Model
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:
Application
We put all kinds of a constant into the model and calculate in MATLAB to obtain RyR2 opening probability in different local calcium concentration. The result shows RyR2 opening probability increased along with the rise of local calcium concentration. And when calcium concentration reaches a certain extent, RyR2 opening probability approach the maximum we assumed before.
So, we get a conclusion that in failing cardiac cells hyperphosphorylation of RyR2 and calcium channel instability lead to RyR2 opening probability higher than normal condition, subsequently results in calcium leak, which increases the number of calcium sparks.
Intrabody AR185 and phosphorylation of RyR2
We make this model mainly for stimulating the competition between the protection by intrabody and phosphorylation by protein kinase. A reversible binding holds the protection on RyR2 of our intracellular expressed antibody——AR185. We think it is the same as the classic antibody-antigen binding course, that consists of binding and separation phases.
Equally, there is a phosphorylation process happened when a cardiac muscle cell is under attack by heart failure. Because of so many kinds of influence factors when heart failure happened, we chose the targeting point of our intrabody and simplified the influence process into a protein kinase-mediated phosphorylation process of a single point S2808 on RyR2.
Two processes compete for each other in an influenced cardiac muscle cell. Under the circumstance without the protection of intrabody AR185, most vulnerable sites on the RyR2 will be phosphorylated step by step and affect its channel function. If our intrabody competitively occupies the key site S2808, it will block the phosphorylation so that a part of RyR2 could maintain its regulated calcium releasing function.
We made many hypotheses in our model because that heart failure caused by different stimulations have different pathophysiologic environments. In our limited knowledge, we assume that the activity of protein kinase A (PKA) increase along with a timeline of the heart failure progression. And the activity is reflected on the formula by the pattern of the concentration of PKA. Besides, the concentration of ATP is considered as a constant. The total quantity of RyR2 includes the bare RyR2 that without any phosphorylated site and intrabody binding, phosphorylated RyR2 and the RyR2 bound by intrabody AR185.
According to the law of mass action, we could calculate the reaction rate in different condition:
Without intrabody AR185:
Collectively, we can calculate the phosphorylating rate of RyR2:
After integral calculation, the phosphorylated RyR2 become a dependent variable changed with time in failing state.
With intrabody AR185:
When AR185 is expressed in cardiac cell, there would be two reaction compete with each other.
It is easy to derive the same equations as above.
The simultaneous equations could be derived as follow:
Application
We respectively compare two conditions, without and with AR185 competing on RyR2, and calculate the equation set in MATLAB.
Without AR185:
With AR185:
Consequently, without the intrabody AR185, RyR2 would be catalyzed by PKA and about 50 percent of RyR2 were phosphorylated in our simulation. However, when AR185 expressed in cell plasma, because of the high affinity between AR185 and S2808 site on RyR2, about only 25 percent RyR2 were finally phosphorylated.
