Computing Characterizations of Drugs for Ion Channels and Receptors Using Markov Models pp 55-69 | Cite as

# Models of Open and Closed State Blockers

## Abstract

So far we have studied a one-dimensional model of calcium-induced calcium release. The analysis started with a stochastic differential equation modeling release from internal storage to the dyad. We found that this model could be analyzed using Monte Carlo simulations or a system of deterministic partial differential equations giving the probability density functions of the open and the closed states. Furthermore, we found analytical solutions of the stationary solutions of the probability density system.

So far we have studied a one-dimensional model of calcium-induced calcium release. The analysis started with a stochastic differential equation modeling release from internal storage to the dyad. We found that this model could be analyzed using Monte Carlo simulations or a system of deterministic partial differential equations giving the probability density functions of the open and the closed states. Furthermore, we found analytical solutions of the stationary solutions of the probability density system.

The aim of the present chapter is to introduce mathematical models of a drug and then show how the parameters defining the drug can be computed so that it works as well as possible. For simplicity, we will focus on closed to open mutations (CO-mutations; see page 16), but it will become clear how to handle open to closed mutations (OC-mutations) in later chapters.

*μ*and replace the reaction rate

*k*

_{ co }by

*μ k*

_{ co },

*μ*= 1 represents the wild type case and the size of

*μ*> 1 gives the strength of the mutation. By recalling what the Markov model means, we see that the mutation increases the probability of going from the closed to the open state and thus the open state probability will increase.

The mathematical problem of finding a suitable theoretical drug is now to find the parameters *k*_{ bc } and *k*_{ cb } for the closed state blockers and *k*_{ bo } and *k*_{ ob } for the open state blockers such that the effect of the mutation is reduced as much as possible. We will see that this problem is much easier using the probability density approach than using Monte Carlo simulations.

To compute optimal drugs for the CO-mutation, we will first consider the equilibrium states of the reactions. For closed state blockers, we can use the equilibrium considerations to reduce the number of free parameters from two to one. In principle, this can also be done for open state blockers, but some averaging is needed in the process and optimality is not obtained. For the closed state blocker, we can use the steady state system derived above to completely characterize both parameters of the drug to obtain optimality and computations will show that the resulting drug is theoretically extremely good and asymptotically perfect in the sense that it completely reverses the effect of the mutation. We are also able to derive a good open state blocker, but the method is less satisfactory and the results are not as good as for the closed state blocker.

## 3.1 Markov Models of Closed State Blockers for CO-Mutations

*k*

_{ cb }and

*k*

_{ bc }must be determined so that the mutated cell behaves as similarly to the wild type cell as possible. We regard these parameters as free and we seek to compute them to obtain optimal efficiency of the theoretical drug. Allow us also to briefly repeat that this is basically our definition of a theoretical drug as discussed on page 18.

### 3.1.1 Equilibrium Probabilities for Wild Type

*o*denote the probability of being in the open state and

*c*the probability of being in the closed state. Suppose the channel just flickers between open and closed and nothing else happens. Then the equilibrium probabilities are characterized by

### 3.1.2 Equilibrium Probabilities for the Mutant Case

*C*to

*O*is increased and we define

*μ*= 1 denotes wild type. The equilibrium open probability of the mutant is given by

*μ*.

### 3.1.3 Equilibrium Probabilities for Mutants with a Closed State Drug

*δ*

_{ c }such that \(o_{\mu,\delta _{c}} \approx o\) and this can clearly be achieved by requiring that

## 3.2 Probability Density Functions in the Presence of a Closed State Blocker

*μ*and the drug is characterized by a single parameter given by

*k*

_{ bc }. For a given value of

*μ*our aim is now to compute the value of

*k*

_{ bc }such that the probability density function of the open state given by this system is as similar as possible to the probability density function of the open state in the case of

*μ*= 1, that is, the wild type solution when no drug is applied.

### 3.2.1 Numerical Simulations with the Theoretical Closed State Blocker

*μ*= 3 and we apply closed state blockers (see reaction scheme (3.5)) with parameters satisfying the relation (3.10). In Fig. 3.1, we show the results of these simulations using the Monte Carlo approach: The lower panel of the figure is the same as the upper panel, except that we focus on concentrations ranging from 80 to 91 \(\upmu\) M. We observe significant differences between the wild type solution and the solution representing the mutation. Furthermore, we observe that the drug works quite well. Similar results are given in Fig. 3.2, where the computations are based on the probability density approach: Here the lower panel focuses on very high concentrations ranging from 89 to 91 \(\upmu\) M. We also see that the closed state drug improves as the value of

*k*

_{ bc }increases. In fact, the result seems to indicate that the drug is asymptotically perfect in the sense that the solution converges toward the wild type solution when \(k_{bc} \rightarrow \infty\). Model parameters for these simulations are given in Table 3.1.

