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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 the indicated four state model when C3 has a mean lifetime of 1 s (A), 1 ms (B), and 1 μs (C). The time constants (and areas) of the exponential components are (A) E3: 3003 ms (33.4%); E2: 2.00 ms (50.0%); E1: 0.667 ms (16.6%); (B) E3: 7.46 ms (62.2%); E2: 1.00 ms (33.3%); E1: 0.536 ms (4.47%); (C) E3: 5.24 ms (72.4%); E2: 0.764 ms (27.6%); E1: 0.001 ms (∼0.00%). The observed distribution of all interval durations (black continuous lines) can be expressed as either the sum of E1 (black dashed lines), E2 (red dashed lines), and E3 (gray dashed lines) or as the sum of {C1} (green lines), {C1C2} (blue lines), and {C1C2C3} (orange lines). E3 is comprised of all intervals from {C1C2C3} plus intervals from {C1C2} and {C1} as needed to complete the E3 exponential. E2 is comprised of any leftover intervals from {C1C2} plus intervals from {C1} as needed to complete the E2 exponential, and E1 is comprised of any leftover intervals from {C1}.
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fig8: Composition of the dwell-time distribution of all intervals for the indicated four state model when C3 has a mean lifetime of 1 s (A), 1 ms (B), and 1 μs (C). The time constants (and areas) of the exponential components are (A) E3: 3003 ms (33.4%); E2: 2.00 ms (50.0%); E1: 0.667 ms (16.6%); (B) E3: 7.46 ms (62.2%); E2: 1.00 ms (33.3%); E1: 0.536 ms (4.47%); (C) E3: 5.24 ms (72.4%); E2: 0.764 ms (27.6%); E1: 0.001 ms (∼0.00%). The observed distribution of all interval durations (black continuous lines) can be expressed as either the sum of E1 (black dashed lines), E2 (red dashed lines), and E3 (gray dashed lines) or as the sum of {C1} (green lines), {C1C2} (blue lines), and {C1C2C3} (orange lines). E3 is comprised of all intervals from {C1C2C3} plus intervals from {C1C2} and {C1} as needed to complete the E3 exponential. E2 is comprised of any leftover intervals from {C1C2} plus intervals from {C1} as needed to complete the E2 exponential, and E1 is comprised of any leftover intervals from {C1}.

Mentions: The above sections examined Scheme 2 in which two connected closed states were followed by an open state. We now examine a model with three closed and one open state in series, C3-C2-C1-O1, which would generate three closed exponential components E1, E2, and E3. Data are presented in Fig. 8 (A–C), where tC1 and tC2 are both 1 ms for all three schemes, and tC3 is 1 s in A, 1 ms in B, and 1 μs in C, changed by altering kC3-C2 as indicated. The transition probabilities PC1-O1, PC1-C2, PC2-C1, and PC2-C3 are the same for the three schemes, with a value of 0.5. For each scheme, intervals from {C1C2C3} generate a convolution type distribution analogous to {C1C2} presented earlier, but with one more closed state contributing to the closed intervals. When tC3 is 1 s (A), E3 and {C1C2C3} have long time courses and very low amplitudes so that they run just above the abscissa and are not readily visible. Shortening tC3 to 1 ms (B) or 1 μs (C) progressively increases the amplitudes of E3 and {C1C2C3} and speeds their decays. For all three lifetimes of C3, E3 superimposes {C1C2C3} at longer times, indicating the E3 arises from {C1C2C3} at longer times. Intervals from {C1C2} and {C1} then fill in the gap between the {C1C2C3} distribution and E3 at shorter times to complete the E3 exponential. The remaining intervals from {C1C2} and some of the intervals from {C1} then generate the E2 exponential, and finally, any remaining intervals in {C1} not used to complete the E3 and E2 exponentials generate E1.


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 the indicated four state model when C3 has a mean lifetime of 1 s (A), 1 ms (B), and 1 μs (C). The time constants (and areas) of the exponential components are (A) E3: 3003 ms (33.4%); E2: 2.00 ms (50.0%); E1: 0.667 ms (16.6%); (B) E3: 7.46 ms (62.2%); E2: 1.00 ms (33.3%); E1: 0.536 ms (4.47%); (C) E3: 5.24 ms (72.4%); E2: 0.764 ms (27.6%); E1: 0.001 ms (∼0.00%). The observed distribution of all interval durations (black continuous lines) can be expressed as either the sum of E1 (black dashed lines), E2 (red dashed lines), and E3 (gray dashed lines) or as the sum of {C1} (green lines), {C1C2} (blue lines), and {C1C2C3} (orange lines). E3 is comprised of all intervals from {C1C2C3} plus intervals from {C1C2} and {C1} as needed to complete the E3 exponential. E2 is comprised of any leftover intervals from {C1C2} plus intervals from {C1} as needed to complete the E2 exponential, and E1 is comprised of any leftover intervals from {C1}.
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fig8: Composition of the dwell-time distribution of all intervals for the indicated four state model when C3 has a mean lifetime of 1 s (A), 1 ms (B), and 1 μs (C). The time constants (and areas) of the exponential components are (A) E3: 3003 ms (33.4%); E2: 2.00 ms (50.0%); E1: 0.667 ms (16.6%); (B) E3: 7.46 ms (62.2%); E2: 1.00 ms (33.3%); E1: 0.536 ms (4.47%); (C) E3: 5.24 ms (72.4%); E2: 0.764 ms (27.6%); E1: 0.001 ms (∼0.00%). The observed distribution of all interval durations (black continuous lines) can be expressed as either the sum of E1 (black dashed lines), E2 (red dashed lines), and E3 (gray dashed lines) or as the sum of {C1} (green lines), {C1C2} (blue lines), and {C1C2C3} (orange lines). E3 is comprised of all intervals from {C1C2C3} plus intervals from {C1C2} and {C1} as needed to complete the E3 exponential. E2 is comprised of any leftover intervals from {C1C2} plus intervals from {C1} as needed to complete the E2 exponential, and E1 is comprised of any leftover intervals from {C1}.
Mentions: The above sections examined Scheme 2 in which two connected closed states were followed by an open state. We now examine a model with three closed and one open state in series, C3-C2-C1-O1, which would generate three closed exponential components E1, E2, and E3. Data are presented in Fig. 8 (A–C), where tC1 and tC2 are both 1 ms for all three schemes, and tC3 is 1 s in A, 1 ms in B, and 1 μs in C, changed by altering kC3-C2 as indicated. The transition probabilities PC1-O1, PC1-C2, PC2-C1, and PC2-C3 are the same for the three schemes, with a value of 0.5. For each scheme, intervals from {C1C2C3} generate a convolution type distribution analogous to {C1C2} presented earlier, but with one more closed state contributing to the closed intervals. When tC3 is 1 s (A), E3 and {C1C2C3} have long time courses and very low amplitudes so that they run just above the abscissa and are not readily visible. Shortening tC3 to 1 ms (B) or 1 μs (C) progressively increases the amplitudes of E3 and {C1C2C3} and speeds their decays. For all three lifetimes of C3, E3 superimposes {C1C2C3} at longer times, indicating the E3 arises from {C1C2C3} at longer times. Intervals from {C1C2} and {C1} then fill in the gap between the {C1C2C3} distribution and E3 at shorter times to complete the E3 exponential. The remaining intervals from {C1C2} and some of the intervals from {C1} then generate the E2 exponential, and finally, any remaining intervals in {C1} not used to complete the E3 and E2 exponentials generate E1.

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