Abstract
The essence of calciuminduced calcium release is once more illustrated in Fig. 5.1. This figure is very similar to Fig. 2.1 on page 24 except that the box surrounded by a thin red line is now slightly extended.
Keywords
 Probability Density Function
 Numerical Scheme
 Equilibrium Solution
 Invariant Region
 Closed Channel
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The essence of calciuminduced calcium release is once more illustrated in Fig. 5.1. This figure is very similar to Fig. 2.1 on page 24 except that the box surrounded by a thin red line is now slightly extended. This is meant to illustrate that the model is now extended to account for changes in the calcium concentration of the junctional sarcoplasmic reticulum (JSR) space (see Fig. 5.1); so we now consider a twodimensional (2D) model where the concentration of the dyad (\(\bar{x} =\bar{ x}(t)\)) and the JSR (\(\bar{y} =\bar{ y}(t)\)) vary, recalling that the bar notation indicates stochastic variables. The concentration of the cytosol and the network sarcoplasmic reticulum (NSR) are still kept constant and we still ignore Ltype calcium currents. An illustration of the mathematical model under consideration is given in Fig. 5.2.
The basic steps of the analysis of the 2D problem follow the steps of the analysis of the onedimensional (1D) problem. We will start our analysis of the 2D problem by formulating a 2 × 2 system of stochastic differential equations giving the dynamics of the calcium concentration of the dyad and of the JSR. This model will be used as a basis for Monte Carlo simulations. By following the steps above, we also derive a 2D deterministic equation describing the probability density functions of the open and closed states. A numerical method for this system will be presented and, again, we will find that it is reasonable to focus on steady state computations. The probability density model will be extended to account for open or closed state blockers and, as above, we will see that we can find very good closed state blockers for COmutations (see page 16).
5.1 2D Calcium Release
The process of calcium release illustrated in Fig. 5.2 can be modeled as follows:
where v _{ r } denotes the rate of release from the JSR to the dyad, v _{ d } denotes the speed of calcium diffusion from the dyad to the cytosol, and v _{ s } denotes the speed of calcium diffusion from the NSR to the JSR. Furthermore, \(\bar{\gamma }(t)\) is a stochastic variable taking on two possible values, zero and one, with (as above) zero denoting a closed channel and one denoting an open channel. The dynamics of \(\bar{\gamma }\) are governed by the Markov model under consideration. Furthermore, we always assume that
For the 2D case, we also assume^{Footnote 1} that
5.1.1 The 1D Case Revisited: Invariant Regions of Concentration
Suppose the speed of diffusion, v _{ s }, from the JSR to the NSR becomes very large. From (5.2), we observe that the limiting case when \(v_{s} \rightarrow \infty\) yields y = c _{1} and thus the problem is in 1D and can be written
which is exactly the problem we discussed in Chap. 2 (page 25). We analyzed this equation and saw that, when the channel is closed (γ = 0), the solution tends toward the equilibrium point represented by
and, when the channel is open, the equilibrium solution is given by
where
Based on this, we concluded that if the initial concentration is in the interval \(\left [c_{0},c_{+}\right ],\) the solution will always remain in this interval. The reason for this is that if the channel is closed, the solution will decrease toward c _{0} and, if the channel is open, the solution will increase toward c _{+}. For closed channels, c _{0} is a stable equilibrium and, similarly, if the channel is open, c _{+} is a stable equilibrium.
5.1.2 Stability of Linear Systems
Before we consider the 2D case, we need to recall some basic properties of linear systems of ordinary differential equations. For a system of the form
where A is a matrix and the unknown x is a vector, we know that the equilibrium solution x = 0 is stable, provided that the real part of all the eigenvalues of A is negative. However, the systems under consideration here are of the form
where b is a known vector. In the case of a nonsingular matrix A, the equilibrium solution is given by
and we are interested in the stability of this solution. To assess the stability, we define
and observe that
and, of course, e = 0 is a stable equilibrium of the system
provided that the real part of all the eigenvalues of A are negative. Therefore, the equilibrium solution (5.6) of the system (5.5) is stable under the same condition. With these observations at hand, we are ready to try to understand the dynamics of the system (5.1) and (5.2).
5.1.3 Convergence Toward Two Equilibrium Solutions
Our aim is now to understand the dynamics of the 2D case and we start by considering the system when the channel is closed.
