Computing Characterizations of Drugs for Ion Channels and Receptors Using Markov Models pp 223-236 | Cite as

# The Burst Mode of the Mutant Sodium Channel

## Abstract

We observed above that the effect of the \(\Delta\) KPQ mutation of the SCN5A gene leading to a delayed sodium current can be modeled by increasing the reaction rates from the inactivated state to the open state and to the permissible state *C*_{0}. The model gave results at least qualitatively similar to the experimental data (see Fig. 12.4).

We observed above that the effect of the \(\Delta\) KPQ mutation of the SCN5A gene leading to a delayed sodium current can be modeled by increasing the reaction rates from the inactivated state to the open state and to the permissible state *C*_{0}. The model gave results at least qualitatively similar to the experimental data (see Fig. 12.4).

*k*

^{ u }(where

*u*stands for up) and the reaction rates from the normal mode to the burst mode are given by

*k*

^{ d }(where

*d*stands for down). We assume

*k*

^{ d }< <

*k*

^{ u }, which means that, for the wild type, the probability of being in the burst mode is very small. The probability of being in the burst mode increases with the mutation severity index

*μ*. As usual,

*μ*= 1 represents the wild type. In the wild type, a channel is basically never in the burst mode and therefore the channel inactivates as it should and no late sodium current is observed. In the mutant case, however, the probability of being in the burst mode is increased. Since there is no inactivated state in the burst mode, the channel fails to inactivate and therefore the probability of being in the open state is increased and therefore we observe a non-negligible late current. This will be illustrated in the numerical computations below.

## 14.1 Equilibrium Probabilities

*μ*; in fact,

## 14.2 The Mean Open Time

## 14.3 An Optimal Theoretical Open State Blocker

*d*is used to remind us that this concerns the case where the theoretical drug has been applied.

*k*

_{ ob }and

*k*

_{ bo }as the unknowns and we want to solve the following 2 × 2 system of equations:

*k*

_{ ob }, of the drug,

*μ*= 1, the drug is completely turned off, which is reasonable. Since

*k*

_{ ob }is known, the off rate of the drug can be computed by solving (14.12). If we define

## 14.4 Numerical Experiments

The purpose of this section is to show how the burst mode can be used to represent impaired inactivation and how the theoretical drug derived above works.

### 14.4.1 Representation of the Late Sodium Current Using the Burst Mode Model

*μ*= 20 seems to represent the late current of Fig. 12.4 fairly well.

### 14.4.2 The Open State Blocker Repairs the Effect of the Mutation

*μ*= 20, and the drug using the optimal open state blocker defined by (14.14) and (14.17). We observe that the late current induced by the mutation is repaired by the open state blocker. The statistics of the open probability density function (for the wild type, the mutant (

*μ*= 20), and the mutant where the drug has been applied) are given in Table 14.2 and the corresponding probability density functions are shown in Fig. 14.4. Again we note that the open blocker repairs the main features of the solution.

Statistics of the stationary probability density functions computed using the Markov model illustrated in Fig. 14.2. The subscript *o* refers to open states and the subscript *n* refers to non-conducting states

| | | | | |
---|---|---|---|---|---|

1 | 0.59 | 0.41 | 31.2 | −54.2 | |

20 | 0.96 | 0.04 | 33.1 | −52.8 | |

50 | 0.98 | 0.02 | 33.1 | −51.3 | |

100 | 0.99 | 0.01 | 33.2 | −49.8 | |

20+OB | 0.46 | 0.54 | 30.2 | −57.9 |

## 14.5 A More Sophisticated Markov Model

*μ*increases. This will lead to a sustained sodium current characteristic of the mutation under consideration.

*b*and

*b*

^{∗}as the sum of the probabilities in the normal and burst modes, respectively. By using the equilibrium probabilities derived above, we obtain

## 14.6 Numerical Experiments Illustrating the Effect of the Burst Mode

| 1,10,30,100 | |
---|---|---|

| 0.1 ms | |

| 0.01 ms |

Probabilities and expected values of the transmembrane potential for open and non-conducting states for increasing values of the mutation severity index *μ*

| 1000 × | | | | |
---|---|---|---|---|---|

1 | 0.05738 | 0.99994 | -53.2 | -84.9 | |

10 | 0.08435 | 0.99992 | -26.1 | -84.9 | |

30 | 0.12109 | 0.99983 | -8.3 | -84.9 | |

100 | 0.22305 | 0.99978 | 10.5 | -84.9 | |

30+OB | 0.05490 | 0.99995 | -57.0 | -84.9 |

## 14.7 A Theoretical Drug for the Mutation Represented by the Burst Mode

*k*

_{ ob }and

*k*

_{ bo }such that

*wt*denotes wild type values. As above, we have two equations for the two unknowns

*k*

_{ ob }and

*k*

_{ bo }and the solution is given by

*μ*= 30), and the mutant case where the optimal open blocker is applied. The blocker repairs the effect of the mutation and the same effect is seen in Fig. 14.11 where the currents are given; the open blocker removes the late sodium current.

## 14.8 Notes

- 1.
- 2.
The form of the model illustrated in Fig. 14.6 is taken from Clancy and Rudy [14], but the functions and parameters of the model are not taken from their paper.

- 3.
As mentioned above, the introduction of a burst mode is a convenient way of modeling the effect of certain mutations. The notion that gating may enter various modes has been considerably extended and studied in the papers by Chakrapani et al. [10, 11, 12] and by Ionescu et al. [37]. In the recent paper by Siekmann et al [83] the concept of modal gating is studied and a method for detecting mode changes based on single channel data is developed.

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