representing a slice through a rat tumor mast cell, which then served as a basis for stochastic modeling of inositol-trisphosphate-mediated calcium responses. The stochastic approach was verified by reaction-diffusion modeling within the same geometry. Local proximity of the endoplasmic reticulum to either the plasma membrane or mitochondria is predicted to differentially impact local inositol trisphosphate receptor transport. The explicit consideration of organelle spatial relationships represents an important step toward building a comprehensive, realistic model of cellular calcium dynamics.

Figure 1

RBL-2H3 3D reconstruction. (A) Tilt series of electron microscopy images. (B) 3D reconstruction. (C–F) Details showing portions of the endoplasmic reticulum closely approaching the plasma membrane (C and E) and mitochondria (D). (F) ER-plasma membrane contact sites. Note the transition from the rough to smooth character of the ER as it comes closer to the plasma membrane (arrow).

Figure 2

Distribution of interorganellar distances. (A–C) Histograms of distances between the ER and plasma membrane (A), ER and mitochondria (B), and plasma membrane and mitochondria (C). Note the presence of close contacts between the ER and the plasma membrane and the ER and mitochondria (A and B) and their absence between the plasma membrane and the mitochondria (C). Note also the two peaks in A corresponding to the cortical ER and the nuclear envelope. To calculate distances between organelles, we wrote a code that runs through all the voxels of a given cellular structure (e.g., the ER membrane A) and computes their distance from the closest voxel of another structure of interest. (D–F) Graphs showing the probability of encountering certain organelles at a certain distance in the cases where the signal starts at the plasma membrane (D), the ER (E), and the mitochondria (F). Again, the close contacts between the ER and plasma membrane (D and E) and the ER and mitochondria are clearly visible. The code used to obtain these plots ran through all the voxels of a given structure (e.g., the plasma membrane in D). There were 20,000 points positioned on 200 concentric spheres with radii ranging from 1 to 200 nm, each containing 100 evenly distributed points. The center of these spheres was set into the center of the particular voxel's surface, facing the cytosol. Each of these 20,000 points was then checked and assigned to a particular structure, depending on the value of the voxel they fell into.

Figure 3

Intracellular geometry limits diffusion of substances compared to free solution. (A) Color coding of individual intracellular compartments: blue, cytosol; yellow, mitochondria; green, nucleus; gray, ER membrane. The resulting mask matrix formed the basis of most of the models presented. (B) One million test particles were placed at a point in the plasma membrane and allowed to diffuse in the cytosol, and their mean-squared displacement was calculated. (C–H) Comparison between the time course of the mean-squared displacement (MSD) in free solution and in individual intracellular compartments (results show MSD in the xy plane). The diffusion coefficient for calcium (223 μm²/s) was used as an example. The particles in this simulation did not react, bind, or cross the boundaries between the compartments. Solid lines, mean; dashed lines, ±SD.

Figure 4

Use of EM images to define 3D environment for stochastic modeling of calcium signaling. (A) Orange and blue spots show the random cytoplasmic distributions of calcium ions and calmodulin molecules, respectively, when superimposed upon the electron microscopy image used to define cell geometry in the stochastic simulation space. Other components of the model (transporters, calreticulin) are not shown for simplicity's sake. Slice thickness, 0.2 μm. This figure is a snapshot of Movie S2. (B) 3D box geometry illustrating stochastic release of calcium ions from a single IP₃R located on one box side. (C) PDE solutions compared with stochastic solutions for [Ca²⁺]_i averaged over a hemisphere centered at the channel mouth (radius 500 nm). The smooth yellow line represents the PDE solution, whereas other colored lines show the variability among many stochastic simulations.

Figure 5

Simulation of IP₃ release, diffusion, and channel activation in a 3D stochastic model. (A) Comparison of compartmental and stochastic modeling of IP₃ release from the plasma membrane and its binding to IP₃Rs in the ER. The stochastic simulation was based upon a fractional 3D cellular geometry with a depth of 266 nm. (B) IP₃ molecules released by phospholipase C diffuse from the plasma membrane and bind to the IP₃Rs in the ER (see also Movie S4). (C) When all four IP₃R subunits are in the activated state (i.e., with bound IP₃ and calcium at their activation site and no calcium at the inhibitory site), the channels open and release calcium. (D) Results of varying the number of IP₃Rs from 7000 to 28,000. This simulation was based upon a more complete cellular geometry (image at right) and a depth of 100 nm.

Figure 6

Spatial inhomogeneity during IP₃R activity. Local domains of high calcium concentration result in much higher activity of transport mechanisms/buffer saturation in close vicinity to the channels. (A) Mitochondria-rich section of the 3D reconstruction. (B) Cytosolic calcium concentration after release from two IP₃R clusters with five (left) and three (right) active IP₃Rs. PDE-based approach (model 2) was used. (C) The same experiment using the stochastic model. (D) Mitochondrial uniporter activity (assuming half-maximum activity at [Ca²⁺]_i = 19 μM) at various sites of the mitochondrial membrane (at time t = 4 ms), expressed as a percentage of the maximal uniporter activity (left y axis). (Upper) Stochastic solution; (lower) PDE solution. Also shown in the lower plot is the increase in mitochondrial matrix calcium concentration (right y axis, also at t = 4 ms), with ordinates as numbered areas of mitochondrial membrane.

Figure 7

Influence of local environment on IP₃R channel activity. Calcium release from a stochastically gated type II IP₃R was modeled using a 1D PDE model and organellar volume fractions (Fig. S2). (A–H) Graphs comparing results for two spatially discrete channels, one in an average location (A–D) and one facing a narrow space formed by an ER fold (E–H). A and E compare the nearby changes in cytosolic calcium concentration; B and F report the nearby ER lumenal calcium concentration; C and G report calcium binding to calmodulin; and D and H report calcium binding to calreticulin. Individual traces represent local concentrations at 20 (blue) and 50 nm (green) from the center of the channel mouth. The resulting increase in cytosolic calcium concentration and calmodulin saturation was much higher in the restricted space. (I and J) IP₃R locations in the ER membrane were randomly assigned. The volume fraction occupied by nearby organelles was used to define three categories: “close to mitochondria”, “close to plasma membrane”, or “far from either organelle”. The cutoff distance for the close categories was ∼75 nm. The IP₃Rs were manually opened at t = 0 and left open for 1 ms, and the time courses of calcium concentration in the cytosol (I) and the ER (J) at 10 nm (solid lines) and 25 nm (dotted lines) from the center of the channel were plotted. (K) Typical ER calcium concentration traces for a channel located at a location far from mitochondria and plasma membrane (blue) and a channel close to mitochondria (green) 50 nm from the center of the channel mouth. Note that opening of channels close to mitochondria resulted in a more pronounced decrease in ER calcium, but due to lower opening probability, the overall release was reduced. (L) Mean (±SE) ER calcium concentration at various distances from the IP₃R channel mouth during 100 ms of IP₃R channel activity.