Phasic activation and state-dependent inhibition: An explicit solution for a three-state ion channel system

Vladimir V. Uteshev, Peter S. Pennefather

Research output: Contribution to journalArticle

5 Citations (Scopus)

Abstract

Ion channels can exist in three broad classes of states: closed (C), open (O), and desensitized or inactivated (I). Many ion channel modulators interact preferentially with one of these states giving rise to use or state dependent effects and often complex interactions with phasic stimulation. Although mathematical descriptions of three-state systems at steady-state or following a single perturbation are well known, a solution to the boundary problem of how such a system interacts with regular phasic perturbations or stimuli has not previously been reported. In physiological systems, ion channels typically experience phasic stimulation and an explicit mathematical description of the interaction between phasic activation and use-dependent modulation within the framework of a three-state system should be useful. Here we present derivations of generalized, recurrent and explicit formulae describing this interaction that allow prediction of the degree of use dependent modulation at any point during a train of repeated stimuli. Each state is defined by two functions of time (y or z) that define the fraction of channels in that state during the alternating stimulation and resting phases, respectively. For a train of repeated stimuli we defined vector Z(2n) that has coordinates Z(2n)(O) and Z(2n)(I) representing the values for O and I states at the end of the n-th resting phase. We then defined a recurrent relationship, Z(2n) = FZ(2n-2) + G. Therefore, for the steady state: Z = (E-F)-1G, where F = (C11C12C21C22), G = (C13C23), E is the identity matrix. Matrix and vector elements, c(ij), are defined in terms of duration of the repeated stimulation and resting phases and the two sets of six rate constants that describe the three-state model during those two phases. Several conclusions can be deduced from the formulation: (1) in order to determine an occupancy of any state under the cyclic stimulus-rest protocol it is necessary to know at least two occupancy levels - either of the same state but related to different phases of the stimulus protocol or of different states at the same point in the stimulus protocol, for instance: Z(2n)(I) = f(Z(2n-2)(O), Z(2n-2)(I)) = h(Z(2n)(O), Z(2n-2)(O)) = g(Z(2n)(O), Z(2n-2)(I)) = ...; (2) the solution Z(2n) can be approximated by a matrix-exponential function, with the precision of the approximation depending on the interval between stimuli; (3) for all steady-state solutions, the matrix F is such that lim n→∞nΠF = ∞ΠF = 0 is a zero-matrix. Application of this approach is illustrated using experimentally derived parameters describing desensitization of GABA(a) receptors and modulation of that process by the anesthetic propofol.

Original languageEnglish
Pages (from-to)11-23
Number of pages13
JournalJournal of Theoretical Biology
Volume181
Issue number1
DOIs
StatePublished - 7 Jul 1996

Fingerprint

Ion Channels
ion channels
Explicit Solution
Activation
Chemical activation
Modulation
Dependent
Ions
GABA Receptors
Interaction
Zero matrix
Propofol
Perturbation
anesthetics
Matrix Exponential
Unit matrix
Anesthetics
Matrix Function
Steady-state Solution
Modulator

Cite this

@article{6594d7173bb342aea7bfb5b399abd0b8,
title = "Phasic activation and state-dependent inhibition: An explicit solution for a three-state ion channel system",
abstract = "Ion channels can exist in three broad classes of states: closed (C), open (O), and desensitized or inactivated (I). Many ion channel modulators interact preferentially with one of these states giving rise to use or state dependent effects and often complex interactions with phasic stimulation. Although mathematical descriptions of three-state systems at steady-state or following a single perturbation are well known, a solution to the boundary problem of how such a system interacts with regular phasic perturbations or stimuli has not previously been reported. In physiological systems, ion channels typically experience phasic stimulation and an explicit mathematical description of the interaction between phasic activation and use-dependent modulation within the framework of a three-state system should be useful. Here we present derivations of generalized, recurrent and explicit formulae describing this interaction that allow prediction of the degree of use dependent modulation at any point during a train of repeated stimuli. Each state is defined by two functions of time (y or z) that define the fraction of channels in that state during the alternating stimulation and resting phases, respectively. For a train of repeated stimuli we defined vector Z(2n) that has coordinates Z(2n)(O) and Z(2n)(I) representing the values for O and I states at the end of the n-th resting phase. We then defined a recurrent relationship, Z(2n) = FZ(2n-2) + G. Therefore, for the steady state: Z = (E-F)-1G, where F = (C11C12C21C22), G = (C13C23), E is the identity matrix. Matrix and vector elements, c(ij), are defined in terms of duration of the repeated stimulation and resting phases and the two sets of six rate constants that describe the three-state model during those two phases. Several conclusions can be deduced from the formulation: (1) in order to determine an occupancy of any state under the cyclic stimulus-rest protocol it is necessary to know at least two occupancy levels - either of the same state but related to different phases of the stimulus protocol or of different states at the same point in the stimulus protocol, for instance: Z(2n)(I) = f(Z(2n-2)(O), Z(2n-2)(I)) = h(Z(2n)(O), Z(2n-2)(O)) = g(Z(2n)(O), Z(2n-2)(I)) = ...; (2) the solution Z(2n) can be approximated by a matrix-exponential function, with the precision of the approximation depending on the interval between stimuli; (3) for all steady-state solutions, the matrix F is such that lim n→∞nΠF = ∞ΠF = 0 is a zero-matrix. Application of this approach is illustrated using experimentally derived parameters describing desensitization of GABA(a) receptors and modulation of that process by the anesthetic propofol.",
author = "Uteshev, {Vladimir V.} and Pennefather, {Peter S.}",
year = "1996",
month = "7",
day = "7",
doi = "10.1006/jtbi.1996.0110",
language = "English",
volume = "181",
pages = "11--23",
journal = "Journal of Theoretical Biology",
issn = "0022-5193",
publisher = "Academic Press Inc.",
number = "1",

}

Phasic activation and state-dependent inhibition : An explicit solution for a three-state ion channel system. / Uteshev, Vladimir V.; Pennefather, Peter S.

