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

Generalized Systems Governing Probability Density Functions

So far we have considered one-dimensional and two-dimensional release processes.

When the channel can take on two states—open or closed—we have seen that the associated probability density functions are governed by22 systems of partial differential equations. When a drug is added to the Markov model, an extra state is introduced associated with either the open or the closed state and we obtain a model for the probability density functions phrased in terms of 33 systems of partial differential equations. In subsequent chapters, we will study situations involving many states and, to do so without drowning in cumbersome notation, we need mathematical formalism to present such models compactly. The compact form we use here is taken from Huertas and Smith [35]. We will introduce the more compact notation simply by providing a couple of examples. These will, hopefully, clarify how to formulate rather complex models in an expedient manner.

7.1 Two-Dimensional Calcium Release Revisited

Let us start by recalling that the two-dimensional process of calcium release illustrated in Fig.5.2on page92can be modeled as

N

x⁰.t/D N.t/vr.yN Nx/Cvd.c₀ Nx/ ; (7.1) N

y⁰.t/D N.t/vr.xN Ny/Cvs.c₁ Ny/ ; (7.2) whereN D N.t/is a stochastic variable governed by a Markov model represented by a reaction scheme of the form

C

koc

kco

O:

© 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

119

120 7 Generalized Systems We have seen (see, e.g., page102) that the probability density functions of the open state.o/and the closed state.c/are governed by the system

@o

@t C @

@x a^x_oo

C @

@y a^y_oo

Dkcockoco; (7.3)

@c

@t C @

@x a^x_cc

C @

@y a^y_cc

Dkocokcoc; (7.4)

where

a^x_o Dvr.yx/Cvd.c0x/ ;

a^y_o Dvr.xy/Cvs.c1y/ ; (7.5) a^x_cDvd.c0x/ ;

a^y_cDvs.c1y/ :

To prepare ourselves for more complex systems, we number the states in this simple system withi D1; 2;whereiD1is for the open state andiD 2is for the closed state. The system can now be written in the form

@i

@t C @

@x a^x_ii

C @

@y a^y_ii

D.K/i;

where.K/idenotes theith component of the matrix vector productK:Here the vectoris given by

D ₁

2

D

o

c

and the matrix is given by

KD

k₁₂ k₂₁ k₁₂ k₂₁

D

koc kco

koc kco

:

Furthermore, we introduce the functions

a^x_i Divr.yx/Cvd.c0x/ ; a^y_i Divr.xy/Cvs.c₁y/ ;

wherei is one for the open state (i.e.,i D 1) and zero for the closed state (i.e., iD2).

7.2 Four-State Model

It useful to illustrate this compact notation for a slightly more complex model based on four states. Suppose that the Markov model governing the stochastic variableN in model (7.1) and (7.2) is based on four states: two open statesO₁andO₂and two closed statesC₁andC₂, as shown in Fig.7.1.

The probability density system associated with the model (7.1) and (7.2) when the Markov model is given by Fig.7.1can now be written in the form

@o₁

@t C @

@x a^x_oo₁

C @

@y a^y_oo₁

Dkc₁o₁c₁.ko₁c₁Cko₁o₂/ o₁Cko₂o₁o₂;

@o₂

@t C @

@x a^x_oo₂

C @

@y a^y_oo₂

Dkc₂o₂c₂.ko₂c₂Cko₂o₁/ o₂Cko₁o₂o₁;

(7.6)

@c₁

@t C @

@x a^x_cc₁

C @

@y a^y_cc₁

Dko₁c₁o₁.kc₁o₁Ckc₁c₂/ c₁Ckc₂c₁c₂;

@c₂

@t C @

@x a^x_cc₂

C @

@y a^y_cc₂

Dkc₁c₂c₁.kc₂c₁Ckc₂o₂/ c₂Cko₂c₂o₂;

where

a^x_o Dvr.yx/Cvd.c₀x/ ;

a^y_o Dvr.xy/Cvs.c₁y/ ; (7.7) a^x_cDvd.c₀x/ ;

a^y_cDvs.c₁y/ :

By defining the statesO₁;O₂;C₁;andC₂to be the states 1, 2, 3, and 4, respectively, we can write the system (7.6) in the more compact form

@i

@t C @

@x a^x_ii

C @

@y a^y_ii

D.K/i (7.8)

Fig. 7.1 Markov model including four possible states:

two open states,O₁andO₂, and two closed states,C₁ andC₂

C1 O1

C2 O2

kc1o1

kc1c2

ko1c1

ko1o2

kc2c1

kc2o2

ko2o1

k

122 7 Generalized Systems

foriD1; 2; 3; 4;where

a^x_i Divr.yx/Cvd.c₀x/ ; a^y_i Divr.xy/Cvs.c₁y/ ;

andD .₁; ₂; ₃; ₄/^T:Hereiis one for the open states (i.e.,i D1andiD 2) and zero for the closed states (i.e.,i D 3 andi D 4). Furthermore, the matrix is given by

KD 0 BB

@

.ko₁c₁Cko₁o₂/ ko₂o₁ kc₁o₁ 0 ko₁o₂ .ko₂c₂Cko₂o₁/ 0 kc₂o₂

ko₁c₁ 0 .kc₁o₁Ckc₁c₂/ kc₂c₁

0 ko₂c₂ kc₁c₂ .kc₂c₁Ckc₂o₂/ 1 CC A;

which in compact notation is

KD 0 BB

@

.k₁₃Ck₁₂/ k₂₁ k₃₁ 0 k₁₂ .k24Ck₂₁/ 0 k₄₂ k₁₃ 0 .k₃₁Ck₃₄/ k₄₃

0 k₂₄ k₃₄ .k43Ck₄₂/ 1 CC A:

7.3 Nine-State Model

We have seen how to formulate probability density systems for two-state and four- state Markov models. For even larger Markov models, it is useful to introduce two- dimensional numbering. This will be illustrated using the nine-state model given in Fig.7.2. HereSij;i;j D 1; 2; 3, denotes the states of the Markov model andK_ij^mn

Fig. 7.2 Markov model

including nine possible states S31 S32 S33

S21 S22 S23

S11 S12 S13

K₃₁³² K₃₁²¹

K₃₂³¹

K₃₂³³ K₃₂²²

K₃₃³²

K₃₃²³ K₂₁³¹

K₂₁²² K₂₁¹¹

K₂₂³² K₂₂²¹

K₂₂²³ K₂₂¹²

K₂₃³³

K₂₃²²

K₂₃¹³ K₁₁²¹

K₁₁¹² K₁₂²² K₁₂¹¹

K₁₂¹³ K₁₃²³ K₁₃¹²

denotes¹the reaction rate from the stateSijto the stateSmn:The system governing the probability density functions of these states can be written in the form

@ij

@t C @

@x a^x_ijij

C @

@y

a^y_ijij

DRij; (7.9)

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:

Here ij denotes the probability density function of the state Sij and we use the convention thatK_ij^mnD0fori;j;m;n… f1; 2; 3g:We also have

a^x_ijDijvr.yx/Cvd.c₀x/ ; a^y_ijDijvr.xy/Cvs.c₁y/ ;

whereij D 1 when the state Sijrepresents an open state and ij D 0 when Sij

represents a closed state.

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1We useK_ijas shorthand forK_i;j;but we use the comma when an index of the formjC1is needed,