Thư viện số Văn Lang: Computing Characterizations of Drugs for Ion Channels and Receptors Using Markov Models

Numerical Drugs for Calcium-Induced Calcium Release

In the previous chapter, we developed models of calcium-induced calcium release (CICR) in terms of both a stochastic release model and a model of the probability density functions of the states involved in the stochastic release model. The models incorporated the effects of mutations in both the ryanodine receptors (RyRs) and the L-type calcium channels (LCCs). The purpose of the present chapter is to introduce theoretical drugs aimed at repairing the effect of mutations of both the LCCs and RyR channels. Model parameters used throughout this chapter are given in Table9.1.

We have seen in previous chapters that, if we ignore the effect of the LCC, we can completely repair the effect of an RyR mutation using a closed state blocker if the mutation is of the CO type. In this chapter, we want to see if this result also holds when the effect of the LCCs is taken into account. Since the transmembrane potentialV enters the model as a parameter, it is sufficient to control the effect of the LCCs for a number of different values ofV. The next issue we want to address is how to repair the effect of LCC mutations. We will find optimal open and closed state blockers.

9.1 Markov Models for CICR, Including Drugs

We consider a situation where the RyR or the LCC may be affected by CO- mutations. Both effects are modeled by Markov models and in this section we introduce theoretical drugs in terms of open and closed state blockers for both the RyR and the LCC.

© The Author(s) 2016

A. Tveito, G.T. Lines,Computing Characterizations of Drugs for Ion Channels and Receptors Using Markov Models, Lecture Notes in Computational Science and Engineering 111, DOI 10.1007/978-3-319-30030-6_9

143

144 9 Numerical Drugs for CICR

Table 9.1 Values of parameters used in simulations in this chapter

vd 1 ms¹ vr 0.1 ms¹ vs 0.01 ms¹ c₀ 0.1 M c₁ 1,000 M

9.1.1 Theoretical Blockers for the RyR

As discussed above, the gating of the release of calcium from the sarcoplasmic reticulum to the dyad is given by the stochastic variableNr D Nr.t/governed by the reaction scheme

Cr k^r_oc k^r_co

Or: (9.1)

Hereis the mutation severity index, which is one in the wild type case. We have seen that open and closed state blockers can be added to the reaction as

B^r_c

k^r_cb

k^r_bc

Cr k^r_oc k_co^r

Or k^r_bo

k^r_ob

B^r_o; (9.2)

whereB^r_c and B^r_o denote the blocked states associated with the closed and open states, respectively. The characteristics of the drugs are given by the constantsk^r_cb andk^r_bc(for the closed state blocker) andk^r_obandk^r_bo(for the open state blocker).

9.1.2 Theoretical Blockers for the LCC

The Markov model governing the stochastic variableNlD Nl.t/of the LCC is given by

Cl k^l_oc k^l_co

Ol; (9.3)

where we have introduced the parameterto indicate a mutation of the LCC. The wild type case is again represented byD 1and any > 1denotes a leaky LCC.

We introduce a theoretical representation of a drug as for the RyR channels:

B^l_c

k^l_cb

k^l_bc

Cl k^l_oc k_co^l

Ol;^k

l

bo k^l_ob

B^l_o; (9.4)

where, in line with the RyR case,B^l_c andB^l_o denote the blocked states associated with the closed and open states, respectively, and the characteristics of the LCC drugs are given by the constantsk^l_cbandk^l_bc (for the closed state blocker) andk^l_ob andk^l_bo(for the open state blocker).

