Computing Characterizations of Drugs for Ion Channels and Receptors Using Markov Models pp 23-54 | Cite as

# One-Dimensional Calcium Release

## Abstract

The contraction of a single cardiac cell is initiated by an increase in the transmembrane potential leading to opening of the so-called L-type calcium channels (LCCs). When these channels are open, calcium flows into a rather small space called the dyadic cleft (often simply referred to as the dyad), leading to a locally increased concentration of Ca^{2+} ions. This increased concentration leads to the opening of the ryanodine receptors (RyRs), which control the flow of calcium from the internal stores referred to as the sarcoplasmic reticulum (SR). This process is referred to as the calcium-induced calcium release (CICR) and is of vital importance in the functioning of the heart. A schematic description of the process is given in Fig. 2.1.

## Keywords

Probability Density Function Markov Model Sarcoplasmic Reticulum Steady State Solution Invariant Region^{2+}ions. This increased concentration leads to the opening of the ryanodine receptors (RyRs), which control the flow of calcium from the internal stores referred to as the sarcoplasmic reticulum (SR). This process is referred to as the calcium-induced calcium release (CICR) and is of vital importance in the functioning of the heart. A schematic description of the process is given in Fig. 2.1.

Our aim is therefore to understand in some detail what is going on in the process illustrated in Fig. 2.1. However, this figure is in itself a huge simplification of the complex CICR process. The cell consists of 10,000 to 20,000 dyads, each dyad having up to 100 RyRs, and human ventricles consist of billions of cells. Our aim is to focus entirely on a very small but essential element in the CICR mechanism.

We model the release of Ca^{2+} ions from the SR to the dyad by formulating a stochastic differential equation governing the concentration of Ca^{2+} ions in the dyad. The model will be studied both numerically and analytically and we show how the solution’s properties depend on the parameters defining the model. Next, we will derive a deterministic partial differential equation (PDE) giving the probability density function of the states of the Markov model. Although the transition from a stochastic model to a deterministic model for the probability density functions is classical by now, we will spend some time deriving the equations in detail because the transition from stochastic to deterministic is such a wonderful piece of insight. Furthermore, we will provide detailed comparisons of Monte Carlo simulations based on the stochastic model and the probability density functions. In subsequent chapters, we will develop the model further by using two small spaces, the dyad and the JSR (see Fig. 2.1), allowing for different concentrations of Ca^{2+} ions. This leads to a two-dimensional (2D) problem.

Finally, we will take the LCCs into account. This leads to a 2D problem depending on one parameter: the transmembrane potential.

In these notes, we will use the concept of dimension in two different, but related, ways. In the first version of the stochastic model of CICR, we will model only the concentration of Ca^{2+} in the dyad and we will refer to the model as one dimensional (1D). When a deterministic model governing the probability density function of the states of the Markov model is derived, that model is also 1D in the sense that it depends on one spatial variable; the concentration of Ca^{2+}. Next we move to two concentrations (in the dyad and the JSR), leading to a 2D stochastic model in the sense that it is a 2 × 2 system of stochastic ordinary differential equations. The associated model governing the deterministic probability density functions is also 2D in the sense that the model depends on two spatial variables: the concentration of Ca^{2+} in the dyad and in the JSR. So the general rule is that the number of different concentrations allowed in the system of stochastic ordinary differential equations carries over to the spatial dimension of the deterministic system of PDEs governing the probability density functions of the states involved in the Markov model. Furthermore, the number of states in the Markov model decides the number of equations in the deterministic system of PDEs.

## 2.1 Stochastic Model of Calcium Release

Suppose that the cytosolic Ca^{2+} concentration is given by *c*_{0} and the SR concentration is given by *c*_{1}; we assume both to be constant and that \(c_{1} \gg c_{0}\). We want to model the concentration \(\bar{x} =\bar{ x}(t)\) in the dyad located between the cytosol and the SR (see Fig. 2.2). Throughout these notes, we will use a bar to indicate stochastic variables.

