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.
You have full access to this open access chapter, Download chapter PDF
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.
The stochastic model considered in this chapter is the same as in Chap. 10
with the parameters given in Table 10.1 on page 154.
11.1 Three-State Markov Model
The reaction scheme of an ion channel taking on the three states O, C, and I is given in Fig. 11.1. To model the properties of the action potential in the way we described above, we need to introduce reaction rates that depend on the transmembrane potential v. At this point, we just want to derive a prototypical model and we therefore, admittedly somewhat arbitrarily, define the following rates:
where
and
By the definition of k io , these rates satisfy the principle of detailed balance (see page 10 and the notes of Chap. 1).
11.1.1 Equilibrium Probabilities
We saw above (see page 8) that the equilibrium state of the reaction shown in Fig. 11.1 is given by
These probabilities are graphed as functions of the transmembrane potential in Fig. 11.2. Note that the open probability in equilibrium is quite small; the channel is basically closed for 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
When the inactivated state is included in the model, as indicated in Fig. 11.1, the system governing the associated probability density functions is given by
where
11.2.1 Numerical Simulations
Again, we want to compare the solution computed by Monte Carlo simulations based on the stochastic differential equation given in (11.1) and the probability density functions defined by the system (11.3)–(11.5). The numerical results are given in their usual form in Fig. 11.3. As expected, the histograms computed using Monte Carlo simulations and the numerical solution of the system (11.3)–(11.5) are quite similar. In these computations, the stochastic simulation ran for 100 s, with \(\Delta t = 0.01\) ms, and we used the mesh size \(\Delta v = 0.01\) in the numerical solution of the system (11.3)–(11.5). It is particularly interesting to see that the tiny boundary layer close to 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.
In mathematical terms, we define
where \(\mu \geqslant 1\) and where 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
We want to derive a theoretical drug repairing the effect of the mutation described in (11.7). In the Markov model illustrated in Fig. 11.5, we have introduced a blocked state associated with the open, closed, and inactivated state and we now want to figure out what the best choice might be. The equilibrium solution of the reaction represented in Fig. 11.5 is characterized by the equations
It is useful to define
and to note that
With this notation, the principle of detailed balance stating that
can be written as
The equations above can now be written as
It is convenient to express all probabilities in terms of the open probability:
Since c + i + o + b c + b o + b i = 1, we have
where
We refer to 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
As discussed above, we are interested in understanding how to define a theoretical drug for mutations affecting the inactivation of the ion channel. We assume that the mutation affects the inactivation in a way that reduces the probability of being in the inactivated state. As mentioned above, this can be modeled by increasing the reaction rates from the inactivated state to both the closed and the open states. We assume that
where \(\mu \geqslant 1\) is the mutation severity index. This gives
and
We assume that the reaction rates between the closed and open states are unaffected by the mutation and therefore
Detailed balance dictates that we should have
which holds regardless of the choice of μ, since the wild type rates satisfy the principle of detailed balance.
The inverse open probability in the presence of the mutations is given by
11.4.2 The Open Probability in the Presence of the Theoretical Drug
When the drug given in Fig. 11.5 is applied, the inverse open probability is
where 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
where 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
holds for all relevant values of the transmembrane potential v. We observe that if we put r cb = r ob = 0, we obtain the condition
and therefore we set
We conclude that we can repair the equilibrium state of the mutation completely by applying a drug consisting of a blocker of the inactivated state, provided that the reaction rates of the drug satisfy
where 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
In Sect. 11.2 above, we derived a system governing the probability density functions of the open, closed, and inactivated states. Here, we want to extend the system to account for the theoretical drug represented by a blocker of the inactivated state. The Markov model of the drug is given in Fig. 11.6. The drug will completely repair the equilibrium state of the Markov model, provided that
where 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
where 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
The associated model of the wild type is given by
All the reactions rates used in the computations are given in (11.2); the computational domain is given by \(\Omega = [0,1]\) and we used 201 mesh points. In Fig. 11.7, we show the difference between the open state probability density function of the wild type, denoted 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
where, as usual,
We observe that, as k bi increases, the drug defined by (11.8) completely repairs the effect of the mutation.
Author information
Authors and Affiliations
Rights and permissions
Open Access This chapter is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by-nc/4.0/), which permits use, duplication, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, a link is provided to the Creative Commons license and any changes made are indicated.
The images or other third party material in this chapter are included in the work’s Creative Commons license, unless indicated otherwise in the credit line; if such material is not included in the work’s Creative Commons license and the respective action is not permitted by statutory regulation, users will need to obtain permission from the license holder to duplicate, adapt or reproduce the material.
Copyright information
© 2016 The Author(s)
About this chapter
Cite this chapter
Tveito, A., Lines, G.T. (2016). Inactivated Ion Channels: Extending the Prototype Model. 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/10.1007/978-3-319-30030-6_11
Download citation
DOI: https://doi.org/10.1007/978-3-319-30030-6_11
Published:
Publisher Name: Springer, Cham
Print ISBN: 978-3-319-30029-0
Online ISBN: 978-3-319-30030-6
eBook Packages: Mathematics and StatisticsMathematics and Statistics (R0)