9.1.3 Combined Theoretical Blockers for the LCC and the RyR

To use the probability density formalism, it is convenient to rewrite the two Markov models as one combined model of the form illustrated in Fig. 9.1.This model consists of 16 separate states given by

B^l_cB^r_c B^l_cCr B^l_cOr B^l_cB^r_o ClB^r_c ClCr ClOr ClB^r_o OlB^r_c OlCr OlOr OlB^r_o B^l_oB^r_c B^l_oCr B^l_oOr B^l_oB^r_o

(9.5)

and the combined LCC and RyR drug is fully specified by

k^r_cb;k^r_bc;k^r_bo;k_ob^r ;k^l_cb;k^l_bc;k^l_bo; andk_ob^l : (9.6)

Fig. 9.1 The Markov model represented in Fig.8.3 extended to account for blockers for the LCC and the RyR

B_c^lB^r_c B^l_cCr B_c^lOr B_c^lB_o^r

ClB^r_c ClCr ClOr ClB_o^r

OlB_c^r OlCr OlOr OlB^r_o

B_o^lB_c^r B^l_oCr B_o^lOr B^l_oB^r_o k^r_bc

k^l_bc k^r_cb

μk^r_co k_bc^l

k_oc^r

k_ob^r k_bc^l

k_bo^r

k_bc^l k_cb^l

k^r_bc ηk_co^l

k_cb^l k^r_cb

μk^r_co ηk^l_co

k^l_cb k_oc^r

k_ob^r ηk^l_co

k^l_cb

k_bo^r

ηk^l_co k^l_oc

k^r_bc k^l_ob

k^l_oc k^r_cb

μk^r_co k_ob^l

k^l_oc k_oc^r

k_ob^r k_ob^l

k_oc^l

k_bo^r

k_ob^l k^l_bo

k^r_bc k^l_bo k^r_cb

μk^r_co k^l_bo k_oc^r

k_ob^r k_bo^l k_bo^r

146 9 Numerical Drugs for CICR

9.2 Probability Density Functions Associated with the 16-State Model

As mentioned in Chap.7 (see page119), it is convenient to use a more compact notation to represent the system of partial differential equations governing the probability density functions when the Markov model consists of numerous states.

By using the notation introduced in Chap.7, we can write the probability density system associated with the Markov model in Fig.9.1in the form

@ij

@t C @

@x a^x_ijij

C @

@y

a^y_ijij

DRij; (9.7)

where

RijDK_iⁱ_;^;_j^j_C1i;jC1CK_iⁱ_C1;^;j _jiC1;jCKⁱ_i;^;_j^j₁i;j1CKⁱ_i1;^;j _ji1;j

K_iⁱ_;^;j_j^C1CK_iⁱ_;^C1;_j ^jCK_iⁱ_;^;_j^j1CK_iⁱ_;^1;_j ^j i;j:

The flux terms are given by

a^x_ijDi^rvr.yx/Cvd.c0x/j^lJl; a^y_ijDi^rvr.xy/Cvs.c1y/ ;

wherei^rD1when the RyR state is open andi^rD0when the RyR state is closed and similarly for^land the LCC.

9.3 RyR Mutations Under a Varying Transmembrane Potential

In this section, we assume that a mutation affects the RyR such that the mutation severity index is increased. This problem has been discussed several times above, but here we also need to take into account that the value of the transmembrane potential may change. In our computations, we use D3and we try to repair the effect of the mutation by adding a closed state blocker to the Markov model of the RyR channel. The Markov model is shown in Fig.9.2.

The closed state drug applied to the RyR channel is represented by the two parametersk_cb^r andk^r_bc:We have seen above that for closed state blockers of the RyR it is reasonable to define

k^r_cbD.1/k^r_bc; where the value ofk^r_bcremains to be decided.

ClB_c^r ClCr ClOr

OlB^r_c OlCr OlOr

k^r_bc k_co^l

k^r_cb

μk^rco

k^l_co koc^r

k^l_co k^l_oc

k^r_bc k^l_oc k^r_cb

μk^rco

k^l_oc

koc^r

Fig. 9.2 The Markov model represented in Fig.8.3extended to include an RyR mutation and a closed state blocker for the RyR