*v*

_{ r }denote the speed of release (when the channel is open) and let

*v*

_{ d }be the speed of diffusion; both are non-negative. Then a stochastic model of the concentration \(\bar{x} =\bar{ x}(t)\) in the dyad is given by

*k*

_{ oc }and

*k*

_{ co }as reaction rates that may depend on the concentration. Markov models were introduced on page 4 but let us recall that the reaction rates

*k*

_{ oc }and

*k*

_{ co }basically indicate the tendency of a channel to change state. So, if the channel is open, the probability that the channel changes from open to closed in a very short time interval \(\Delta t\) is given by \(\Delta tk_{oc}\) and, similarly, if the channel is closed, \(\Delta tk_{co}\) is the probability that it becomes open in the time interval \(\Delta t\). This means that the higher the rate

*k*

_{ co }, the more likely it is that the channels are open. This property will be used repeatedly in what follows.

### 2.1.1 Bounds of the Concentration

*t*=

*t*

_{0}, the channel is closed (

*γ*= 0), that the concentration is given by

*x*(

*t*

_{0}) =

*x*

_{0}, and that the channel remains closed for \(t\leqslant t_{0} + \Delta t.\) Then, in the interval \(t_{0}\leqslant t\leqslant t_{0} + \Delta t,\) the dynamics are given by the deterministic equation

^{1}

*x*(

*t*) of the dyad approaches

*c*

_{0}(the cytosolic concentration) at an exponential rate. The decay is faster for larger values of the diffusion velocity

*v*

_{d. }By consulting Fig. 2.2 we see that this is quite reasonable; if we close the release from the SR, the concentration of the dyad will gradually approach the concentration of the cytosol.

*x*(

*t*

_{0}) =

*x*

_{0}. We can rewrite this in the form

*x*(

*t*) of the dyad approaches

*c*

_{+}at an exponential rate. Furthermore, we note that the rate increases with \(v_{r} + v_{d}\). Note also that

*c*

_{+}<

*c*

_{1}and when it is closed, the concentration approaches

*c*

_{0}.

*c*

_{0}or larger than

*c*

_{+}. We therefore have

*c*

_{+}approaches

*c*

_{1}if

*v*

_{ r }fixed and we let

*v*

_{ d }approach zero. Then

*c*

_{+}approaches

*c*

_{1}, which is reasonable since calcium will be poured into the dyad, but the connection to the cytosol is almost closed and thus the dyadic concentration will increase until it reaches an equilibrium with the SR concentration.

### 2.1.2 An Invariant Region for the Solution

The invariant region (2.5) deserves a comment, since it will become quite useful later. Suppose that the initial concentration of the dyad is somewhere in the interval defined by *c*_{0} and *c*_{+}. Then, we have seen that if the channel is either closed or open, the solution remains in this interval as long as the channel does not change state. When the channel changes state, say, at time \(t = \Delta t\), we have a new initial condition in the interval *c*_{0} and *c*_{+} and we can solve the equation deterministically once more and the solution will remain in the interval. The process can be repeated over and over and the solution will always remain in the interval *c*_{0} and *c*_{+}. This property is useful, because it directly implies that the probability of being outside this interval is zero, which is what we need when we want to define boundary conditions for the model defining probability density functions.

### 2.1.3 A Numerical Scheme

*γ*

_{ n }takes on the value zero (closed) or one (open). The value of

*γ*

_{ n }is computed as follows: Let \(\sigma _{n}\) be a random number in the unit interval. Assume that \(\gamma _{n-1} = 0\). Then, if \(k_{co}\Delta t>\sigma _{n}\), we set

*γ*

_{ n }= 1, but if this condition does not hold, we set

*γ*

_{ n }= 0. Similarly, assume that \(\gamma _{n-1} = 1\). Then, if \(k_{oc}\Delta t>\sigma _{n}\), we set

*γ*

_{ n }= 0, but if this condition does not hold, we set

*γ*

_{ n }= 1.

### 2.1.4 An Invariant Region for the Numerical Solution

### 2.1.5 Stochastic Simulations

## 2.2 Deterministic Systems of PDEs Governing the Probability Density Functions

We have seen that model (2.6) can be studied using Monte Carlo simulations based on the numerical scheme (2.7). Such simulations clearly give some insight into the dynamics. In addition to the simulations shown above, we can use the numerical scheme to see the effect of changing the rates of the Markov model and the other parameters of the model. However, it is tricky to compare solutions of simulations based on stochastic processes because the results vary from simulation to simulation anyway. So we are faced with the following question: Is the difference in solutions from one computation to another due to stochastic effects or are they due to changes of parameters? This matter becomes especially pertinent when we introduce theoretical drugs, because we want to compare solutions with and without application of the theoretical drug. It is tempting to derive some sort of statistics based on the simulation results and then compare the solutions computed based on two sets of parameters based on the statistics.

