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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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It is the tC2/tC1 ratio in Scheme 2 rather than the transition probabilities that determine the paradoxical shifts in the linkage between components and states. (A–C) Plots of τE1 and τE2 against k(C2-C1) in Scheme 2 as k(C2-C1) is changed over six orders of magnitude. These changes in k(C2-C1) change tC2 from 1 s to 1 μs as tC1 remains constant at 1 ms. The resulting change in the tC2/tC1 ratio is plotted at the bottom of the figure. Plots are presented for three different transition probabilities ratios for PC1-C2/PC1-O1 of 0.999/0.001 (A), 0.5/0.5 (B), and 0.001/0.999 (C). For all three transition probability ratios, τE1 tracks tC1 and τE2 tracks tC2 (with an offset in A and B) when tC2 >> tC1 and then τE1 tracks tC2 and τE2 tracks tC1 (with an offset in A and B) when tC2 << tC1. The inset in C shows the switch in tracking follows the same pattern as in A and B. (D–F) Areas of E1, E2, {C1}, and {C1C2} as a function of kC1-C2 and the resulting tC2/tC1 ratio. In D, a log scale is used so that the change in the small area of E1 can be seen. The corresponding change in the area of E2 is too small compared with the large area of E2 to be seen. Note that the paradoxical shifts in time constants and areas of the exponential components as a function of the tC2/tC1 ratio are still observed for a 106-fold change in transition probabilities.
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fig6: It is the tC2/tC1 ratio in Scheme 2 rather than the transition probabilities that determine the paradoxical shifts in the linkage between components and states. (A–C) Plots of τE1 and τE2 against k(C2-C1) in Scheme 2 as k(C2-C1) is changed over six orders of magnitude. These changes in k(C2-C1) change tC2 from 1 s to 1 μs as tC1 remains constant at 1 ms. The resulting change in the tC2/tC1 ratio is plotted at the bottom of the figure. Plots are presented for three different transition probabilities ratios for PC1-C2/PC1-O1 of 0.999/0.001 (A), 0.5/0.5 (B), and 0.001/0.999 (C). For all three transition probability ratios, τE1 tracks tC1 and τE2 tracks tC2 (with an offset in A and B) when tC2 >> tC1 and then τE1 tracks tC2 and τE2 tracks tC1 (with an offset in A and B) when tC2 << tC1. The inset in C shows the switch in tracking follows the same pattern as in A and B. (D–F) Areas of E1, E2, {C1}, and {C1C2} as a function of kC1-C2 and the resulting tC2/tC1 ratio. In D, a log scale is used so that the change in the small area of E1 can be seen. The corresponding change in the area of E2 is too small compared with the large area of E2 to be seen. Note that the paradoxical shifts in time constants and areas of the exponential components as a function of the tC2/tC1 ratio are still observed for a 106-fold change in transition probabilities.

Mentions: The observations in 3–5 and Table III suggest that the relative contribution of the {C1} and {C1C2} intervals to E1 and E2 shifts with the tC2/tC1 ratio. To investigate these shifts further, Fig. 6 B plots the time constants of E1 and E2, τE1 and τE2, and the lifetimes of C1 and C2, tC1 and tC2, and Fig. 6 E plots the areas of E1, E2, {C1}, and {C1C2} as kC2-C1 in Scheme 2 is changed over six orders of magnitude to change the tC2/tC1 ratio from 103 to 10−3 (see bottom of Fig. 6). This change in kC2-C1 changes tC2 from 1 s to 1 μs (Fig. 6 B, red dashed line) while having no effect on tC1, which remains constant at 1 ms (Fig. 6 B, black continuous line). As tC2 decreases, decreasing the tC2/tC1 ratio, τE1 first tracks tC1 and then switches to track tC2 (Fig. 6 B, black dashed line). The switch in tracking occurs as the tC2/tC1 ratio passes through 1, with τE1 equal to tC1 when tC2 >> tC11 and then equal to tC2 when tC2 << tC1.