10⁻⁵ 10⁰

2 3 4 5

x 10⁻³ π_oo

kbc WT MT Drugged

10⁻⁵ 10⁰

0.04 0.06 0.08 0.1 0.12

π_co

kbc

10⁻⁵ 10

0

4 5 6 7 8

x 10⁻³ π_oc+π_ob

kbc

10⁻⁵ 10⁰

0.8 0.85 0.9 0.95 1 1.05

π_cc+π_cb

kbc

Fig. 9.3 Total probabilities based on the model for the probability density functions associated with the Markov model in Fig.9.2. A closed state blocker is applied, the mutation severity index isD3, and the transmembrane potential isVD0mV. Theplotsshow the total probability of being in the state OO, (OC+OB), CO, or (CC+CB) as a function ofk^r_bc. In theupper left plot, the total probability of being in the OO state is higher for the mutant than for the wild type. This is repaired by the closed state drug. Similar results are shown for the other states

9.3.1 Theoretical Closed State Blocker Repairs the Open Probabilities of the RyR CO-Mutation

Numerical results using the closed state drug shown in Fig.9.2are given in Fig.9.3.

Note that we aim to repair the probability of being in the open state and are not

148 9 Numerical Drugs for CICR

45 50 55

500 600 700 800

Eoo

x (μM)

y (μM)

25 30 35 40 45

300 400 500 600

Eco

x (μM)

y (μM)

18.5 19 19.5 20

600 700 800 900

Eoc+ob

x (μM)

y (μM)

1.4 1.6 1.8 2

600 700 800 900

Ecc+cb

x (μM)

y (μM)

Fig. 9.4 Based on the probability density functions of the states OO, (OC+OB), CO, and (CC+CB), we can compute the expected concentrations of the dyad (x) and the JSR (y). The wild type is denoted byıand the RyR mutation indexincreases from one to three along thesolid line. In thedashed line, we keepD3and increase the value ofk^r_bcfrom 0 to 100 ms¹. We observe that ask_bc^r increases, the expected concentrations are completely repaired. The experiment is carried out for the case ofVD0mV

interested in whether the channel is in a blocked state or in a closed state. The probability of being in a closed or blocked state is therefore added in the graphs.

We observe from the graphs that the mutant channel is completely repaired by the closed state blocker.

In Fig. 9.4, we show the development of the expected concentrations of the dyad (x) and the junctional sarcoplasmic reticulum (JSR) (y) and observe that the expected concentrations are repaired by a sufficiently strong version of the blocker associated with the closed state of the RyR channel.

9.3.2 The Open State Blocker Does Not Work as Well as the Closed State Blocker for CO-Mutations in RyR

In Table9.2, we report on the performance of the open and closed state blockers for the RyR mutation. Recall that the probabilityoo, the expected dyad concentration (E^x_oo), and the expected JSR concentration (E^y_oo) are defined on page72. The closed blocker clearly is best suited to repair this mutation.

Table 9.2 Properties of the probability density function (oo) of being in the state OO withD3 (andD1). The closed state blocker works fine in the sense that it is well suited for repairing a CO-mutation of the RyR. The open state blocker is unable to completely repair the effect of the mutation. The open state blocker is found using Matlab’sFminsearch, with a cost function defined to minimize the difference between the wild type and the mutation when the drug is applied. In this table, WT and MT mean wild type and mutant, respectively, andV D0mV is used in the simulations

WT MT Optimal closed blocker Optimal open blocker

10³oo 2:75 5:11 2:74 0:81

E^x_oo 51:87 45:62 51:92 52:46

E^y_oo 751:76 544:30 751:99 713:89

Fig. 9.5 The Markov model represented in Fig.8.3 extended to include an LCC mutation and a closed state blocker for the LCC

Bc^lCr Bc^lOr

ClCr ClOr

OlCr OlOr

k^r_co k_bc^l

k^r_oc

k_bc^l k^l_cb

k^rco

ηk^l_co k_cb^l

k^r_oc

ηk^l_co k_oc^l

k^r_co k^l_oc

k^r_oc Fig. 9.6 The Markov model

represented in Fig.8.3 extended to include an LCC mutation and an open state blocker for the LCC

ClCr ClOr

OlCr OlOr

Bo^lCr B^loOr

k^r_co ηk^l_co

k^r_oc

ηk^l_co k_oc^l

k^rco

k^l_ob k_oc^l

k^roc

k_ob^l k_bo^l

k^r_co k^l_bo k^r_oc

9.4 LCC Mutations Under a Varying Transmembrane Potential

Next, we address the problem of defining a theoretical drug for LCC mutations. We consider closed state LCC blockers of the form given in Fig.9.5and open state blockers of the form given in Fig.9.6.