By running numerous simulations, we can add the results and compute probability density functions based on the stochastic simulations. Exactly how this can be done will be explained below. However, it turns out that the probability density functions can also be computed by solving a deterministic system of PDEs. In this section we show how to derive this system of PDEs. We will see below that this is quite useful, because it is much easier to compare solutions of deterministic differential equations than stochastic solutions. By analyzing the deterministic system of PDEs we can also, analytically, derive properties of the process that would be very hard to derive based on direct analysis of the stochastic model (2.6).

### 2.2.1 Probability Density Functions

*t*, the probability of the channel being open and the concentration \(\bar{x} =\bar{ x}(t)\) being in the interval \((x,x + \Delta x)\) is given by

*ρ*

_{ c }is the probability density function of the channel being in the closed state. Note that

The probability density functions *ρ*_{ o } and *ρ*_{ c } contain a great deal of information about the process under consideration. At every point in time, we can understand how likely it is that the concentration is in a certain interval for a given state of the channel. It is therefore of great interest to be able to compute these functions.

### 2.2.2 Dynamics of the Probability Density Functions

Now, we are interested in understanding how *ρ*_{ o } and *ρ*_{ c } change dynamically. Consider *ρ*_{ o } and suppose that, for a given *x* and *t*, the density *ρ*_{ o }(*x*, *t*) is known. Over a small time interval, several things can happen that will affect the density: a) the channel can change from open to closed (reducing *ρ*_{ o }), b) the channel can change from closed to open (increasing *ρ*_{ o }), and, finally, c) the concentration can move from outside the interval \((x,x + \Delta x)\) to inside this interval or the concentration can move from inside the interval \((x,x + \Delta x)\) to outside this interval.

Here cases a) and b) are handled by the Markov model and we will return to that issue below, but we will start by taking care of the change in probability density due to changes in concentration. It turns out that this part will be governed by an advection^{2} equation and we will start by considering two very special cases illustrating how the probability is advected in the absence of a Markov model.

### 2.2.3 Advection of Probability Density

We start by considering two very special cases in which we just assume that the channel is always open or the channel is always closed.

#### 2.2.3.1 Advection in a Very Special Case: The Channel Is Kept Open for All Time

*t*= 0 and that it is given by a very simple function,

*ρ*

_{ o }= 0 for values of

*x*outside the interval \(\tilde{\Omega }\). Here

*h*is assumed to be a given positive number and \(\tilde{c} = \frac{1} {2}(c_{0} + c_{+})\), where we recall that

*ρ*

_{ c }= 0 for all values of

*x*and, since we have somehow forced the channel to remain open, nothing will happen to

*ρ*

_{ c }.

*x*

_{0}in the interval \(\tilde{\Omega }\), we know that the concentration will develop according to the ordinary differential equation

*t*for ten values of initial data

*x*

_{0}in the interval \(\tilde{\Omega }\), using

*h*= 20 \(\upmu\) M. The figure illustrates that the probability density function

*ρ*

_{ o }, in this special case of a forced open channel, is simply advected in time and the advection is clearly governed by the speed of

*x*=

*x*(

*t*), which is given by \(x^{{\prime}}(t) = a_{o}(x)\).

#### 2.2.3.2 Advection in Another Very Special Case: The Channel Is Kept Closed for All Time

*ρ*

_{ c }= 0 for values of

*x*outside the interval \(\tilde{\Omega }\). Again we pick any initial concentration

*x*

_{0}in the interval \(\tilde{\Omega }\) and recall that the concentration evolves as

*x*

_{ c }(

*t*;

*x*

_{0}) as a function of

*t*for ten values of initial data

*x*

_{0}in the interval \(\tilde{\Omega }\). Again we observe that the probability density function is simply advected according to the speed of

*x*=

*x*(

*t*), which is given by \(x^{{\prime}} = a_{c}(x)\).

