Linking Exponential Components to Kinetic States in Markov Models for Single-Channel Gating

Ion channels are ubiquitously distributed proteins that control the passive flux of ions through cell membranes by opening and closing (gating) their pores (Hille, 2001). As gatekeepers, ion channels play key roles in many physiological processes, including generation and propagation of action potentials, synaptic transmission, and sensory reception (Hille, 2001). Ion channels gate their pores by passing through a series of conformational states (Jiang et al., 2002; Blunck et al., 2006; Tombola et al., 2006; Purohit et al., 2007). The gating can be described in terms of kinetic reaction schemes that give the number of open and closed states entered during gating, the transition pathways among the states, the rate constants for the transitions, and the voltage and ligand modulation of the rate constants (Colquhoun and Hawkes, 1982, 1995b). Such discrete state Markov models have proven highly useful for describing the underlying gating mechanisms (Horn and Vandenberg, 1984; Zagotta et al., 1994; Cox et al., 1997; Schoppa and Sigworth, 1998; Horrigan et al., 1999; Cox and Aldrich, 2000; Rothberg and Magleby, 1998, 2000; Gil et al., 2001; Zhang et al., 2001; Sigg and Bezanilla, 2003; Chakrapani et al., 2004), and critical tests of single-channel gating for BK channels (McManus and Magleby, 1989) and NMDA receptors (Gibb and Colquhoun, 1992) are consistent with Markov gating.

Single channel recordings from ion channels indicate transitions between open and closed states by characteristic step changes in the single-channel current level. Ion channels can also make transitions among states with the same conductance, such as transitions among closed states and transitions among open states. Connected states of the same conductance are referred to as compound states, and transitions among compound states are hidden because the current level does not change. Nevertheless, information about these hidden transitions is contained in the interval durations, which are lengthened by such transitions.

A standard method used to display data recorded from single channels is to plot the number of observed intervals against their durations, giving open and closed dwell-time histograms, also referred to as dwell-time distributions, or open and closed period distributions. Normalizing the area of the distribution to 1.0 by dividing by the number of intervals in the distribution gives a probability density function, where the area under the curve between any two time values gives the probability of observing an interval with a lifetime (dwell time) between those values (Colquhoun and Hawkes, 1994, 1995b).

Markov models used to describe single channel kinetics predict that the open and closed dwell-time distributions are comprised of the sums of exponential components (more correctly mixtures because the areas sum to 1.0), with the total number of open and closed exponential components equal to the number of open and closed states, respectively (Colquhoun and Hawkes, 1982, 1995b). Consequently, the experimentally observed dwell-time distributions are typically fit with sums of exponential components to describe the data, such that

where f(t) is the dwell-time distribution, w_i and τ_i are the magnitude and time constant of each exponential component i, respectively, and t is interval duration. The area of each component, a_i, which gives the number of intervals in that component, is given by a_i = w_iτ_i. It is the exponential components that are typically listed in tables and discussed in papers on single channel kinetics, and the exponential components are often the output (solutions) of gating mechanism calculated with analytical or Q matrix methods (Colquhoun and Hawkes, 1981, 1982, 1995a).

In spite of the emphasis on the exponential components and the many hundreds of papers published with plotted dwell-time distributions and tables of exponential components, there is little practical understanding of how the components relate to specific states in kinetic gating mechanisms (Colquhoun and Hawkes, 1994, 1995b). The reason for this is that all of the rate constants that determine the lifetimes of any of the states in a compound state also contribute to each of the exponential components generated by those states (Colquhoun and Hawkes, 1982, 1995b). Consequently, it is well known for gating mechanisms with compound states that the time constants of the exponential components cannot simply be interpreted as the mean lifetimes of certain states and that the areas of the components cannot be interpreted as the numbers of sojourns to those states (Colquhoun and Hawkes, 1994, 1995b). The problem is further compounded because the methods used to calculate the exponential components from gating mechanisms give little practical information about the relationships between specific components and states. For analytic solutions, which can be derived for models with a limited number of states, the relationship between components and states is obscured in the equations, as shown in the Appendix and Covernton et al., (1994) for a three state model, and in Colquhoun and Hawkes (1977, 1981), Magleby and Pallotta (1983), and Jackson (1997) for more complex models. For the numeric methods that can be used to solve any gating mechanism (Colquhoun and Hawkes, 1981, 1982), there is even less practical information about the contributions of specific states to the various exponential components because of the matrix methods used in the calculations (Horn and Lange, 1983; Colquhoun and Hawkes, 1995a; Colquhoun et al., 1996; Qin et al., 1997).

Hence, the standard dogma is that it is not possible to place physical interpretations on the time constants and magnitudes of the exponential components (Colquhoun and Hawkes, 1995b) except in special cases with extreme differences in some of the rate constants (Colquhoun and Hawkes, 1994), although it should be mentioned that some information relating observed exponentials in experimental data to the underlying states can be obtained when the starting state is known, by examining either first latencies to the next opening/shutting interval or the rise times of macroscopic currents following step changes in agonist concentration or voltage (Edmonds and Colquhoun, 1992; Colquhoun et al., 1996; Wyllie et al., 1998; Horrigan and Aldrich, 2002).

We now present an approach to resolve the relationship between components and states for a model with one open and two closed states in series. We examine simulated gating to determine directly the contributions of the various states to the exponential components, and quantify the contributions in terms of linkage. Our systematic analysis reveals both intuitive and highly paradoxical relationships between components and states, depending on the lifetime ratios of the closed states. Nevertheless, both the intuitive and paradoxical results can be described within a consistent framework.