Parameter values for the undrugged case

| 1 ms | |
---|---|---|

| 0.1 ms | |

| 0.1 \(\upmu \mathrm{M}\) | |

| 1,000 \(\upmu \mathrm{M}\) | |

| 0. 1 | |

| 1 ms |

## 3.3 Asymptotic Optimality for Closed State Blockers in the Stationary Case

In the simulations above, we observed that the closed state blocker worked well and that the drug became more effective as the value of *k*_{ bc } increased. Our aim is now to indicate that, when \(k_{bc} \rightarrow \infty\), the drug will completely repair the mutation. It is worth mentioning that the possibility of making a drug with \(k_{bc} = \infty\) is quite unlikely, but the asymptotic result is still of theoretical interest.

*a*

_{ c }< 0 for \(x \in \left (c_{0},c_{+}\right )\). Now, the system (3.14)–(3.16) can be rewritten in the form

*k*

_{ bc }becomes very large and we therefore note that, in the limit as \(k_{bc} \rightarrow \infty,\) the second equation yields

*completely*repairs the probability density functions of the mutated cell.

## 3.4 Markov Models for Open State Blockers

*μ*= 1 is the wild type case, we note that in that case

*δ*

_{ o }= 0 is the optimal drug, which makes sense; there is no need to drug the wild type. However, for mutant cells, we have

*μ*> 1 and the characterization (3.28) of

*δ*

_{ o }depends on the dyad calcium concentration,

*x*. We will therefore use direct optimization to find suitable open state blockers.

### 3.4.1 Probability Density Functions in the Presence of an Open State Blocker

*k*

_{ ob }and

*k*

_{ bo }be free parameters and use the

*Fminsearch*function in Matlab to optimize these parameters by minimizing the discrete

*l*

_{2}difference

^{1}between the wild type and mutant

*ρ*

_{ o }. The resulting parameters are

*k*

_{ ob }= 0. 28 ms

^{−1}, and

*k*

_{ bo }= 1. 63 ms

^{−1}and the associated numerical results are given in Fig. 3.3, marked as

*opt*.

## 3.5 Open Blocker Versus Closed Blocker

*opt*in Fig. 3.3) and closed state blocker, using

*k*

_{ bc }= 1, 000 ms

^{−1}(see Fig. 3.2). We clearly see that the closed state blocker is better; in fact, at this resolution of the graphs, it is hard to distinguish the wild type solution from the solution of the mutant case where the closed state blocker is applied.

Parameter values for the drugs used in Fig. 3.4

| 0.28 ms | |
---|---|---|

| 1.63 ms | |

| 2,000 ms | |

| 1,000 ms |

## 3.6 CO-Mutations Does Not Change the Mean Open Time

## 3.7 Notes

- 1.
In this chapter we focused on CO-mutations (see page 16) and, for such mutations, closed state blockers are best suited from a theoretical perspective. We will see later that OC-mutations are more easily repaired using open state blockers.

- 2.
The argument of asymptotic optimality given on page 63 is not a rigorous proof. To prove it mathematically, we have to take the boundary layer into consideration. Our derivation assumes smooth solutions but that assumption does not hold at the boundary.

- 3.
In this section, we used probability density formulations for systems with more than two states. The general case of many states is presented in Appendix C of Huertas and Smith [35].

- 4.
The mean open time will be introduced and analyzed in Chap. 13. In the present chapter we just used very basic properties.

- 5.
We mentioned above that we used the function

*Fminsearch*in Matlab to solve a minimization problem; see page 66. The*Fminsearch*function uses the Melder-Nead [58] algorithm studied by Lagarias et al. [46]. The method is very powerful and will be used routinely in these notes. - 6.
It is an underlying assumption for Markov models that the states of the model correspond to the conformational states of the channel protein. This should not be interpreted literally; rather, it has proved to be a useful modeling technique. A thorough discussion on the modeling of ion channels using Markov models and the models relation to the states of the protein is provided by Rudy and Silva [75].

## Footnotes

- 1.
The discrete

*l*_{2}difference between two vectors is given by \(\left \|u - v\right \|_{2} = (\sum _{i}(u_{i} - v_{i})^{2})^{1/2}\).

## References

- 35.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)MathSciNetCrossRefGoogle Scholar - 46.J.C. Lagarias, J.A. Reeds, M.H. Wright, P.E. Wright, Convergence properties of the Nelder–Mead simplex method in low dimensions. SIAM J. Optim.
**9**(1), 112–147 (1998)MathSciNetCrossRefMATHGoogle Scholar - 58.J.A. Nelder, R. Mead, A simplex method for function minimization. Comput. J.
**7**(4), 308–313 (1965)CrossRefMATHGoogle Scholar - 75.Y. Rudy, J.R. Silva, Computational biology in the study of cardiac ion channels and cell electrophysiology. Q. Rev. Biophys.
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