#### 2.2.3.3 Advection: The General Case

We have seen how the probability density functions evolve in two very special cases. Next we consider the general case of how the probability density functions are advected when the state of the channel is kept fixed, and we focus on the probability density function of the open state.

*J*

_{ o }(

*x*,

*t*) denote the flux per time of the probability across the point

*x*at time

*t*. A positive flux at

*x*indicates a flux of probability into the domain \(\left (x,x + \Delta x\right )\) and a positive flux at \(x + \Delta x\) indicates a flux of probability out of the interval. This gives

### 2.2.4 Changing States: The Effect of the Markov Model

*t*. If we ignore the advection of concentration, handled above, we find that the probability density changes as follows from time

*t*to time \(t + \Delta t:\)

### 2.2.5 The Closed State

### 2.2.6 The System Governing the Probability Density Functions

#### 2.2.6.1 Boundary Conditions

The boundary conditions are set up to avoid the leak of probability across the boundary. Hence we need the fluxes \(a_{o}\rho _{o}\) and \(a_{c}\rho _{c}\) to be zero for *x* = *c*_{0} and \(x = c_{+}.\) Note that \(a_{o}(c_{+}) = a_{c}(c_{0}) = 0,\) so we require that \(\rho _{o}(c_{0}) = 0\) and \(\rho _{c}(c_{+}) = 0.\)

These conditions are fine as long as we know that the concentration is always in the interval bounded by *c*_{0} and *c*_{+}. However, we may be interested in studying initial concentrations outside this interval.^{3} Then we can extend the computational domain and use zero Dirichlet boundary conditions on the new computational domain.

## 2.3 Numerical Scheme for the PDF System

*a*and

*h*are smooth functions of

*x*. We let \(\rho _{i}^{n}\) denote an approximation of

*ρ*at time \(t = n\Delta t\) for \(x \in [x_{i-1/2},x_{i+1/2})\), where \(x_{i} = c_{0} + i\Delta x\), with

*M*> 1. The numerical approximation is defined by the scheme

## 2.4 Rapid Convergence to Steady State Solutions

*ρ*

_{ c }(

*x*,

*t*). Here we have plotted the logarithm of the distribution to highlight the small but significant probability densities for the channel being closed at high concentrations and again we note rapid convergence toward equilibrium.

## 2.5 Comparison of Monte Carlo Simulations and Probability Density Functions

*t*

^{∗}= 1 s. The Monte Carlo-based solution is computed by dividing the interval \([c_{0},c_{+}]\) into 100 intervals and then counting the number of open states in each interval. The counting is performed over a period of time where we assume that the histogram has reached a stationary shape. In Fig. 2.9 the counting is based on the time interval running from \(t = t^{{\ast}}/2\) to

*t*=

*t*

^{∗}, with

*t*

^{∗}= 1 s. By considering the simulations shown in Fig. 2.7, we know that in this interval the probability density functions have reached their steady state solutions. In the figure, the histogram is computed running 500 Monte Carlo simulations. The figure clearly shows that the probability density approach gives the average of a large number of Monte Carlo simulations. We will see this repeated over and over in this text.

At steady state, we observe that it is quite unlikely that we have a low concentration combined with an open channel and it is quite likely that we have a large concentration (close to \(c_{+} = 91\ \mathrm{\upmu M}\)) combined with an open channel. There is a boundary layer close to the upper possible concentration, which means that the channel tends to be open and the concentration tends to be close to its maximum value.

*c*

_{+}of the calcium concentration and the channel tends to be open.

## 2.6 Analytical Solutions in the Stationary Case

*a*

_{ c }< 0 and

*a*

_{ o }> 0 for \(x \in \left (c_{0},c_{+}\right )\) and thus we have

*A*=

*A*(

*x*) as

*c*is a constant. We can find

*c*by observing that

## 2.7 Numerical Solution Accuracy

Since we have a steady state analytical solution, we can evaluate the accuracy of the numerical method under consideration. However, to do so, we will first clarify how we compute stationary solutions using the numerical scheme.