Our observations should facilitate an understanding of single channel data by providing a physical basis for the origins of the exponential components and of the relationship between components and states. Our observations should also provide sufficient insight to prevent incorrect conclusions when interpreting dwell-time distributions in terms of underlying states and transition probabilities.

Commonly used abbreviations are listed in Table I.

Colquhoun and Hawkes (1982, 1994, 1995b) have presented detailed methods for calculating the exponential components that sum to describe the dwell-time distributions generated by discrete state Markov models (Colquhoun and Hawkes, 1982, 1994, 1995b). We use their Q-matrix methods (Colquhoun and Hawkes, 1995a) and also their analytical approach (equations in the Appendix) to calculate the exponential components for the models examined. The first step we use to examine the relationship between the exponential components and the underlying states is to determine the specific contributions of the individual states and compound states to the distribution of all closed intervals. Whereas such information can be obtained by the Laplace transform, convolution, and Q matrix methods of Colquhoun and Hawkes (1982), we have chosen to obtain this information by simulating the process by which a hypothetical channel gates, as we found this approach more transparent for revealing the underlying physical basis for the various intervals. This section describes how the constituent dwell-time distributions that sum to form the dwell-time distribution of all intervals were generated.

The probability for a given gating sequence among states in a kinetic scheme is the product of the probabilities for each of the individual gating steps in the sequence. The probability of a transition from state i to state j, P_ij, is given by

where k_ij is the rate constant from state i to state j (Colquhoun and Hawkes, 1995b).

Consider the following gating mechanism

where the rate constants in this scheme (and all following schemes) are in units of per second, and C₂, C₁, and O₁ represent two closed and one open state connected in series, with C₂-C₁ forming a compound state. From this scheme and Eq. 2 the probabilities of various gating transitions and sequences can be calculated. P_O1-C1, the probability of the transition from O₁ to C₁ is 1, as there is only one possible route away from O_1, P_C1-O1 is 0.5, P_C1-C2 is 0.5, and P_C2-C1 is 1.0 Thus, the probability of the gating sequence O₁-C₁-O₁ is 1 × 0.5 = 0.5. The probability of the gating sequence O₁-C₁-C₂-C₁-O₁ is: 1.0 × 0.5 × 1.0 × 0.5 = 0.25. Because closed intervals are always initiated by transitions from O₁ to C₁ and always terminate by transitions from C₁ back to O₁, the general case for any gating sequence in the closed states can be abbreviated as _C1-(C2-C1-)n, where n indicates the number of transitions from C₂ to C₁. The probability of a gating sequence with n transitions from C₂ to C₁, referred to as gating sequence n, is

where n can have integer values ranging from 0 to infinity. For a sample size of N intervals for all possible gating sequences, each specific constituent distribution {C₁-(C₂-C₁)_n} for n = 0 to effectively infinity (see below) was simulated with N×(P_C1-C2)ⁿ×P_C1-O1 random intervals of duration

where d_C1 and d_C2 are random dwell times described by

where t_C1 and t_C2 are the mean lifetimes of states C₁ and C₂, and Rnd is a random number between 0 and 1. N is typically 10⁷ for the simulations.

When n = 0, the constituent distribution includes all unitary sojourns to C₁ and is designated {C₁}; there are no transitions to C₂. In contrast, for values of n between 1 and infinity, each interval results from the sum of 2n+1 exponentially distributed dwell times. Consequently, the constituent distribution {C₁-(C₂-C₁-)_n} for each value of n is described by the convolution of 2n+1 exponential distributions. (Convolutions are discussed in Colquhoun and Hawkes (1995b).) Unlike exponentials, which have a maximum amplitude at zero time, convolutions have a zero magnitude at zero time, increase to a maximum, and then decay (Colquhoun and Hawkes, 1995b).

The sum of all the constituent distributions for values of n from 1 to infinity will be designated as {C₁C₂}, as all intervals in this distribution arise from one or more sojourns to both C₁ and C₂. {C₁C₂} is calculated with an algorithm that sums all of the constituent distributions.

Because the {C₁} and {C₁C₂} constituent distributions include the closed intervals from all possible gating sequences, the sum of {C₁} and {C₁C₂} will give the dwell-time distribution for all observed closed intervals. This is the frequency histogram that would be observed experimentally, assuming that all closings are detected. Dividing the number of intervals in each constituent distribution by N, the total number of closed intervals in all constituent distributions, gives the fraction of all intervals in each constituent distribution. Dividing the number of intervals in each bin of the distribution of all closed intervals by N converts the distribution to a probability density function with an area of 1.

In theory, n should go to infinity in Eq. 6, but in practice, to include all gating sequences with a probability of occurrence of >10⁻⁹, the maximum needed value of n is given by: −9/(log10(P_C1-C2)). When P_C1-C2 = 0.5, n_max ∼30. Note the parallel between the analytical Eqs. 149 and 150 of Colquhoun and Hawkes (Colquhoun and Hawkes, 1995b) and the approach described above to generate the various distributions by simulation. The above example of simulating the dwell-time distributions of intervals for each specific gating sequence for a three-state model is also extended to a four state model and could be extended to any gating sequence. The methods used to simulate the single channel current records have been described previously (Blatz and Magleby, 1986).

To approach the question of the relationship between components and states, we start with the simplest possible model for a channel that can gate its pore, having one open and one closed state (Scheme 1). Infinite frequency response is assumed so that all intervals are detected.

In this example, both the opening and closing rate constants are 1,000/s, giving mean lifetimes (dwell times) of 1 ms for both the open and closed states.

