Linking exponential components to kinetic states in Markov models for single-channel gating.
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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.
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Affiliation: Department of Physiology and Biophysics and the Neuroscience Program, University of Miami, Miller School of Medicine, Miami, FL 33136, USA.
ABSTRACT
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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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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. |
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Affiliation: Department of Physiology and Biophysics and the Neuroscience Program, University of Miami, Miller School of Medicine, Miami, FL 33136, USA.