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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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Simulated single-channel data and closed dwell-time distributions for the three-state model described by Scheme 2. (A) Simulated single-channel current record. Channel openings are shown as upward steps. (B) Sigworth and Sine plot of the closed-dwell time distribution for 106 simulated intervals. (C) Same data as in B on linear coordinates. The dashed lines in both plots indicate the fast E1 and slow E2 exponential components. The time constants (arrows) and areas, which are identical in both B and C, are listed.
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fig2: Simulated single-channel data and closed dwell-time distributions for the three-state model described by Scheme 2. (A) Simulated single-channel current record. Channel openings are shown as upward steps. (B) Sigworth and Sine plot of the closed-dwell time distribution for 106 simulated intervals. (C) Same data as in B on linear coordinates. The dashed lines in both plots indicate the fast E1 and slow E2 exponential components. The time constants (arrows) and areas, which are identical in both B and C, are listed.

Mentions: To determine the effect of a compound state on the relationship between components and states, we examined a linear gating mechanism with two closed states in series, as described by Scheme 2.(SCHEME 2)As with Scheme 1, each state has a mean lifetime of 1 ms. The two connected closed states C1 and C2 in Scheme 2 form a compound closed state. Compound states arise when transitions can occur directly between two or more states of indistinguishable conductance. Simulated single channel records from Scheme 2 are shown in Fig. 2 A, where there are brief duration closed intervals, as in Fig. 1 A, and also longer duration closed intervals. As was the case for Scheme 1, which also had one open state, the open dwell-time distribution would be described by a single exponential component with a time constant identical to the mean lifetime of the open state and would be identical to the distributions in Fig. 1 (B and C). The closed dwell-time distribution from Scheme 2 is shown in Fig. 2 B for the Sigworth and Sine transform and in Fig. 2 C for linear coordinates. In contrast to the single exponential for Scheme 1, the closed dwell-time distribution for Scheme 2 (continuous line) is now described by the sum of two exponential components, E1 and E2 (dashed lines), with time constants of 0.586 ms and 3.41 ms (arrows) and areas of 0.146 and 0.854, respectively. Neither of these time constants match the 1-ms mean lifetime of either closed state. Hence, when a kinetic scheme contains a compound state, exponential components are not necessarily directly linked to states, as previously noted (Colquhoun and Hawkes, 1994, 1995b).


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

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

Simulated single-channel data and closed dwell-time distributions for the three-state model described by Scheme 2. (A) Simulated single-channel current record. Channel openings are shown as upward steps. (B) Sigworth and Sine plot of the closed-dwell time distribution for 106 simulated intervals. (C) Same data as in B on linear coordinates. The dashed lines in both plots indicate the fast E1 and slow E2 exponential components. The time constants (arrows) and areas, which are identical in both B and C, are listed.
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Related In: Results  -  Collection

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getmorefigures.php?uid=PMC2483338&req=5

fig2: Simulated single-channel data and closed dwell-time distributions for the three-state model described by Scheme 2. (A) Simulated single-channel current record. Channel openings are shown as upward steps. (B) Sigworth and Sine plot of the closed-dwell time distribution for 106 simulated intervals. (C) Same data as in B on linear coordinates. The dashed lines in both plots indicate the fast E1 and slow E2 exponential components. The time constants (arrows) and areas, which are identical in both B and C, are listed.
Mentions: To determine the effect of a compound state on the relationship between components and states, we examined a linear gating mechanism with two closed states in series, as described by Scheme 2.(SCHEME 2)As with Scheme 1, each state has a mean lifetime of 1 ms. The two connected closed states C1 and C2 in Scheme 2 form a compound closed state. Compound states arise when transitions can occur directly between two or more states of indistinguishable conductance. Simulated single channel records from Scheme 2 are shown in Fig. 2 A, where there are brief duration closed intervals, as in Fig. 1 A, and also longer duration closed intervals. As was the case for Scheme 1, which also had one open state, the open dwell-time distribution would be described by a single exponential component with a time constant identical to the mean lifetime of the open state and would be identical to the distributions in Fig. 1 (B and C). The closed dwell-time distribution from Scheme 2 is shown in Fig. 2 B for the Sigworth and Sine transform and in Fig. 2 C for linear coordinates. In contrast to the single exponential for Scheme 1, the closed dwell-time distribution for Scheme 2 (continuous line) is now described by the sum of two exponential components, E1 and E2 (dashed lines), with time constants of 0.586 ms and 3.41 ms (arrows) and areas of 0.146 and 0.854, respectively. Neither of these time constants match the 1-ms mean lifetime of either closed state. Hence, when a kinetic scheme contains a compound state, exponential components are not necessarily directly linked to states, as previously noted (Colquhoun and Hawkes, 1994, 1995b).

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