150 9 Numerical Drugs for CICR

10⁻⁵ 10⁰

3 4 5 6 7 8 9

x 10⁻³ π_oo

kbc WT MT Drugged

10⁻⁵ 10⁰

0.03 0.04 0.05 0.06 0.07

π_co+π_bo

kbc

10⁻⁵ 10⁰

0.01 0.015 0.02

π_oc

kbc

10⁻⁵ 10⁰

0.85 0.9 0.95 1 1.05

π_cc+π_bc

kbc

Fig. 9.7 Total probabilities based on the model for the probability density functions associated with the Markov model in Fig.9.5, where an LCC-type closed state blocker is included. The LCC mutation severity index isD3and the transmembrane potential isVD0mV. Theplotsshow the total probability of being in the state OO, OC, (CO+BO), or (CC+BC) as a function ofk^l_bc. In theupper left plot, the total probability of being in the OO state is higher for the mutant than for the wild type. This is repaired by the closed state drug. Similar results are shown for the other states

For the closed state blockers, we need to determine the two parametersk^l_bcand k_cb^l and for the open state blockers we must determinek^l_bo andk^l_ob:For the closed state blockers we define

k^l_cbD.1/k^l_bc and we consider various values ofk_bc^l :

9.4.1 The Closed State Blocker Repairs the Open Probabilities of the LCC Mutant

The results of applying the theoretical closed state blocker associated with the closed state (see Fig. 9.5) of the LCC are given in Figs. 9.7,9.8, and 9.9. In the first figure, we show how the closed state blocker repairs the total probabilities and in the

46 48 50 52 54 500

600 700 800

Eoo

x (μM)

y (μM)

34 36 38 40

450 500 550 600

Eco+bo

x (μM)

y (μM)

19.5 20 20.5 21

600 700 800 900

Eoc

x (μM)

y (μM)

1 1.5 2 2.5

600 700 800 900

Ecc+bc

x (μM)

y (μM)

Fig. 9.8 Using the probability density functions of the states OO, OC, (CO+BO), and (CC+BC), we compute the expected concentrations of the dyad (x) and the JSR (y). The wild type is denoted byıand the LCC mutation indexincreases from one to three along thesolid line. In thedashed line, we keep D3and increase the value ofk_bc^l from 0 to 100 ms¹. We observe that ask^l_bc increases, the expected concentrations are completely repaired. The simulations are performed usingVD0mV

second figure we consider the expected concentrations. In Fig.9.9, we show how the expected concentrations are repaired for six values of the transmembrane potential.

152 9 Numerical Drugs for CICR

38 40 42 44

700 800 900

V=20

x (μM) y (μM)

45 50 55

400 600 800

V=0

x (μM)

y (μM)

70 72 74 76

700 800 900

V=−20

x (μM)

y (μM)

102.6 102.8 103 103.2 890

900 910

V=−40

x (μM)

y (μM)

132.02 132.025 132.03 873.6

873.8 874

V=−60

x (μM)

y (μM)

161.32 161.34 161.36 824.38

824.39 824.4

V=−80

x (μM)

y (μM)

Fig. 9.9 The expected concentration of the dyad (x) and the JSR (y) for the OO state for the transmembrane potential changing from80 to 20 mV. The wild type is denoted byıand the LCC mutation indexincreases from one to three along thesolid line. In thedashed line, we keep D 3and increase the value ofk^l_bc from 0 to 100 ms¹. In all cases, the closed state blocker repairs the effect of the mutation

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