5.1.3.1 Equilibrium Solution for Closed Channels
In this case, the system (5.1) and (5.2) is quite simple, since there is no communication between the dyad and the JSR. The system is
and the stable equilibrium solution of this system is given by
5.1.3.2 Equilibrium Solution for Open Channels
The more interesting case is when the channel is open. Then the system reads
and the equilibrium solution is given by
where
It is useful, but not surprising, to note that
since c _{1} is assumed to be larger than c _{0} (see (5.3)).
5.1.3.3 Stability of the Equilibrium Solution
Whether the equilibrium solution for open channels is stable remains to be seen. As noted above, this can be determined by invoking the eigenvalues of the system matrix, which, in this case, are given by
Since the matrix is symmetric, the eigenvalues are real, so it is sufficient to see if they are always nonpositive. The eigenvalues are given by
where obviously \(\lambda _{} <0\) for any \(v_{r},v_{d},v_{s}> 0.\) Hence, \(\lambda _{+} <0\) also remains to be shown. To this end, we start by assuming that \(\lambda _{+}> 0;\) so we assume that
with
and
We can safely multiply both sides of this inequality with something positive such as \(\sqrt{u} + v\) and we therefore find that
and, since \(v_{r},v_{d},v_{s}> 0\), this is a contradiction and we conclude that \(\lambda _{+} <0\) for all \(v_{r},v_{d},v_{s}> 0.\)
5.1.4 Properties of the Solution of the Stochastic Release Model
We have found that when the channel is closed, the equilibrium solution is given by
which is stable. Similarly, when the channel is open, the equilibrium solution is given by
and this solution is also stable. The solution of the model given by the system (5.1) and (5.2) will therefore tend toward \((x_{c},y_{c})\) whenever the channel is closed and toward \((x_{o},y_{o})\) whenever the channel is open. This will be illustrated in numerical simulations below.
5.1.5 Numerical Scheme for the 2D Release Model
To perform 2D stochastic simulations, we use the numerical scheme
where γ is computed according to the Markov model given by the reaction scheme
(see page 28), where k _{ oc } and k _{ co } are reaction rates that may depend on both the concentrations represented by x = x(t) and y = y(t).
5.1.5.1 Simulations Using the 2D Stochastic Model
We use the numerical scheme given by (5.11) and (5.12), where the parameters and functions involved are described in Table 5.1. The numerical solutions are given in Figs. 5.3 and 5.4. In the latter figure, we also indicate when the channel is open and closed (upper panel).
5.1.6 Invariant Region for the 2D Case
We observed in the 1D case that an invariant region for the numerical scheme used to compute approximate solutions of the stochastic model was useful for the probability density system, since it defined the interval in which to solve the system. Similarly, we will derive an invariant region for numerical solutions generated by the scheme (5.11) and (5.12) and this invariant region will define the geometry we will use to solve the probability density system.
Let us start by recalling the assumptions (5.3) and (5.4) and let us also assume that the time step \(\Delta t> 0\) is chosen such that
Define
and observe that
It now follows from the assumptions (5.3) and (5.4) that we have
Our aim is now to show that \(\Omega = (x_{},x_{+}) \times (y_{},y_{+})\) is an invariant region for solutions of the scheme (5.11) and (5.12). Because of (5.19), this means, in particular, that under the assumptions (5.3) and (5.4) the lowest possible calcium concentration of the JSR will always be larger than (or equal to) the highest calcium concentration of the dyad.
The numerical scheme (5.11, 5.12) can be written in the form
where
We will consider the properties of the functions F and G for x and y in the domain
and for 0 ≤ γ ≤ 1. Note that
by condition (5.14). In addition, we have
and
where we use (5.19) and (5.20). Similarly, we have
which is also positive by condition (5.14). Finally, we have
and
by (5.19) and (5.20). We now assume that
Then,
and
So we conclude that
Similarly,
and
so we conclude that
We have seen that under the assumptions (5.3), (5.4), and (5.14), it follows that, if
then also
and we therefore conclude that \(\Omega\) is an invariant region for the scheme of (5.11) and (5.12). This means that the probability density system will be solved in the domain defined by \(\Omega.\)
5.2 Probability Density Functions in 2D
In the 1D case considered above, we derived a model for the probability density functions. In the 2D case, we can follow exactly the same steps and arrive at a system of partial differential equations of the form
where
As in 1D, ρ _{ o } and ρ _{ c } denote the open and closed probability density functions, respectively, satisfying the integral condition
Here the domain can be taken to be
and the boundary conditions are again defined to ensure that there is no flux of probability out of the domain (see page 37).