Fig. 1 A presents an example of simulated single-channel data for the gating mechanism described by Scheme 1. The wide range of durations of the open and closed intervals reflect natural stochastic variation arising from the exponentially distributed dwell times in states of Markov models (Colquhoun and Hawkes, 1995b). As a typical first step in analysis, single-channel current records like that in Fig. 1 A, but of much longer duration, are sampled to determine the durations of the open and closed intervals. These durations are then binned into frequency histograms (dwell-time distributions) and fitted with sums of exponential components to quantify the description of the data. For Scheme 1, the open and closed dwell-time distributions are the same because of identical closing and opening rates, so only the closed distribution will be shown. Fig. 1 (B and C) plots the closed dwell-time distribution in two different ways often used in single-channel analysis. Both distributions use log binning so that bin width increases geometrically with time. Log binning gives the ability to quantify interval durations ranging from picoseconds to the age of the universe with constant minimal error in just a few hundred bins (McManus et al., 1987). Fig. 1 B presents the data plotted with the Sigworth and Sine (1987) transform, in which the square root of the number of intervals per bin is plotted against mean bin time on a log scale. The log binning gives a constant apparent bin width on the logarithmic abscissa. Fig. 1 C presents the data displayed on linear coordinates, where the abscissa indicates the mid time of each bin and the ordinate indicates the numbers of intervals per microsecond of bin width, rather than intervals per bin, to transform the log-binned data to the appearance it would have on linear coordinates with constant bin width.

The distributions using either the linear or the Sigworth and Sine transforms are described by a single exponential (continuous lines) with a time constant of 1 ms (arrows). Whereas the Sigworth and Sine plots are highly useful in indicating the time constant of the distribution of intervals by the time at the peak of the distribution, it needs to be remembered in the interpretation of single-channel data that such plots are transforms. The actual distribution of dwell times from a discrete state are like that in Fig. 1 C; the shorter the duration of the interval the greater the frequency of occurrence. It is the exponentially distributed dwell times shown in Fig. 1 (B and C) that give rise to the wide variation in interval durations in Fig. 1 A.

For Scheme 1 with one open and one closed state and perfect time resolution, the closed exponential component would arise entirely from and include all sojourns to C₁, and the open exponential component would arise entirely from and include all sojourns to O₁. Hence, there is perfect linkage between the exponential components and states.

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.

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).

To explore the relationship between exponential components and states, the origin of the intervals in the closed dwell-time distribution generated by Scheme 2 was examined. Each closed interval arises from either a unitary sojourn to C₁ or a compound sojourn that includes both C₁ and C₂. In a unitary sojourn, the closed interval is initiated by entry from O₁ into C₁ and is then terminated by a transition from C₁ to O₁ without ever transitioning to C₂, as indicated by gating sequence 0 in Table II. The constituent dwell-time distribution of all such unitary sojourns when n = 0 is designated {C₁} and can be calculated as described in the Materials and methods. For Scheme 2 the probability of a unitary sojourn is 0.5 (Table II), indicating that half of all closed intervals are in {C₁}.

For a compound sojourn, the initiation of the closed interval starts the same as for a unitary sojourn, by a transition from O₁ to C₁. Each closed interval is then extended by one or more repeated transitions from C₁ to C₂ and back to C₁ before termination by a transition to O₁. The gating sequences and also the probabilities of compound sojourns arising from 1, 2, or 3 repeated sojourns to C₂, together with the general case gating sequence for n repeated sojourns, are listed in Table II. The constituent dwell-time distribution for each specific gating sequence can be calculated as described in the Materials and methods. The sum of all the constituent dwell-time distributions from all gating sequences for n = 1 to infinity in Table II is designated {C₁C₂} and can be calculated using Eq. 6 in the Materials and methods. For Scheme 2 the probabilities of the compound gating sequences for n = 1 to infinity sum to 0.5, indicating that half of all the closed intervals are in {C₁C₂} (Table II).

The {C₁} and {C₁C₂} distributions are plotted in Fig. 3 (A and B) on linear and semilogarithmic coordinates, respectively, together with the E₁ and E₂ exponential components from Fig. 2. (Recall that an exponential on a plot with a logarithmic ordinate and linear abscissa gives a straight line.) E₁ together with {C₁} and E₂ together with {C₁C₂} are also plotted in Fig. 3 (C and D), respectively, for ease of comparison. {C₁} is a single exponential (green lines) with maximum amplitude at zero time and a time constant of decay of 1 ms, equal to t_C1, the mean lifetime of state C₁. In contrast, {C₁C₂} has a zero magnitude at zero time, rises with a slight inflection to reach a peak at ∼2.5 ms, and then decays, with the decay becoming exponential for durations longer than ∼6 ms (blue lines). The {C₁C₂} distribution has some characteristics in common with distributions arising from convolutions of exponential functions, because it is comprised of the sum of an infinite number of constituent distributions, each arising from convolutions of exponentially distributed dwell times. Each gating sequence in Table II, as n goes from 1 to infinity, contributes a constituent distribution.

The various constituent distributions for n = 1 to 6 in Table II are plotted as numbered purple lines in Fig. 3 B. As n increases, the time to the peak increases, the amplitude of the peak decreases, and the decay after the peak is slower. The increased time to peak and slower decay reflects the increased numbers of sojourns through C₂-C₁ contributing to each closed interval. The decreased amplitudes as n increases reflect that each successive distribution has 50% fewer intervals than the previous one (Table II) and that the interval durations are spread over a greater range (more dwell times contribute to each interval) so that there are fewer intervals of any specific duration. Interestingly, none of the constituent distributions for the individual gating sequences for n = 1 to infinity decay exponentially after reaching their peaks, as indicated by the curved decays of the purple lines in Fig. 3 B. However, the sum of all the constituent distributions for the individual gating sequences for n = 1 to infinity does decay exponentially, as indicated by the straight line decay of {C₁C₂} in Fig. 3 B (blue line) after ∼6 ms.

The {C₁} and {C₁C₂} dwell-time distributions shown in Fig. 3 (A–D) would not be apparent as individual distributions in the experimental data. Rather, {C₁} and {C₁C₂} sum to form the distribution of all experimentally observed intervals, referred to as the closed dwell-time distribution (continuous black lines in Fig. 3, A and B).

To describe the data, the experimentally observed dwell-time distribution would be fitted with the sum of fast and slow exponential components (as in Fig. 2) indicated as E₁ (black dashed lines) and E₂ (red dashed lines) in Fig. 3 (A–D). The predicted dwell-time distribution that would be calculated for Scheme 2 using either Q-matrix or analytical methods would also be given as the sum of the exponential components E₁ and E₂. Hence, both the description of the data and the predicted gating of Scheme 2 would be expressed in terms of the exponential components E₁ and E₂ rather than in terms of the distributions {C₁} and {C₁C₂} that reflect the actual underlying gating of the channel.

In the interpretation of single-channel data it is sometimes inferred that the {C₁} sojourns generate the fast exponential component. A comparison of the {C₁} and E₁ distributions in Fig. 3 (A–C), shows that this is not the case for Scheme 2. The area of E₁ is 0.146 and of {C₁} is 0.5. Thus, no more than 29.2% of the {C₁}sojourns could contribute to the E₁ component. In addition, the E₁ intervals have a mean duration of 0.586 ms compared with a mean duration of 1 ms for {C₁} sojourns. Hence, E₁ intervals from {C₁} would have to be selectively drawn from the briefer intervals in {C₁}.

In the interpretation of single-channel data it is also sometimes inferred that {C₁C₂} sojourns (those sojourns to the compound state C₁C₂) generate the slow exponential component. A comparison of the {C₁C₂} and E₂ distributions in Fig. 3 (A, B, and D) indicates that this is also not the case for Scheme 2. Intervals from {C₁C₂} do not generate an exponential, but a distribution with zero amplitude at zero time compared with maximum amplitude at 0 time for the E₂ exponential. Consequently, there is a severe deficit of intervals in {C₁C₂} at short times compared with E₂ (Fig. 3 D, gray area). For durations >6 ms, however, intervals in {C₁C₂} are sufficient to account for the tail of the slow exponential component, as indicated by the superposition of the decay of {C₁C₂} and E₂ at longer times (Fig. 3, A, B, and D). Hence, the relationship between components and states changes with the duration of the intervals. At very short times, E₂ arises almost exclusively from {C₁}, whereas at very long times, E₂ arises almost exclusively from {C₁C₂}. The lack of direct correspondence between {C₁} and E₁ and also between {C₁C₂} and E₂ clearly shows that exponential components and kinetic states are not directly linked for Scheme 2.

Although components and states are not directly linked for Scheme 2, they can be related to each other through the experimentally observed dwell-time distribution of all closed intervals (continuous black lines in Fig. 3, A and B). This distribution can be described in two different ways: by the sum of the two exponential components E₁ and E₂, and also by the sum of {C₁} and {C₁C₂}. Thus, for each interval duration in these distributions

and by rearrangement

Fig. 3 D shows that the {C₁C₂} and E₂ distributions are identical at longer interval durations but that {C₁C₂} is less than E₂ at shorter interval durations. E₂ – {C₁C₂} then gives the number of “missing intervals” (Fig. 3 D, shaded area) that would be required to fill in the gap between {C₁C₂} and E₂ to complete the E₂ exponential component. Because all intervals in the exponential components arise from {C₁} and {C₁C₂}, the observation in Fig. 3 D that there are insufficient intervals in {C₁C₂} to complete E₂ indicates that the missing intervals come from {C₁}, as there are no other intervals available.

Fig. 3 C shows that the {C₁} distribution is greater than the E₁ distribution for all interval durations. Hence, {C₁} – E₁ indicates the number of “excess intervals” in {C₁} that are not required for E₁ (Fig. 3 C, shaded area). Eq. 8 shows that the missing intervals in Fig. 3 D should exactly equal the excess intervals in Fig. 3 C at every point in time. Fig. 3 E shows that this is the case because the lines plotting the numbers of missing and excess intervals superimpose.

Further rearrangement of Eq. 7 indicates the composition of the exponential components

Thus, E₂ is comprised of all the {C₁C₂} intervals plus those excess intervals in {C1} required to fill in the gap between {C1C2} and E₂ to complete the E₂ exponential, and E₁ is comprised of the leftover intervals in {C₁} not used to fill in the E₂ exponential. Because intervals arising from transitions through the compound state C₁C₂ will always form a convolution type of distribution with too few intervals at brief times to complete the E₂ exponential, then some intervals from {C₁} will always be required to fill in the E₂ exponential. The fraction of {C₁} intervals required to fill in the gap at any point in time depends on interval duration, ranging from 0.5 at zero time to essentially 0 at very long times for Scheme 2 (Fig. 3 D).

To explore the effect of changing the lifetime of C₂, t_C2 on the relationship between components and states, t_C2 in Scheme 2 was altered by changing k_C2-C1, the rate constant for the transition from C₂ to C_1. Changing t_C2 in this manner did not change the lifetime of C₁, t_C1 (which remained at 1 ms), did not change the probability of entering C₂ from C₁ (which remained at 0.5), did not change the probability of the transition from C₂ to C₁ (which remained at 1), and did not change the relative areas of {C₁} and {C₁C₂}, both of which remained at 0.5. Changing t_C2 without changing any other aspects of the gating was found to have profound effects on the relationship between components and states.

Results are shown in Fig. 4 for t_C2 of 5 ms, and in Fig. 5 for t_C2 of 0.2 ms. These changes in t_C2 were obtained by changing k_C2-C1 in Scheme 2 from 1,000/s to either 200/s or 5,000/s, respectively. The findings in Figs. 4 and 5 should be compared with those in Fig. 3 where t_C2 was 1 ms. Table III lists the time constants and areas of E₁ and E₂ for these and other values of t_C2. Calculations over a wide range of state lifetimes for C₁ and C₂ showed that it is the lifetime ratio t_C2/t_C1 rather than the absolute values of the lifetimes that determines the relationship between components and states when the transition probabilities are fixed (not depicted). Consequently, the observations will be discussed in terms of the t_C2/t_C1 ratio in order to make them more general. The t_C2/t_C1 ratios for 3–5,⁴⁵ are 1, 5, and 0.2, respectively.

The key observations to be made from a systematic examination of 3–5,⁴⁵ are as follows.

(a) {C₁} (continuous green lines) is identical in each figure (A, B, and C), with a time constant of 1 ms, because changing k_C2-C1 has no effect on t_C1 or on the fraction of intervals in {C₁}, which remains constant at 0.5.

(b) Increasing t_C2 fivefold compared with t_C1 decreases the peak amplitude of {C₁C₂} while increasing the time to peak and greatly slowing the decay (compare Fig. 4 to Fig. 3, A, B, and D). These changes in {C₁C₂} greatly decrease the deficit of intervals required to fill in the gap between {C₁C₂} and E₂ to complete the E₂ exponential at shorter times (compare gray area in Fig. 4 D to Fig. 3 D). Consequently, because fewer {C₁} intervals are required to fill in the gap when t_C2 >> t_C1, most of the {C₁} intervals go to E₁ (compare Fig. 4 C to Fig. 3 C). As a result, the time constant and area of E₁ approach that of {C₁} when t_C2 >> t_C1 (Fig. 4, A–C; Table III),

(c) In contrast, decreasing t_C2 fivefold compared with t_C1 increases the peak amplitude of {C₁C₂}, while decreasing the time to peak and accelerating the decay (compare Fig. 5 to Fig. 3, A, B, and D). These changes in {C₁C₂} greatly increase the number of intervals required to fill in the gap between {C₁C₂} and E₂ to complete the E₂ exponential at shorter times (compare gray area in Fig. 5 D to Fig. 3 D). Consequently, because most of the {C₁} intervals are required to fill in the missing intervals when t_C2 << t_C1, then very few of the {C₁} intervals go to E₁ (compare Fig. 5 C to Fig. 3 C). As a result, the time constant and area of E₁ become markedly less than that of {C₁} when t_C2 << t_C1 (Fig. 5, A–C; Table III), so that E₁ becomes uncoupled from {C₁}.

(d) That {C₁} intervals mainly go to E₁ when t_C2 >> t_C1 and to E₂ when t_C2 << t_C1 is readily seen by comparing Fig. 4 (C–E) to Fig. 5 (C–E), respectively.

The observations in 3–5,⁴⁵ and Table III suggest that the relative contribution of the {C1} and {C₁C₂} intervals to E₁ and E₂ shifts with the t_C2/t_C1 ratio. To investigate these shifts further, Fig. 6 B plots the time constants of E₁ and E₂, τ_E1 and τ_E2, and the lifetimes of C₁ and C₂, t_C1 and t_C2, and Fig. 6 E plots the areas of E₁, E₂, {C₁}, and {C₁C₂} as k_C2-C1 in Scheme 2 is changed over six orders of magnitude to change the t_C2/t_C1 ratio from 10³ to 10⁻³ (see bottom of Fig. 6). This change in k_C2-C1 changes t_C2 from 1 s to 1 μs (Fig. 6 B, red dashed line) while having no effect on t_C1, which remains constant at 1 ms (Fig. 6 B, black continuous line). As t_C2 decreases, decreasing the t_C2/t_C1 ratio, τ_E1 first tracks t_C1 and then switches to track t_C2 (Fig. 6 B, black dashed line). The switch in tracking occurs as the t_C2/t_C1 ratio passes through 1, with τ_E1 equal to t_C1 when t_C2 >> t_C11 and then equal to t_C2 when t_C2 << t_C1.

Just as there is a shift in the tracking of τ_E1 from t_C1 to t_C2 as the t_C2/t_C1 ratio passes through 1, there is also a shift in the tracking τ_E2 from t_C2 to t_C1. τ_E2 (Fig. 6 B, red continuous line) first tracks t_C2 when t_C2 >> t_C1 and then switches to track t_C1 when t_C2 << t_C1. This tracking occurs with an offset. τ_E2 is twice t_C2 when t_C2 >> t_C1 and then switches to become twice t_C1 when t_C2 << t_C1.

These paradoxical shifts in the tracking of the time constants are also associated with dramatic shifts in the areas of E₁ and E_2,a_E1 and a_E2 (Fig. 6 E). When t_C2 >> t_C1, a_E1 and a_E2 approach 0.5, essentially the same as the 0.5 areas of {C₁} and {C₁C₂} (Fig. 6 E, left; Table III). As the t_C2/t_C1 ratio decreases so that t_C2 << t_C1, then a_E1 approaches 0 and a_E2 approaches 1 (Fig. 6 E, right; Table III). Note that the dramatic shifts in the time constants and areas of E₁ and E₂ occur even though the areas of {C₁} and of {C₁C₂} remain constant at 0.5 (Fig. 6, B and E; Table III).

The plotted areas in Fig. 6 E quantify the observations shown in 3–5,⁴⁵ (C and D). When t_C2 >> t_C1, the areas (and distributions) of E₁ and {C₁} are essentially identical and the areas (and distributions) of E₂ and {C1C2} are also essentially identical. When t_C2 << t_C1, then the area of E₁ approaches 0 and the area of E₂ approaches the area of {C₁} + {C₁C₂}. Hence, when t_C2 >> t_C E₁ is comprised of essentially all of the C₁ intervals and E₂ is comprised of essentially all {C₁C₂} intervals. The shift in the {C1} intervals from E₁ to E₂ as the lifetime ratio shifts is shown by the decrease in a_E1 and increase in a_E2, such that when t_C2 << t_C1 essentially all of the {C₁} and {C₁C₂} intervals go to E₂.

The paradoxical shifts in the time constants and areas of E₁ and E₂ as the t_C2/t_C1 ratio passes through 1 (Fig. 6, B and E) follow directly from the graphical origins of the exponential components shown in 3–5,⁴⁵ and from the equations in the Appendix. The shifts do not arise from a swapping of the fast and slow exponential components between Eqs. A2 and A3 and Eqs. A4 and A6 in the Appendix, but are self contained in the equation for each component. This is shown graphically in Fig. 6 B, where τ_E1 is always faster than τ_E2,, and in Fig. 6 B by the smooth functions for changes in area. The shifts can be explained visually from the graphical origins of the exponential components detailed in 3–5,⁴⁵. As the t_C2/t_C1 ratio decreases, the shape of {C₁C₂} changes so that an increasing number of {C₁} intervals are required to fill in the gap between {C₁C₂} and E₂ to complete the E₂ exponential, with any leftover {C₁} intervals going to generate E₁. It is this shift of {C₁} intervals from E₁ to E₂ that shifts the areas and time constants of E₁ and E₂.

The time constant of E₂ tracks t_C2 (with an offset) when t_C2 >> t_C1 (Fig. 6 B) because under these conditions the number of intervals required to fill in the gap between {C₁C₂} and E₂ is negligible so that essentially all of the intervals in E₂ arise from {C₁C₂} (Fig. 4 and Fig. 6 B), where the duration of C₁ is negligible because t_C2 >> t_C1. That the offset for τ_E2 is twice t_C2 when t_C2 >> t_C1 (Fig. 6 B) is readily calculated from Table II by setting t_C1 to zero and then calculating the mean closed interval duration (which gives the time constant of E₂) for gating sequences of n = 1 to infinity. Note that n starts at 1 because there are essentially no {C₁} intervals in E₂ when t_C2 >> t_C1. The tracking occurs with an offset equal to twice the duration of t_C2 because the average number of sojourns through C₂ for intervals generated by gating sequences 1 to infinity is 2.

As t_C2 becomes less than t_C1, τ_E2 switches over to track t_C1 (Fig. 6 B). The tracking now occurs with a time constant equal to twice t_C1 rather than t_C2, because when t_C2 << t_C1, all of the {C₁} and {C₁C₂} intervals go to E₂ (Fig. 5 and Fig. 6 E), with the sojourns to C₂ having such brief durations that the dwell time in C₂ does not contribute to interval duration. That τ_E2 is twice t_C1 when t_C2 << t_C1 (Fig. 6 B) is readily calculated from Table II by setting t_C2 to zero and calculating the mean closed interval duration for n = 0 to infinity, with n starting at 0 because essentially all {C₁} and {C₁C₂} intervals go to E₂. Thus, the paradoxical shift in the tracking of τ_E2 from twice t_C2 when t_C2 >> t_C1 to twice t_C1 when t_C2 << t_C1, as determined by the equations in the Appendix, is readily accounted for mechanistically as well as analytically.

The time constant of E₁ directly tracks t_C1 when t_C2 >> t_C1 (Fig. 6 B), because under these conditions an insignificant number of intervals in {C₁} are required to fill in the gap between {C₁C₂} and E₂, so (essentially) all {C₁} intervals go to E₁ (Fig. 4 and Fig. 6 B). Consequently, when t_C2 >> t_C1, E₁ and {C₁} become synonymous (they contain the same numbers and durations of intervals) so that τ_E1 directly tracks and is equal to t_C1. As t_C2 becomes less than t_C1, τ_E1 switches over to track t_C2 (Fig. 6 B) because the majority of the {C₁} intervals now go to fill in the gap between {C₁C₂} and E₂ to complete the E₂ exponential so that they are no longer available for E1 (Figs. 5 and 6). Interestingly, the few remaining intervals in {C₁} left to generate E₁ have a lifetime equal to t_C2. It is not readily apparent why this is the case, but it can be shown by numerical substitution into Eq. A2 (Appendix) that when k₊₁ >> (β + k_-1), i.e., when t_C2 << t_C1, then τ_E1 ∼ 1/(k₊₁), i.e., τ_E1 ∼ t_C2.

