Model
Project safety
Based on the brilliant works of the wet group, as well as for providing a direction for further works, we built two corresponding mathematical models, which could simulate some parts of our experiments in an extent. These two models are respectively counting numbers of calcium sparks and calcium releasing extent affected by phosphorylation of RyR2, and the relationship between phosphorylation level of RyR2 and intrabody AR185. These two models, on the one hand, give us a direction for further works to improve experiment designing and evaluate the therapeutic outcome. On the other hand, the experimental results equally feedback to the models to adjust parameters.
RyR2 phosphorylation and calcium sparks
Hypotheses
To simplify the calculation model, we made some simplification to models as well as ensuring its accuracy to an extent. On the ground of previous studies and combining with relative theoretical researches, we made following hypotheses as the basis of our models:
1. The opening of each RyR2 is a stochastic process only affected by local calcium concentration; 2. The opening of each RyR2 is persistent; 3. Calcium current induced by the opening of RyR2 is instantaneous; 4. The location of each RyR2 could be labeled by a coordinate system (X, Y, Z) that cell center as the origin;
Signs
| Model Parameters | Definition | Value | Unit |
|---|---|---|---|
| F | Faraday constant | 96485.33289±0.00059 | C/mol |
| Pmax·open | RyR2 maximum open probability | 0.3 | ms-1 |
| n | Hill coefficient of RyR2 open probability | 1.6 | |
| Ksensitivity | Ca2+ sensitivity coefficient | 15 | μM |
| Topen | The opening time of RyR cluster | 8.0 | ms |
| ICa2+ | Calcium ion current released by each RyR2 | 20 | pA |
Modeling
According to the assumption that opening probability of RyR2 is determined by local calcium gradient and whether to open the channel is stochastic, we choose to use the Sigmoid function as an initial model. Its base expression as follow:
In our model, the local calcium gradient is set as an independent variable, expressed by [Ca2+]local. Considering the cellular physiological condition the opening probability of RyR2 could not reach one hundred percent, we made a maximum limitation of opening probability, expressed by Pmax·open. The sensitivity of RyR2 to calcium gradient could affect the opening probability as well, and the sensitivity degree is a constant in normal condition, expressed by Ksensitivity. Besides, RyR2 is a tetramer located on sarcoplasmic reticulum so that we bring in a Hill coefficient, expressed as n.
Collectively, we get the opening probability of RyR2 as follow:
Meanwhile, we build a function-valued 1 and 0 to express opening state. We define a random number g ranging from 0~0.3 to compare with Popen. When g is greater than Popen, that function is valued 1, indicating channel open, vice versa. The function is expressed as follow:
Opening process is a one-unit impulse function and its constraint condition as follow:
To calculate each calcium flux released from RyR2 opening, we found from relative articles that each calcium current caused by RyR2 opening (expressed as ICa2+)is about 20pA and duration is about 8ms (expressed as T0pen).
According to:
We got equivalent calcium flux expression from each opening of RyR2:
Above all, calcium current at each opening moment expressed as:
Total calcium current in cell express as follow:
