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Linking exponential components to kinetic states in Markov models for single-channel gating.

Shelley C, Magleby KL - J. Gen. Physiol. (2008)

Bottom Line: Lond.The relationship between components and states is found to be both intuitive and paradoxical, depending on the ratios of the state lifetimes.The approach used here allows the exponential components to be interpreted in terms of underlying states for all possible values of the rate constants, something not previously possible.

View Article: PubMed Central - PubMed

Affiliation: Department of Physiology and Biophysics and the Neuroscience Program, University of Miami, Miller School of Medicine, Miami, FL 33136, USA.

ABSTRACT
Discrete state Markov models have proven useful for describing the gating of single ion channels. Such models predict that the dwell-time distributions of open and closed interval durations are described by mixtures of exponential components, with the number of exponential components equal to the number of states in the kinetic gating mechanism. Although the exponential components are readily calculated (Colquhoun and Hawkes, 1982, Phil. Trans. R. Soc. Lond. B. 300:1-59), there is little practical understanding of the relationship between components and states, as every rate constant in the gating mechanism contributes to each exponential component. We now resolve this problem for simple models. As a tutorial we first illustrate how the dwell-time distribution of all closed intervals arises from the sum of constituent distributions, each arising from a specific gating sequence. The contribution of constituent distributions to the exponential components is then determined, giving the relationship between components and states. Finally, the relationship between components and states is quantified by defining and calculating the linkage of components to states. The relationship between components and states is found to be both intuitive and paradoxical, depending on the ratios of the state lifetimes. Nevertheless, both the intuitive and paradoxical observations can be described within a consistent framework. The approach used here allows the exponential components to be interpreted in terms of underlying states for all possible values of the rate constants, something not previously possible.

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Composition of the dwell-time distribution of all intervals for Scheme 2 in which kC2-C1 is 5,000/s, giving a tC2/tC1 ratio of 0.2. See legend of Fig. 3 for plot details. Compared with Fig. 3 where the tC2/tC1 ratio 1, decreasing tC2/tC1 fivefold greatly increases the number of {C1} intervals needed to fill in the gap between {C1C2} and E2 to complete E2 (gray areas in C and D and plot in E). Consequently, most of the {C1} intervals go to E2, leaving very few {C1} intervals to generate E1. The net result is that E1 has a low magnitude and fast time constant (A–C).
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fig5: Composition of the dwell-time distribution of all intervals for Scheme 2 in which kC2-C1 is 5,000/s, giving a tC2/tC1 ratio of 0.2. See legend of Fig. 3 for plot details. Compared with Fig. 3 where the tC2/tC1 ratio 1, decreasing tC2/tC1 fivefold greatly increases the number of {C1} intervals needed to fill in the gap between {C1C2} and E2 to complete E2 (gray areas in C and D and plot in E). Consequently, most of the {C1} intervals go to E2, leaving very few {C1} intervals to generate E1. The net result is that E1 has a low magnitude and fast time constant (A–C).

Mentions: Composition of the dwell-time distribution of all intervals for Scheme 2 in which the tC2/tC1 ratio is 1. (A) The observed distribution of all interval durations (black line) is comprised of all {C1} intervals (green line) plus all {C1C2} intervals (blue line) and is also described by the sum of the exponential components E1 (black dashed line) plus E2 (red dashed line). (B) Semilogarithmic plots of the distributions shown in A plus the constituent distributions (purple lines) for specific gating sequences n = 1–6 in Table II, where n indicates the number of C1 to C2 transitions for each interval in that distribution. The constituent distributions for n = 1 to infinity sum to generate {C1C2}. (C) The difference between {C1} and E1 (shaded area) indicates the “excess” intervals in {C1} over those required for E1. (D) The difference between E2 and {C1C2} (shaded area) indicates the “missing” intervals needed to fill in the gap between {C1C2} and E2 to complete E2. (E) A plot of the “missing” intervals, E2-{C1C2}, exactly superimposes a plot of the “excess” intervals, {C1}-E1, for all interval durations, indicating that the excess intervals are exactly sufficient to fill in the missing intervals at each point in time. Clearly, E1 is not equal to {C1} and E2 is not equal to {C1C2} when the ratio of tC2/tC1 is 1. 3–5 and 8 can be converted into probability density functions by dividing the values on the ordinate by 2.


Linking exponential components to kinetic states in Markov models for single-channel gating.