3–5,⁴⁵ and Fig. 6 (B and E) examined the relationship between components and states as a function of the t_C2/t_C1 ratio for the specific case of equal transition probabilities away from state C₁ in Scheme 2 where P_C1-C2 is equal to P_C1-O1, with both equal to 0.5. This section examines whether the same general relationship between components and states holds when the ratio of the two transition probabilities away from C₁ is changed over six orders of magnitude. Data are presented for transition probability ratios of P_C1-C2/P_C1-O1 of 0.999/0.001 (Fig. 6, A and D) and of 0.001/0.999 (Fig. 6, C and F) for comparison to data for the transition probability ratio of 0.5/0.5 in Fig. 6, B and E).

A comparison of the data for these three markedly different transition probability ratios shows that the paradoxical shifts in the relationship between time constants of exponential components and state lifetimes occurs independently of the transition probability ratio of P_C1-C2/P_C1-O1. For the three transition probability ratios considered that span six orders of magnitude (upper, middle, and lower parts) and for changes in t_C2/t_C1 also over six orders of magnitude (abscissa), τ_E2 first tracks t_C2 and then switches to track t_C1, whereas τ_E1 first tracks t_C1 and then switches to track t_C2. The only differences in the plots are that the magnitudes of the offset of τ_E2, first from t_C2 and then from t_C1, decreases as the transition probability ratio P_C1-C2/P_C1-O1 decreases (see below) and the switch in tracking occurs more rapidly. Thus, the same paradoxical shifts in the tracking of the exponential components to the state lifetimes as the t_C2/t_C1 ratio passes through 1 still occur when the transition probability ratio of P_C1-C2/P_C1-O1 is changed a million fold. A decreased offset of τ_E2 from the state lifetimes would be expected as P_C1-C2/P_C1-O1 decreases because the average number of repeated transitions through C₁C₂ contributing to each closed interval would decrease, leading to a decreased time constant of E₂. For example, when P_C1-C2/P_C1-O1 is 0.999/0.001 so that 999 out of 1,000 transitions away from C₁ are to C₂, then the time constant of E₂ is ∼1,000-fold greater than t_C2 when t_C2 >> t_C1 and ∼1,000-fold greater than t_C1 when t_C2 << t_C1 (Fig. 6 A). At the other extreme, when P_C1-C2/P_C1-O1 is 0.001/0.999 so that only 1 out of 1,000 transitions away from C₁ go to C₂, then the time constant of E₂ is within 0.1% of t_C2 when t_C2 >> t_C1 and within 0.1% of t_C1 when t_C2 << t_C1 (Fig. 6 C).

As more transitions from C₁ are directed to either C₂ or O₁ due to different P_C1-C2/P_C1-O1 ratios, the areas of {C₁} and {C₁C₂} change, as would be expected. For P_C1-C2/P_C1-O1 ratios of 0.999/0.001, 0.5/0.5, and 0.001/0.999, the area of {C₁C₂} is 0.999, 0.5, and 0.001, and the area of {C₁} is 0.001, 0.5, and 0.999, respectively (Fig. 6, D, E, and F, dotted straight lines). These areas remain constant as k_C2-C1 is changed. Just as the paradoxical shifts in time constants occur independently of the P_C1-C2/P_C1-O1 ratio as the t_C2/t_C1 ratio passes through 1, the paradoxical shifts a_E1 and a_E2 also occur independently of the P_C1-C2/P_C1-O1 ratio, that is, independently of whether most of the closed intervals arise from {C₁} or {C₁C₂}. When P_C1-C2/P_C1-O1 is 0.999/0.001, a_E1 is small, containing <0.1% of the intervals when t_C2 >> t_C1 (Fig. 6 D, left). Yet, these few intervals in E₁ still shift to E₂ as the t_C2/t_C1 ratio passes through 1, as indicated by the decrease in a_E1 in Fig. 6 D that is apparent because of the log ordinate. The accompanying increase in a_E2 is not apparent because the fractional increase is small compared with initial large size of a_E2. For the reverse situation in which P_C1-C2/P_C1-O1 is 0.001/0.999, a_E1 contains 99.9% of the area and a_E1 only 0.001% when t_C2 >> t_C1 (Fig. 6 F, left). This distribution of areas then fully reverses as the t_C2/t_C1 ratio passes through 1 (Fig. 6 F, right).

The results in Fig. 6 then show that the paradoxical shifts in the relationship between exponential components and states is determined by the lifetime ratio t_C2/t_C1 rather than by the specific lifetimes of the states or the specific transition probabilities.

If the duration of intervals in an exponential component is determined mainly by the dwell times arising from sojourns through a particular state, then a fractional change in the lifetime of that state should produce the same fractional change in the time constant of the exponential component. Eq. 11 incorporates this rational to quantify the linkage between components and states, Lτ, such that

where τ_Ei is the time constant of exponential component i when the mean lifetime of state j is t_Cj, and τ_Ei′ is the time constant of exponential component i after the lifetime of state j is changed a small fractional amount to t_Cj′. The lifetime of state j is changed without changing the transition probabilities among any of the states by increasing (or decreasing) all of the rate constants leading away from state j by the same small fractional amount (typically 10⁻⁵), with τ_Ei′ and τ_Ei calculated using analytical (Appendix) or Q-matrix methods (Colquhoun and Hawkes, 1995a).

Fig. 7 plots linkage as a function of the t_C2/t_C1 ratio for the same three kinetic schemes that were examined in Fig. 6 encompassing a 10⁶-fold change in the transition probabilities away from state C₁. When t_C2 >> t_C1, there is near perfect linkage of τ_E1 to t_C1, and of τ_E2 to t_C2, as indicated by values for Lτ approaching 1, and essentially no linkage of τ_E2 to t_C1, and of τ_E1 to t_C2, as indicated by values for Lτ approaching 0. The linkages then reverse when t_C2 << t_C1, so there is near perfect linkage of τ_E2 to t_C1, and of τ_E1 to t_C2 and no linkage of τ_E1 to t_C1, and of τ_E2 to t_C2. The quantified linkage in Fig. 7 is consistent with the observations and mechanisms discussed in the previous figures.

