Modelling calcium microdomains using homogenisation

Abstract

Microdomains of calcium (i.e., areas on the nanometer scale that have qualitatively different calcium concentrations from that in the bulk cytosol) are known to be important in many situations. In cardiac cells, for instance, a calcium microdomain between the L-type channels and the ryanodine receptors, the so-called diadic cleft, is where the majority of the control of calcium release occurs. In other cell types that exhibit calcium oscillations and waves, the importance of microdomains in the vicinity of clusters of inositol trisphosphate receptors, or between the endoplasmic reticulum (ER) and other internal organelles or the plasma membrane, is clear. Given the limits of computational power, it is not currently realistic to model an entire cellular cytoplasm by incorporating detailed structural information about the ER throughout the entire cytoplasm. Hence, most models use a homogenised approach, assuming that both cytoplasm and ER coexist at each point of the domain. Conversely, microdomain models can be constructed, in which detailed structural information can be incorporated, but, until now, methods have not been developed for linking such a microdomain model to a model at the level of the entire cell. Using the homogenisation approach we developed in an earlier paper [Goel, P., Friedman, A., Sneyd, J., 2006. Homogenization of the cell cytoplasm: the calcium bidomain equations. SIAM J. Multiscale Modeling Simulation, in press] we show how a multiscale model of a calcium microdomain can be constructed. In this model a detailed model of the microdomain (in which the ER and the cytoplasm are separate compartments) is coupled to a homogenised model of the entire cell in a rigorous way. Our method is illustrated by a simple model of the diadic cleft of a cardiac half-sarcomere.

Figure 1

In the homogenised region we assume the structure is given by this periodic unit. The inner network, which has a square cross section, represents the SR, and the remaining space between the SR and the outer cube represents the cytosol. ε is the period of the SR network divided by the assumed length of the homogenised region, and so is dimensionless.

Figure 2

Schematic diagram of a coupled homogenised/separated model.

Figure 3

Calcium transport in a half-sarcomere: Calcium enters through the L-type channels, stimulating release from the RyR. Some of the calcium in the cytosol and SR is bound to calcium buffers. The increase in calcium concentration in the cytosol allows calcium to bind to the myofilaments, which activates contraction. Calcium is removed from the cytosol by the SERCA pump, SL calcium pump and the NCX.

Figure 4

The geometry of the half-sarcomere as used in the model. A. The cylindrical geometry formed by rotating the plane in B. B. Roman numerals denote regions where different model equations are used. I: homogenised region, II: cytosol in the non-homogenised region, III: SR in the non-homogenised region. Lowercase letters denote the boundaries. a: L-type channels, b: NCX, SL pump and background flux, c: RyR, d: SERCA pump, e: Homogenised/non-homogenised cytosolic boundary, f: Homogenised/non-homogenised SR boundary. The numbered points show where readings of calcium concentration were taken (see Results).

Figure 5

Results when using the non-buffering SERCA pump. The numbers in parentheses in the graph headings refer to the positions given in Fig. 4 B.

Figure 6

Fluxes integrated over time when using the non-buffering SERCA pump: A: The time integral of the fluxes involved in relaxation, and the percentage they contribute. These are close to the percentages found experimentally from a rabbit, as given in [4]. B: The time integral of the flux through the L-type channels and the flux through the RyR.

Figure 7

Calcium concentration gradients along the cross sections given in diagram E, after 5 ms, 50 ms and 200 ms. Plots are not given along those cross sections where the calcium concentration does not vary significantly. A: Cytosolic calcium concentration along cross sections 1 and 2. B: Cytosolic calcium concentration along cross sections 3 and 4. C: Cytosolic calcium concentration along cross section 7 and SR calcium concentration along cross section 3. C: Cytosolic calcium concentration along cross section 8 and SR calcium concentration along cross section 8. E: The positions of the cross sections.

Figure 8

Diagram of the compartments and fluxes in the compartmental model.

Figure 9

The compartmental model results compared to the three-dimensional model results.

Figure 10

A comparison of the compartmental model results and the three-dimensional model results, where the membrane voltage is fixed at −84.39 mV and the L-type flux is as given in the figure.

Figure 11

A comparison of the compartmental model results and the three-dimensional model results, where the membrane voltage is fixed at −84.39 mV and the L-type flux is as given in the figure.

Figure 12

A comparison of the compartmental model results and the three-dimensional model results, where the L-type flux is as given in Fig. 17 and the membrane voltage is as given in the figure.

Figure 13

The three-dimensional model results with the RyR and L-type channels shifted as shown in the figure, compared to the compartmental model results from Fig. 9 and compartmental model results where the cleft is modeled as two separate compartments.

Figure 14

Results when using the non-buffering SERCA pump, the buffering SERCA pump with P_c = 52.7 μmol/L Cyt and the buffering SERCA pump with P_c = 105.4 μmol/L Cyt. The calcium concentration has been measured at position 4 in Fig. 4 B. A: Cytosolic calcium concentration. B: SR calcium concentration.

Figure 15

A: State diagram of the SERCA pump. X₁ gives the surface density of pump protein on the cytosolic side with no calcium bound, X₂ gives the surface density of pump protein on the cytosolic side with 2 calcium ions bound and Y₁ and Y₂ are analogous on the SR side. B: Reduced state diagram of the SERCA pump, formed by assuming the rate constants k₁, k₋₁, k₃ and k₋₃ are fast. X gives the surface density of pump protein on the cytosolic side, and Y gives the surface density on the SR side.

Figure 16

State diagram of the NCX model: E denotes the carrier protein, A denotes the carrier protein with 3 bound sodium ions and B denotes the carrier with 1 bound calcium ion. The subscript o refers to the outside of the cell, and i to the inside.

Figure 17

The flux through the L-type calcium channels integrated over boundary a, which is used as an input into the model. The LabHEART model has been used to generate the data. [19].

Figure 18

The RyR gating scheme given by Shannon et al. [24]. O gives the open probability of the RyR and there is no calcium release in the other three states. c gives the calcium concentration in the cleft.

Figure 19

Voltage across the SL membrane, which is used as an input into the model. The NCX current and background flux are dependent upon the membrane voltage. The LabHEART model has been used to generate the data. [19].