5.2.1 Numerical Method for Computing the Probability Density Functions in 2D
To solve the system (5.21) and (5.22), we need to define a numerical method. For the 1D model (see page 37), we used an upwind scheme as presented by LeVeque [48]. Here, we use the 2D version of the same numerical method. Consider the partial differential equation
where a, b, and g are smooth functions of x and y. We let \(\rho _{i,j}^{n}\) denote an approximation of ρ at time \(t = n\Delta t\) for \((x,y) \in [x_{i1/2},x_{i+1/2}) \times [y_{j1/2},y_{j+1/2})\), where \(x_{i} = x_{} + i\Delta x,\,y_{j} = y_{+} + j\Delta y\), and
Here M _{ x } and M _{ y } denote the number of grid points along the x and y axes, respectively. The numerical approximation is defined by the scheme
where
In our simulations, this scheme is used for both equations (5.21) and (5.22) above, where the righthand sides are given by \(k_{co}\rho _{c}  k_{oc}\rho _{o}\) and \(k_{oc}\rho _{o}  k_{co}\rho _{c}\), respectively.
As pointed out above, the probability densities integrates to one (see (5.24)), and the discrete version of this condition reads,
where \(\rho =\rho _{o} +\rho _{c}\). Note that the initial conditions must be chosen such that this condition holds. The discretization parameters used throughout this chapter are given in Table 5.2.
5.2.2 Rapid Decay to Steady State Solutions in 2D
We observed in 1D that the timedependent probability density functions converge rapidly toward steady state solutions. This is illustrated in Fig. 2.7 on page 39. In Fig. 5.5, we show snapshots of the open probability density function at times 1, 2, 3, 5, 10, 20, 40, 70, and 100 ms and we observe that the solution converges toward an equilibrium solution with time. This is verified in Fig. 5.6, where we plot the (weighted) norm between the dynamic and stationary solutions for time t ranging from 0 to 150 ms and we see that the solution is quite close to equilibrium at t = 100 ms. This observation is useful because it implies that when we assess the effect of various theoretical drugs, it is sufficient to consider steady state solutions.
5.2.3 Comparison of Monte Carlo Simulations and Probability Density Functions in 2D
As in 1D, we want to compare the probability densities ρ _{ o } and ρ _{ c } computed by solving the probability density system (5.21) and (5.22) using the scheme (5.26) with Monte Carlo simulations based on the stochastic differential equations (5.1) and (5.2) solved by the numerical scheme (5.11) and (5.12). The comparison is undertaken in the same manner as in 1D. We simply run a number of Monte Carlo simulations for a long time and count the number of open states in small rectangles. The procedure is a direct generalization of the method used in 1D (see page 40).
The numerical solution of the probability density system is given in Fig. 5.7 and the associated solution based on Monte Carlo simulations is given in Fig. 5.8. As in 1D, we observe that the solutions are quite similar. In both these figures, we observe that the solutions stay inside a region bounded by a red curve. The red curve is computed by solving (5.7) and (5.8) for the closed state and (5.9) and (5.10) for the open state.
5.2.4 Increasing the Open to Closed Reaction Rate in 2D
In 1D, we observed that if we increased the reaction rate k _{ oc } from open to closed, the steady state probability density functions changed considerably (see page 46). We observed that the open probability decreased and the closed probability increased significantly. In Fig. 5.9, we study the same effect in 2D and again we observe that the open probability density function is considerably decreased when k _{ oc } is increased from one to three. The statistics of the solutions are given in Table 5.3 and we note that the total open probability is reduced considerably when k _{ oc } is increased from one to three. The expected dyad concentration (x) does not change very much, but the expected JSR concentration (y) increases significantly and this observation holds for both open and closed channels.
5.3 Notes

1.
The 2D stochastic model and the associated probability density functions are taken from Huertas and Smith [35], but some of the parameters are changed.
Notes
 1.
This is a technical assumption needed in an argument below.
References
M.A. Huertas, G.D. Smith, The dynamics of luminal depletion and the stochastic gating of Ca^{2+}activated Ca^{2+} channels and release sites. J. Theor. Biol. 246(2), 332–354 (2007)
R.J. LeVeque, Finite Volume Methods for Hyperbolic Problems (Cambridge Texts in Applied Mathematics, Cambridge, 2002)
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Tveito, A., Lines, G.T. (2016). TwoDimensional Calcium Release. In: Computing Characterizations of Drugs for Ion Channels and Receptors Using Markov Models. Lecture Notes in Computational Science and Engineering, vol 111. Springer, Cham. https://doi.org/10.1007/9783319300306_5
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DOI: https://doi.org/10.1007/9783319300306_5
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