Computing Characterizations of Drugs for Ion Channels and Receptors Using Markov Models pp 119-123 | Cite as

# Generalized Systems Governing Probability Density Functions

## Abstract

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 by 2 × 2 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 3 × 3 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.

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 by 2 × 2 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 3 × 3 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

*ρ*

_{ o }) and the closed state (

*ρ*

_{ c }) are governed by the system

*i*= 1, 2, where

*i*= 1 is for the open state and

*i*= 2 is for the closed state. The system can now be written in the form

*i*th component of the matrix vector product

*K ρ*. Here the vector

*ρ*is given by

*γ*

_{ i }is one for the open state (i.e.,

*i*= 1) and zero for the closed state (i.e.,

*i*= 2).

## 7.2 Four-State Model

*O*

_{1}and

*O*

_{2}and two closed states

*C*

_{1}and

*C*

_{2}, 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. 7.1 can now be written in the form

*O*

_{1},

*O*

_{2},

*C*

_{1}, and

*C*

_{2}to be the states 1, 2, 3, and 4, respectively, we can write the system (7.6) in the more compact form

*i*= 1, 2, 3, 4, where

*ρ*= (

*ρ*

_{1},

*ρ*

_{2},

*ρ*

_{3},

*ρ*

_{4})

^{ T }. Here

*γ*

_{ i }is one for the open states (i.e.,

*i*= 1 and

*i*= 2) and zero for the closed states (i.e.,

*i*= 3 and

*i*= 4). Furthermore, the matrix is given by

## 7.3 Nine-State Model

*S*

_{ ij },

*i*,

*j*= 1, 2, 3, denotes the states of the Markov model and

*K*

_{ ij }

^{ mn }denotes

^{1}the reaction rate from the state

*S*

_{ ij }to the state

*S*

_{ mn }. The system governing the probability density functions of these states can be written in the form

*ρ*

_{ ij }denotes the probability density function of the state

*S*

_{ ij }and we use the convention that

*K*

_{ ij }

^{ mn }= 0 for \(i,j,m,n\notin \left \{1,2,3\right \}.\) We also have

*γ*

_{ ij }= 1 when the state

*S*

_{ ij }represents an open state and

*γ*

_{ ij }= 0 when

*S*

_{ ij }represents a closed state.

## Footnotes

- 1.
We use

*K*_{ ij }as shorthand for*K*_{i, j}, but we use the comma when an index of the form*j*+ 1 is needed, that is we write*K*_{i, j+1}.

### References

- 35.M.A. Huertas, G.D. Smith, The dynamics of luminal depletion and the stochastic gating of Ca
^{2+}-activated Ca^{2+}channels and release sites. J. Theor. Biol.**246**(2), 332–354 (2007)MathSciNetCrossRefGoogle Scholar

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