### 2.7.1 Stationary Solutions Computed by the Numerical Scheme

*t*= 0, it will hold for all subsequent time steps. More precisely, if we define

*r*

^{0}= 1, then, by the construction of the scheme, we have

*r*

^{ n }= 1 for all

*n*≥ 1. Since the solution we are considering converges rapidly to a stationary solution, it is useful to be able to compute the stationary solution directly. The stationary version of the scheme reads

*r*

^{ n }= 1 is added to obtain a unique solution. When this condition is added, the stationary version of the system can be written in the form

*A ρ*= 0. Therefore, using Matlab terminology, we can find the stationary solution by first computing

### 2.7.2 Comparison with the Analytical Solution: The Stationary Solution

*δ x*is the mesh parameter \(\Delta x\) used in the coarsest simulation in the convergence study. The difference between the analytical solution

*ρ*and numerical solution \(\hat{\rho }\) is measured by

*i*runs over the nodes in the inner interval.

Error of the numerical computations as the mesh is refined. The convergence is first order

\(\Delta\) x | Error | Error/\(\Delta x\) | |
---|---|---|---|

0.909 | 0.086 | 0.095 | |

0.455 | 0.036 | 0.078 | |

0.227 | 0.016 | 0.072 | |

0.114 | 0.008 | 0.069 | |

0.057 | 0.004 | 0.066 | |

0.028 | 0.002 | 0.064 | |

0.014 | 0.001 | 0.063 |

## 2.8 Increasing the Reaction Rate from Open to Closed

*k*

_{ oc }from one to three. This means that the channel is much more prone to be closed and we see that this changes the probability density function

*ρ*

_{ o }considerably. For completeness, we also plot the closed probability density functions (lower panel) and observe that, when

*k*

_{ oc }is increased, there is a high probability of the channel being closed and the concentration being quite low. All the other parameters used in the model are as specified on page 29.

## 2.9 Advection Revisited

In the derivation of system (2.22) above governing the probability density functions of the states of the Markov model, we found it useful to consider a case representing the pure advection of probability density. Let us now see that we can find the same solution using system (2.22), that is,

*ρ*

_{ c }(

*x*, 0) = 0 and that

*ρ*

_{ o }= 0 for values of

*x*outside the interval \(\tilde{\Omega }\); for other notation see page 32. Furthermore, we assume that

*k*

_{ oc }= 0 ms

^{−1}(if the channel is open, it remains open) and

*k*

_{ co }= 1 ms

^{−1}. Then, the solution of system (2.41) with the given initial conditions is given by

^{4}

*r*solves the pure advection equation

*a*(

*x*) =

*a*

_{ o }(

*x*) and the initial condition

*r*(

*x*, 0) =

*ρ*

_{ o }(

*x*, 0).

*ρ*

_{ o }of this problem in the left panel and in the right panel we repeat the solution given in Fig. 2.5, where the pure advection case was studied by solving a series of ordinary differential equations; see page 33.

*k*

_{ co }= 0 ms

^{−1}and

*k*

_{ oc }= 1 ms

^{−1}and we use the initial conditions given by (2.14). In Fig. 2.14 we show (left panel) the solution

*ρ*

_{ c }of this problem computed by solving the pure advection problem

*a*(

*x*) =

*a*

_{ c }(

*x*) and

*r*(

*x*, 0) =

*ρ*

_{ c }(

*x*, 0). We also show (right panel) the solution of the pure advection problem computed by solving a series of ordinary differential equations, as explained on page 34.

## 2.10 Appendix: Solving the System of Partial Differential Equations

In this chapter, we derived the system

### 2.10.1 Operator Splitting

*ρ*

^{ n }at time \(t_{n} = n\Delta t.\) Then the first step is to solve the system

*t*=

*t*

_{ n }to \(t = t_{n} + \Delta t\) using \(\rho (t_{n}) =\rho ^{n}\) as the initial condition. Next we define the initial condition \(u(t_{n}) =\rho (t_{n+1})\) (which we just computed) and then solve the system of ordinary differential equations given by

*t*=

*t*

_{ n }to \(t = t_{n} + \Delta t\). Finally, we define

Now the problem of solving system (2.47) is reduced to solving a linear hyperbolic problem of the form (2.51) and a linear system of ordinary differential equations of the form (2.52). Methods for solving the latter can be found in any introductory text in numerical methods for PDEs. The explicit and implicit Euler methods are particularly popular because of their simplicity (see, e.g., [96]). In our computations, we use either the explicit or the implicit Euler method or we use the ODE15s method provided by Matlab (www.mathworks.com).