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

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

It is the tC2/tC1 ratio in Scheme 2 rather than the transition probabilities that determine the paradoxical shifts in the linkage between components and states. (A–C) Plots of τE1 and τE2 against k(C2-C1) in Scheme 2 as k(C2-C1) is changed over six orders of magnitude. These changes in k(C2-C1) change tC2 from 1 s to 1 μs as tC1 remains constant at 1 ms. The resulting change in the tC2/tC1 ratio is plotted at the bottom of the figure. Plots are presented for three different transition probabilities ratios for PC1-C2/PC1-O1 of 0.999/0.001 (A), 0.5/0.5 (B), and 0.001/0.999 (C). For all three transition probability ratios, τE1 tracks tC1 and τE2 tracks tC2 (with an offset in A and B) when tC2 >> tC1 and then τE1 tracks tC2 and τE2 tracks tC1 (with an offset in A and B) when tC2 << tC1. The inset in C shows the switch in tracking follows the same pattern as in A and B. (D–F) Areas of E1, E2, {C1}, and {C1C2} as a function of kC1-C2 and the resulting tC2/tC1 ratio. In D, a log scale is used so that the change in the small area of E1 can be seen. The corresponding change in the area of E2 is too small compared with the large area of E2 to be seen. Note that the paradoxical shifts in time constants and areas of the exponential components as a function of the tC2/tC1 ratio are still observed for a 106-fold change in transition probabilities.
© Copyright Policy
Related In: Results  -  Collection

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

fig6: It is the tC2/tC1 ratio in Scheme 2 rather than the transition probabilities that determine the paradoxical shifts in the linkage between components and states. (A–C) Plots of τE1 and τE2 against k(C2-C1) in Scheme 2 as k(C2-C1) is changed over six orders of magnitude. These changes in k(C2-C1) change tC2 from 1 s to 1 μs as tC1 remains constant at 1 ms. The resulting change in the tC2/tC1 ratio is plotted at the bottom of the figure. Plots are presented for three different transition probabilities ratios for PC1-C2/PC1-O1 of 0.999/0.001 (A), 0.5/0.5 (B), and 0.001/0.999 (C). For all three transition probability ratios, τE1 tracks tC1 and τE2 tracks tC2 (with an offset in A and B) when tC2 >> tC1 and then τE1 tracks tC2 and τE2 tracks tC1 (with an offset in A and B) when tC2 << tC1. The inset in C shows the switch in tracking follows the same pattern as in A and B. (D–F) Areas of E1, E2, {C1}, and {C1C2} as a function of kC1-C2 and the resulting tC2/tC1 ratio. In D, a log scale is used so that the change in the small area of E1 can be seen. The corresponding change in the area of E2 is too small compared with the large area of E2 to be seen. Note that the paradoxical shifts in time constants and areas of the exponential components as a function of the tC2/tC1 ratio are still observed for a 106-fold change in transition probabilities.
Mentions: The observations in 3–5 and Table III suggest that the relative contribution of the {C1} and {C1C2} intervals to E1 and E2 shifts with the tC2/tC1 ratio. To investigate these shifts further, Fig. 6 B plots the time constants of E1 and E2, τE1 and τE2, and the lifetimes of C1 and C2, tC1 and tC2, and Fig. 6 E plots the areas of E1, E2, {C1}, and {C1C2} as kC2-C1 in Scheme 2 is changed over six orders of magnitude to change the tC2/tC1 ratio from 103 to 10−3 (see bottom of Fig. 6). This change in kC2-C1 changes tC2 from 1 s to 1 μs (Fig. 6 B, red dashed line) while having no effect on tC1, which remains constant at 1 ms (Fig. 6 B, black continuous line). As tC2 decreases, decreasing the tC2/tC1 ratio, τE1 first tracks tC1 and then switches to track tC2 (Fig. 6 B, black dashed line). The switch in tracking occurs as the tC2/tC1 ratio passes through 1, with τE1 equal to tC1 when tC2 >> tC11 and then equal to tC2 when tC2 << tC1.

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