In: Journal of Theoretical Biology, Vol. 181, No. 1, 07.07.1996, p. 11-23.

Research output: Contribution to journalArticle

TY - JOUR

T1 - Phasic activation and state-dependent inhibition

T2 - An explicit solution for a three-state ion channel system

AU - Uteshev, Vladimir V.

AU - Pennefather, Peter S.

PY - 1996/7/7

Y1 - 1996/7/7

N2 - Ion channels can exist in three broad classes of states: closed (C), open (O), and desensitized or inactivated (I). Many ion channel modulators interact preferentially with one of these states giving rise to use or state dependent effects and often complex interactions with phasic stimulation. Although mathematical descriptions of three-state systems at steady-state or following a single perturbation are well known, a solution to the boundary problem of how such a system interacts with regular phasic perturbations or stimuli has not previously been reported. In physiological systems, ion channels typically experience phasic stimulation and an explicit mathematical description of the interaction between phasic activation and use-dependent modulation within the framework of a three-state system should be useful. Here we present derivations of generalized, recurrent and explicit formulae describing this interaction that allow prediction of the degree of use dependent modulation at any point during a train of repeated stimuli. Each state is defined by two functions of time (y or z) that define the fraction of channels in that state during the alternating stimulation and resting phases, respectively. For a train of repeated stimuli we defined vector Z(2n) that has coordinates Z(2n)(O) and Z(2n)(I) representing the values for O and I states at the end of the n-th resting phase. We then defined a recurrent relationship, Z(2n) = FZ(2n-2) + G. Therefore, for the steady state: Z = (E-F)-1G, where F = (C11C12C21C22), G = (C13C23), E is the identity matrix. Matrix and vector elements, c(ij), are defined in terms of duration of the repeated stimulation and resting phases and the two sets of six rate constants that describe the three-state model during those two phases. Several conclusions can be deduced from the formulation: (1) in order to determine an occupancy of any state under the cyclic stimulus-rest protocol it is necessary to know at least two occupancy levels - either of the same state but related to different phases of the stimulus protocol or of different states at the same point in the stimulus protocol, for instance: Z(2n)(I) = f(Z(2n-2)(O), Z(2n-2)(I)) = h(Z(2n)(O), Z(2n-2)(O)) = g(Z(2n)(O), Z(2n-2)(I)) = ...; (2) the solution Z(2n) can be approximated by a matrix-exponential function, with the precision of the approximation depending on the interval between stimuli; (3) for all steady-state solutions, the matrix F is such that lim n→∞nΠF = ∞ΠF = 0 is a zero-matrix. Application of this approach is illustrated using experimentally derived parameters describing desensitization of GABA(a) receptors and modulation of that process by the anesthetic propofol.

AB - Ion channels can exist in three broad classes of states: closed (C), open (O), and desensitized or inactivated (I). Many ion channel modulators interact preferentially with one of these states giving rise to use or state dependent effects and often complex interactions with phasic stimulation. Although mathematical descriptions of three-state systems at steady-state or following a single perturbation are well known, a solution to the boundary problem of how such a system interacts with regular phasic perturbations or stimuli has not previously been reported. In physiological systems, ion channels typically experience phasic stimulation and an explicit mathematical description of the interaction between phasic activation and use-dependent modulation within the framework of a three-state system should be useful. Here we present derivations of generalized, recurrent and explicit formulae describing this interaction that allow prediction of the degree of use dependent modulation at any point during a train of repeated stimuli. Each state is defined by two functions of time (y or z) that define the fraction of channels in that state during the alternating stimulation and resting phases, respectively. For a train of repeated stimuli we defined vector Z(2n) that has coordinates Z(2n)(O) and Z(2n)(I) representing the values for O and I states at the end of the n-th resting phase. We then defined a recurrent relationship, Z(2n) = FZ(2n-2) + G. Therefore, for the steady state: Z = (E-F)-1G, where F = (C11C12C21C22), G = (C13C23), E is the identity matrix. Matrix and vector elements, c(ij), are defined in terms of duration of the repeated stimulation and resting phases and the two sets of six rate constants that describe the three-state model during those two phases. Several conclusions can be deduced from the formulation: (1) in order to determine an occupancy of any state under the cyclic stimulus-rest protocol it is necessary to know at least two occupancy levels - either of the same state but related to different phases of the stimulus protocol or of different states at the same point in the stimulus protocol, for instance: Z(2n)(I) = f(Z(2n-2)(O), Z(2n-2)(I)) = h(Z(2n)(O), Z(2n-2)(O)) = g(Z(2n)(O), Z(2n-2)(I)) = ...; (2) the solution Z(2n) can be approximated by a matrix-exponential function, with the precision of the approximation depending on the interval between stimuli; (3) for all steady-state solutions, the matrix F is such that lim n→∞nΠF = ∞ΠF = 0 is a zero-matrix. Application of this approach is illustrated using experimentally derived parameters describing desensitization of GABA(a) receptors and modulation of that process by the anesthetic propofol.

UR - http://www.scopus.com/inward/record.url?scp=0030573547&partnerID=8YFLogxK

U2 - 10.1006/jtbi.1996.0110

DO - 10.1006/jtbi.1996.0110

M3 - Article

C2 - 8796187

AN - SCOPUS:0030573547

VL - 181

SP - 11

EP - 23

JO - Journal of Theoretical Biology

JF - Journal of Theoretical Biology

SN - 0022-5193

IS - 1

ER -