130 CHAPTER 6. NUMERICAL RESULTS FOR THE INTRACELLULARCA2+ DYNAMICS
Figure 6.24: Solution of cytosolic concentration of 100 clusters with stochastic chan-nel transition of chanchan-nels at different time steps t = 5.139571 s, 5.439938 s, 5.689859s, 6.089905s, 6.339917s and 6.489918s.
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Figure 6.25: Solution of cytosolic concentration of 100 clusters with stochastic chan-nel transition of chanchan-nels at different time steps t = 6.889927 s, 7.189863 s, 7.589875s, 8.589918s, 10.589987s and 14.089910 s.
132 CHAPTER 6. NUMERICAL RESULTS FOR THE INTRACELLULARCA2+ DYNAMICS
Figure 6.26: The zoom of the spatial grid near the channels of level 9 and level 12.
UG [12], see Figure 4.7. A close up view of the local resolution at the ER membrane of the unstructured finite element mesh of level 9 and 12 are shown in Figure 6.26.
The number of nodes, elements and the minimum volume of the element in the domain at different levels is presented in Table 6.3. It can be observed that the minimum volume of elements is very small with finer meshes where we want to put a fine mesh at the membrane.
levels nodes elements min volume of element
0 729 3,072 1.04167·1008
8 26,403 146,627 3.10441
9 28,683 159,463 0.388051
10 30,963 172,299 0.0485064
11 33,243 186,135 0.0060633
12 35,523 197,971 0.000757912
Table 6.3: The number of nodes, elements and minimum volume of element in the domain, i.e. the element present at the membrane, at different mesh levels in cube.
In our numerical simulations we considered only one cluster arrangement. The numeri-cal results of different grid structures with deterministic opening of one channel are tested.
The left hand side of Figure 6.27 shows the average cytosolic Ca2+ against the time until t = 0.1 s and the maximum cytosolic Ca2+ over the number of nodes is presented in the right hand side of Figure 6.27. From these results, one can observe that mesh level 9 is
6.2. NUMERICAL RESULTS IN 3D 133
0 0.02 0.04 0.06 0.08 0.1
0.05 0.055 0.06 0.065 0.07 0.075
average cytosolic [Ca2+] [µM]
time [s]
level − 8 level − 9 level − 10 level − 11 level − 12
2.6 2.8 3 3.2 3.4 3.6
x 104 114
116 118 120 122 124 126
maximum cytosolic [Ca2+] [µM]
number of nodal points
Figure 6.27: The average solution of cytosolic Ca2+ at different levels versus time for an one open channel at left and at right the maximum cytosolic Ca2+ versus levels at the stationary solution.
10−1 100 101 102 103 104
10−2 10−1 100 101 102 103
distance from channel center [nm]
cytosolic calcium [µM]
Figure 6.28: The stationary Ca2+ concentration for an open channel with a distance of 200nm from the channel center directly at the ER membrane (solid) and perpendicular to the membrane (dashed).
sufficient to consider for achieving the numerically convergent solutions. Therefore, the level 9 is considered in all other simulations.
In Figure 6.28, the steady state Ca2+ concentration for an open channel with distance of 4000nm from channel center directly at the ER membrane and perpendicular to the membrane. From this result, one can observe that the spatial profile of the Ca2+ concen-tration is similar at the vertical and the horizontal directions. The solution is plotted using log-log scale. The maximum Ca2+ concentration at the channel center is 114.400 µM.
134 CHAPTER 6. NUMERICAL RESULTS FOR THE INTRACELLULARCA2+ DYNAMICS
Figure 6.29: The spatial profile of the Ca2+ concentration at left and the zoom around the channel mouth at the ER membrane at right at time t= 0.1 s.
The spatial profile of the Ca2+ concentration is plotted in the left hand side of Fig-ure 6.29 at time t = 0.1 s and the zoom around the channel mouth at the ER membrane is depicted in the right hand side of Figure 6.29. Here, we can see a strong profile of the Ca2+ concentration from the channel center to some hundreds of nano meters. Usually this strong localization restricts the time steps in numerical simulations.
Next, the numerical results with the hybrid simulation are presented, which solves the deterministic and stochastic equations simultaneously. The algorithm is based on a recently introduced approach for simulating hybrid models of chemical reaction kinetics in spatially homogenous systems [2], as explained in Section 3.3. Here, the numerical results are presented based on the hybrid method for a single channel system. Analogously we can generalize to multi channel systems.
A stochastic model is adopted for the gating of subunits. This stochastic model is based on the DeYoung-Keizer-model for the subunit dynamics [29]. An IP3R consists of four identical subunits. There are three binding sites on each subunit in the framework of that model: An activating site for Ca2+, an inhibiting Ca2+ site and an IP3 binding site. The three binding sites allow 8 different states Xijk of each subunit. The index i indicates the state of the IP3 site, j the one of the activating Ca2+ site andk the state of the inhibiting Ca2+ site. An index is 1, if an ion is bound and 0 if not. In 3D simulations, one extra state XACT is considered where it is assumed that the channel is open, if at least three of the subunits are in XACT, i.e. they have bound to Ca2+ and IP3 at the activating site. This additional state enables us to fit short mean open and mean close times. These could not be fitted using standard DeYoung-Keizer-model. For more results regarding open
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probability, mean and close times can be found in paper by R¨udiger et al. [80]. Then we associate stochastic variablesX000,X001,. . .,XACT to each channel. These variables count the number of subunits which are in the respective states.
0 1 2 3 4 5
0 0.2 0.4 0.6 0.8 1
time [s]
open channels
Figure 6.30: The opening and closing of the channel in one cluster versus time.
0 1 2 3 4 5
0.05 0.055 0.06 0.065 0.07
time [s]
average cytosolic [Ca2+] [µM]
0 1 2 3 4 5
0 20 40 60 80 100 120 140
time [s]
maximum cytosolic [Ca2+] [µM]
Figure 6.31: The cytosolic Ca2+ concentration over time;left: average value, right: max-imum value at the channel mouth.
In Figure 6.30, the opening and closing of single channel against the time until t= 5s are depicted. Here we can observe the rapid changings of the channel opening and closings.
The corresponding average and the maximum cytosolic Ca2+concentrations at the channel mouth are depicted in Figure 6.31.
136 CHAPTER 6. NUMERICAL RESULTS FOR THE INTRACELLULARCA2+ DYNAMICS