Computing Characterizations of Drugs for Ion Channels and Receptors Using Markov Models pp 165-175 | Cite as

# Inactivated Ion Channels: Extending the Prototype Model

## Abstract

Experimental evidence suggests that some ion channels can take on three main states: open (O), closed (C), or inactivated (I). Here both C and I mean that the channel is non-conducting, but when the channel is inactivated, it is harder to open again than when the channel is in the closed state. This feature is useful in modeling an action potential. In the action potential of a cardiac cell, the upstroke is driven mainly by the sodium current. When the upstroke is completed, the sodium channels are inactivated to avoid spurious new upstrokes before the cell has undergone a restitution period. Certain mutations impair the ability of the channel to deactivate, which may lead to arrhythmias. We will return to this topic below. Here, it suffices to state that we need to introduce an inactivated state in the prototype model discussed above.

Experimental evidence suggests that some ion channels can take on three main states: open (O), closed (C), or inactivated (I). Here both C and I mean that the channel is non-conducting, but when the channel is inactivated, it is harder to open again than when the channel is in the closed state. This feature is useful in modeling an action potential. In the action potential of a cardiac cell, the upstroke is driven mainly by the sodium current. When the upstroke is completed, the sodium channels are inactivated to avoid spurious new upstrokes before the cell has undergone a restitution period. Certain mutations impair the ability of the channel to deactivate, which may lead to arrhythmias. We will return to this topic below. Here, it suffices to state that we need to introduce an inactivated state in the prototype model discussed above.

## 11.1 Three-State Markov Model

*v*. At this point, we just want to derive a prototypical model and we therefore, admittedly somewhat arbitrarily, define the following rates:

*k*

_{ io }, these rates satisfy the principle of detailed balance (see page 10 and the notes of Chap. 1).

### 11.1.1 Equilibrium Probabilities

*v*close to zero and it is inactivated for large values of

*v*.

## 11.2 Probability Density Functions in the Presence of the Inactivated State

### 11.2.1 Numerical Simulations

*v*= 0 for the probability density function of the inactivated state is captured using both the Monte Carlo and the probability density function approaches.

## 11.3 Mutations Affecting the Inactivated State of the Ion Channel

Certain mutations of the sodium channel are known to impair the channel’s ability to deactivate. We introduce a mutation severity index *μ* and assume that the reaction rates of the mutant are changed such that both the probabilities of moving from the inactivated to the closed state and from the inactivated to the open state are increased. The effect of these changes will clearly be to lower the probability of the channel being in the inactivated state.

*k*

_{ ic }and

*k*

_{ io }are the wild type reaction rates given by (11.2). It should be noted that the new reaction rates still satisfy the principle of detailed balance. In Fig. 11.4, we show the equilibrium probability density functions of the open, closed, and inactivated states for the wild type and for three values of the mutation severity index

*μ*.

## 11.4 A Theoretical Drug for Mutations Affecting the Inactivation

*c*+

*i*+

*o*+

*b*

_{ c }+

*b*

_{ o }+

*b*

_{ i }= 1, we have

*p*as the inverse open probability and we note that for the wild type it is given by

### 11.4.1 Open Probability in the Mutant Case

*μ*, since the wild type rates satisfy the principle of detailed balance.

### 11.4.2 The Open Probability in the Presence of the Theoretical Drug

*r*

_{ cb },

*r*

_{ ib }, and

*r*

_{ ob }are used to characterize the drug. Our aim is to now use these parameters to tune the drug such that

*p*is the inverse open probability of the wild type. More precisely, we want to determine the constants

*r*

_{ cb },

*r*

_{ ib }, and

*r*

_{ ob }such that

*v*. We observe that if we put

*r*

_{ cb }=

*r*

_{ ob }= 0, we obtain the condition

*μ*is the severity index of the mutation. This means that we have reduced the problem of finding a drug to a single parameter given by

*k*

_{ bi }. This remaining degree of freedom will be addressed below.

## 11.5 Probability Density Functions Using the Blocker of the Inactivated State

*μ*is the mutation severity index of the mutation (see (11.7)). The stationary probability density functions of the states in the Markov model of Fig. 11.6 are governed by the system

*ρ*

_{ o },

*ρ*

_{ c },

*ρ*

_{ i }, and

*ρ*

_{ b }denote the probability density functions of the open, closed, inactivated, and blocked states, respectively, and where the flux terms are given by

*ρ*

_{ o }, computed by solving the system (11.13)–(11.15), and the mutant where the drug is applied, computed by solving (11.9)–(11.12), denoted by

*ρ*

_{ o }

^{∗}. The difference is defined by the norm

*k*

_{ bi }increases, the drug defined by (11.8) completely repairs the effect of the mutation.

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