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.
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Keywords
- Closed-blocked State
- Probability Density Approach
- Calcium-induced Calcium Release
- Theoretical Drug
- Equilibrium Probability
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.
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.
Let us start by recalling that the Markov model governing the states of the channel is given by
When a CO-mutation is present, we introduce the mutation severity index μ and replace the reaction rate k co by μ k co ,
Obviously, 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.
In this chapter, we will study theoretical open and closed state blockers. We recall from Chap. 1 that open and closed state blockers can be presented in the forms
and
respectively. The reasoning behind this way of modeling the effect of a drug was discussed on page 18 above; see in particular Fig. 1.9 Basically, we assume that the drug introduces a new conformational state of the channel protein that can be attained via the open state (for open state blockers) or via the closed state (for closed state blockers). The blocked states are always assumed to be non-conducting.
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
We start the derivation of theoretical drugs by considering closed state blockers. The reaction scheme of a closed state blocker takes the form
where the reaction rates of the drug given by 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
Consider the Markov model given by
and let 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
This means that the channel keeps on flickering in equilibrium and the probabilities of the open and closed states satisfy the relation (3.6). From this relation it follows that
and then, since \(o + c = 1,\) we obtain
3.1.2 Equilibrium Probabilities for the Mutant Case
In the CO-mutation case, we assume that the rate from C to O is increased and we define
where \(\mu \geqslant 1\) and μ = 1 denotes wild type. The equilibrium open probability of the mutant is given by
which clearly increases with increasing values of μ.
3.1.3 Equilibrium Probabilities for Mutants with a Closed State Drug
The equilibrium probabilities of reaction (3.5) are characterized by
so
and, since \(o + c + b = 1,\) we obtain
So
Define
and note that, in equilibrium, the wild type open probability is given by
and the drugged mutant open probability is given by
Now, we want to choose the drug characterization δ c such that \(o_{\mu,\delta _{c}} \approx o\) and this can clearly be achieved by requiring that
or
So we obtain the characterization
This means that, for the closed state blocker, we reduced the number of parameters characterizing the blocker from two to one. We will use the probability density approach to determine the remaining degree of freedom.
3.2 Probability Density Functions in the Presence of a Closed State Blocker
The probability density approach to the stochastic model in the presence of a closed state drug is
where
From (3.9), the parameters of the drug are related by
so the system is
In the stationary case, we obtain the system
In this system, the mutation severity is given by 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
We consider a mutation characterized by 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.
Numerical solutions of the steady state probability density functions defined by the system (3.11)–(3.13), where the reaction rates are related by (3.10) and the mutation severity index is given by μ = 3. The lower panel focuses on higher levels of concentrations. Note that the concentration axis of this figure is different from that of the lower panel of Fig. 3.1
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.
Consider the steady state system
By adding all the equations, we obtain
From the boundary conditions, we obtain
and therefore
where we recall that a c < 0 for \(x \in \left (c_{0},c_{+}\right )\). Now, the system (3.14)–(3.16) can be rewritten in the form
We are interested in solutions of this system as k bc becomes very large and we therefore note that, in the limit as \(k_{bc} \rightarrow \infty,\) the second equation yields
and therefore the first equation becomes
So
Recall that the wild type model is
(see (2.36)). By comparing (3.26) and (3.27), we see that when the drug is chosen to be of the form
and when we let \(k_{bc} \rightarrow \infty,\) the drug completely repairs the probability density functions of the mutated cell.
3.4 Markov Models for Open State Blockers
Next, we want to consider models of open state blockers. The reaction scheme of an open state blocker for the mutant reads
The equilibrium probabilities are now characterized by
so
and since \(o + c + b = 1,\) we have
We now define the open state blocker characterization
and note that the open probability is given by
Since the wild type open probability is given by
we want to choose the drug such that \(o_{\mu,\delta _{o}} \approx o\) and we therefore require
or
where we recall that the mutation severity index \(\mu \geqslant 1.\) Since 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
The probability density model in the presence of an open state drug is
where we recall that
In the stationary case, we obtain the system
We let both 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 differenceFootnote 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
In Fig. 3.4, we compare the results of the best open state blocker (referred to as 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.
Comparison of the best open state blocker and closed state blocker using k bc = 1, 000 ms−1 (see Table 3.2 for all the parameters of the two drugs). It is hard to distinguish between the wild type solution and the solution of the mutant case where the closed state blocker is applied
3.6 CO-Mutations Does Not Change the Mean Open Time
To understand why the closed state blocker is much better than the open state blocker for CO-mutations, it is useful to recall that the mean open time of the Markov model
is given by
Thus the mean open time is independent of the mutation. If a closed state blocker is introduced as
we clearly see that the mean open time is still given by
On the other hand, for an open state blocker of the form
the mean open time is changed and reads
With a closed state blocker used to repair a CO-mutation, the mean open time is kept constant, as it should, but it is changed using an open state blocker. Consequently, it is hard to see how to derive an efficient open state blocker for a CO-mutation.
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].
Notes
- 1.
The discrete l 2 difference between two vectors is given by \(\left \|u - v\right \|_{2} = (\sum _{i}(u_{i} - v_{i})^{2})^{1/2}\).
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Y. Rudy, J.R. Silva, Computational biology in the study of cardiac ion channels and cell electrophysiology. Q. Rev. Biophys. 39(1), 57–116 (2006)
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Tveito, A., Lines, G.T. (2016). Models of Open and Closed State Blockers. 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_3
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