### 2.10.2 The Hyperbolic Part

*a*has a uniform sign. This is obviously true for \(a = a_{c} = v_{d}(c_{0} - x)\) since \(x \in \left (c_{0},c_{+}\right )\), where we recall that

*a*

_{ c }≤ 0 for all relevant values of

*x*. Similarly,

*a*

_{ o }≥ 0 for all relevant values of

*x*.

*ρ*

_{ o }with

*a*=

*a*

_{ o }≥ 0, we obtain

*ρ*

_{ c }with

*a*=

*a*

_{ c }≤ 0, we obtain

### 2.10.3 The Courant–Friedrichs–Lewy Condition

## 2.11 Notes

- 1.
Figure 2.1 is taken from Winslow et al. [105]. The figure will be used many times in this text as we gradually consider more complex models of CICR. A detailed description of the CICR mechanism and associated models is given by Winslow, Greenstein, Tankskanen, and Chen in [105] and [104].

- 2.
A review of possible pathological changes arising in the vicinity of the dyad is given by Louch et al. [55] and calcium signaling in the developing cardiomyocyte is reviewed by Louch et al. [54]. Cardiac calcium signaling is reviewed by Bers [5].

- 3.
The goal of the calcium dynamics of a cardiac cell is to enable the well coordinated contraction of cardiac muscle. Cardiac excitation contraction is reviewed by Bers [3, 4].

- 4.
A detailed model of a calcium release unit is presented by Hake et al. [30] and Chai et al. [9] used the largest computer in the world (in 2013) to simulate the calcium dynamics of a single sarcomere at the nanometer scale. Simulations of the calcium dynamics of a whole cardiac cell are presented by Nivala et al. [60] and Li et al. [49, 51]. The dynamics was analyzed in [98] using a model developed by Swietach et al. [95].

- 5.
The derivation in Sect. 2.2 of the system of deterministic differential equations based on the stochastic release equations is motivated by the derivation of Nykamp and Tranchina [63].

- 6.
The probability density function approach used to model calcium concentrations is taken from Huertas and Smith [35].

- 7.
As mentioned in the beginning of this chapter, the model illustrated in Fig. 2.2 relies on a series of simplifying assumptions. One additional simplification underlying the model given in (2.1) is that we assume that there is just one channel. In reality, the RyRs come in clusters of 10–20 channels, but here we assume that the effect of these channels can be added together in one big channel taking on the states of the Markov model in question. This is a major simplification that makes it possible to deal with the problem. The case of many interacting channels is dealt with by Bressloff [6] (page 112) for the case of a Markov model consisting of only two states (closed and open).

- 8.
For readers who need to refresh basic notions of differential equations, we recommend a look at the books by Logan [53], Strauss [91] or [96, 100]. As mentioned several times above, we recommend LeVeque [48] for an introduction to the numerical solution of hyperbolic problems.

- 9.
Systems of PDEs written in the form (2.22) appear in many different applications; see Bressloff [6], where other methods of analysis are also presented.

- 10.
An introduction to operator splitting and an explanation of why it works are given by, for example, LeVeque [48]. Operator splitting for the monodomain equation of electrophysiology was used by Qu and Garfinkel [70] and the accuracy was analyzed by Schroll et al. [80]. Application to the bidomain model was presented by Keener and Bogar [45] and by Sundnes et al. [94].

## Footnotes

- 1.
Note that when we consider the case of a given value

*γ*, the model becomes deterministic and we remove the overbar that indicates a variable is stochastic. - 2.
Advection means the transport of a conserved quantity.

- 3.
We have seen above that the interval bounded by

*c*_{0}and*c*_{+}is invariant in the sense that if the initial condition of the stochastic model (2.1) is in this interval, then the solution remains in the same interval for all time. We may, of course, however, pick an initial condition outside that interval, which motivates examination of the probability density functions using a larger domain. In these notes, however, we will stick to the invariant region. - 4.

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