Shelley C, Magleby KL - J. Gen. Physiol. (2008)

Composition of the dwell-time distribution of all intervals for Scheme 2 in which kC2-C1 is 5,000/s, giving a tC2/tC1 ratio of 0.2. See legend of Fig. 3 for plot details. Compared with Fig. 3 where the tC2/tC1 ratio 1, decreasing tC2/tC1 fivefold greatly increases the number of {C1} intervals needed to fill in the gap between {C1C2} and E2 to complete E2 (gray areas in C and D and plot in E). Consequently, most of the {C1} intervals go to E2, leaving very few {C1} intervals to generate E1. The net result is that E1 has a low magnitude and fast time constant (A–C).
© Copyright Policy
Related In: Results  -  Collection

License 1 - License 2
Show All Figures
getmorefigures.php?uid=PMC2483338&req=5

fig5: Composition of the dwell-time distribution of all intervals for Scheme 2 in which kC2-C1 is 5,000/s, giving a tC2/tC1 ratio of 0.2. See legend of Fig. 3 for plot details. Compared with Fig. 3 where the tC2/tC1 ratio 1, decreasing tC2/tC1 fivefold greatly increases the number of {C1} intervals needed to fill in the gap between {C1C2} and E2 to complete E2 (gray areas in C and D and plot in E). Consequently, most of the {C1} intervals go to E2, leaving very few {C1} intervals to generate E1. The net result is that E1 has a low magnitude and fast time constant (A–C).
Mentions: Composition of the dwell-time distribution of all intervals for Scheme 2 in which the tC2/tC1 ratio is 1. (A) The observed distribution of all interval durations (black line) is comprised of all {C1} intervals (green line) plus all {C1C2} intervals (blue line) and is also described by the sum of the exponential components E1 (black dashed line) plus E2 (red dashed line). (B) Semilogarithmic plots of the distributions shown in A plus the constituent distributions (purple lines) for specific gating sequences n = 1–6 in Table II, where n indicates the number of C1 to C2 transitions for each interval in that distribution. The constituent distributions for n = 1 to infinity sum to generate {C1C2}. (C) The difference between {C1} and E1 (shaded area) indicates the “excess” intervals in {C1} over those required for E1. (D) The difference between E2 and {C1C2} (shaded area) indicates the “missing” intervals needed to fill in the gap between {C1C2} and E2 to complete E2. (E) A plot of the “missing” intervals, E2-{C1C2}, exactly superimposes a plot of the “excess” intervals, {C1}-E1, for all interval durations, indicating that the excess intervals are exactly sufficient to fill in the missing intervals at each point in time. Clearly, E1 is not equal to {C1} and E2 is not equal to {C1C2} when the ratio of tC2/tC1 is 1. 3–5 and 8 can be converted into probability density functions by dividing the values on the ordinate by 2.

Bottom Line: Lond.The relationship between components and states is found to be both intuitive and paradoxical, depending on the ratios of the state lifetimes.The approach used here allows the exponential components to be interpreted in terms of underlying states for all possible values of the rate constants, something not previously possible.

View Article: PubMed Central - PubMed

Affiliation: Department of Physiology and Biophysics and the Neuroscience Program, University of Miami, Miller School of Medicine, Miami, FL 33136, USA.

ABSTRACT
Discrete state Markov models have proven useful for describing the gating of single ion channels. Such models predict that the dwell-time distributions of open and closed interval durations are described by mixtures of exponential components, with the number of exponential components equal to the number of states in the kinetic gating mechanism. Although the exponential components are readily calculated (Colquhoun and Hawkes, 1982, Phil. Trans. R. Soc. Lond. B. 300:1-59), there is little practical understanding of the relationship between components and states, as every rate constant in the gating mechanism contributes to each exponential component. We now resolve this problem for simple models. As a tutorial we first illustrate how the dwell-time distribution of all closed intervals arises from the sum of constituent distributions, each arising from a specific gating sequence. The contribution of constituent distributions to the exponential components is then determined, giving the relationship between components and states. Finally, the relationship between components and states is quantified by defining and calculating the linkage of components to states. The relationship between components and states is found to be both intuitive and paradoxical, depending on the ratios of the state lifetimes. Nevertheless, both the intuitive and paradoxical observations can be described within a consistent framework. The approach used here allows the exponential components to be interpreted in terms of underlying states for all possible values of the rate constants, something not previously possible.

Show MeSH