Knowledge of the paradoxical shifts shown in Figs. 6 and 7 and their underlying mechanisms can prevent possible misinterpretation of the origin of the exponential components. For example, solving for the exponential components for Scheme 2 when k_C2-C1 = 10⁵/s gives time constants of 0.01 ms for E₁ and 2.01 ms for E₂ (Fig. 6 B, right side, and Table III, far right column). Since t_C2 is 0.01 ms, the same as τ_E1, it might be tempting to speculate that E₁ arises in some manner from single sojourns to C₂, rather than from leftover {C₁} intervals, as shown in Fig. 5. However, this cannot be the case, as every sojourn to C₂ requires two sojourns through the 1 ms lifetime C₁ in this example, yielding the slower {C₁C₂} distribution (Fig. 5; Table II). Furthermore, the {C₁C₂} distribution has a magnitude of 0 at time 0, whereas the magnitude of E₁ is maximal at time 0 (3–5,⁴⁵). Consequently, intervals that include a transition through C₂, i.e., {C₁C₂} intervals, cannot be the basis for the very fast E₁ exponential component in this example, no matter how brief the lifetime of C₂. The E₁ component always arises from the same underlying mechanism, no matter what the lifetimes of C₁ and C₂, from the leftover intervals in {C₁} not required to fill in the {C₁C₂} distribution to complete the E₂ exponential.

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 t_C1 and t_C2 are both 1 ms for all three schemes, and t_C3 is 1 s in A, 1 ms in B, and 1 μs in C, changed by altering k_C3-C2 as indicated. The transition probabilities P_C1-O1, P_C1-C2, P_C2-C1, and P_C2-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 t_C3 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 t_C3 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.

The fraction of intervals in {C₁C₂} that go to fill in E₃ and E₂ is highly dependent on the t_C3/t_C2 ratio. When t_C3 >> t_C2 (Fig. 8 A), then both E₃ and {C₁C₂C₃} are of long duration and very low amplitude so that very few of the {C₁C₂} and {C₁} intervals are needed to fill in E₃ at shorter times. Consequently, most {C₁C₂} intervals go to E₂, with the decay of E₂ superimposing the decay of {C₁C₂} at longer times. Intervals from {C₁} then fill in the gap between {C₁C₂} and E₂ at shorter times to complete E₂, with the leftover intervals from {C₁} going to generate E₁. This distribution of intervals is very similar to Fig. 3 A, except for the addition of the very low amplitude long duration {C₁C₂C₃} distribution and E₃ component in Fig. 8 A.

In contrast, when t_C3 << t_C1 (Fig. 8 C), then the {C₁C₂C₃} distribution has a faster decay and a much higher peak amplitude than in Fig. 8 A, which leads to a major deficit of intervals at shorter times in {C₁C₂C₃} compared with E₃. Consequently, large numbers of intervals from {C₁C₂} and also from {C₁} are required to fill in the gap between {C₁C₂C₃} and E₃ at shorter times to complete the E₃ exponential. The consequence of using so many {C₁C₂} and also {C₁} intervals to complete the E₃ exponential is that there are few leftover {C₁C₂} intervals to contribute to E₂. Consequently, E₂ is comprised mainly of the briefer duration {C₁} intervals and decays much faster than {C₁C₂}. Because of the large number of {C₁} intervals used for E₃ and E₂ there are essentially no {C₁} intervals left to generate E₁, which essentially disappears, having a very fast time constant and essentially no area.

In Fig. 8 B when t_C3 is 1 ms, intermediate in duration (log scale) between the 1-s lifetime in A and the 1-μs lifetime in part C, then the response is intermediate between those in A and C, with sufficient leftover {C₁} intervals to generate a small but detectable E₁. Thus, the same types of paradoxical shifts and underlying mechanisms that generate the exponential components when there are two closed states in series also apply when there are three closed states in series, but with the additional requirement that some of the {C₁C₂} and {C₁} intervals go to fill in the gap between {C₁C₂C₃} and E₃ at shorter times, leaving fewer intervals for E₂ and E₁.

Frequency histograms of the number of open and closed intervals vs. their durations are a major means of presenting data recorded from single channels. These dwell-time distributions are typically characterized by fitting with sums of exponential components, as the Markov models used to describe the gating of ion channels predict that such dwell-time distributions would be described by sums of exponential components, with the numbers of components equal to the number of states in the gating mechanism (Colquhoun and Hawkes, 1981, 1982; Magleby and Pallotta, 1983; Colquhoun and Hawkes, 1995a; Jackson, 1997) and Appendix. In spite of the central importance of exponential components to the description of single channel data, little is known about the specific contributions of the various states to each of the exponential components. The question is not whether components can be calculated for a given kinetic scheme, as this is readily accomplished through analytical and Q-matrix methods (Colquhoun and Hawkes, 1982, 1995b), nor is the question the detection of components in histograms, as kinetic mechanisms are typically determined by maximum likelihood fitting of rate constants to data, with the numbers of components implicit in the mechanism being fitted (Horn and Lange, 1983; McManus and Magleby, 1991; Colquhoun et al., 1996). Rather, the question is the physical basis for the exponential components, e.g., what is the state contribution to each component? As long as any discussion of exponential components in terms of underlying gating mechanism is avoided, no specific knowledge is needed. However, in order to relate exponential components to the underlying gating process, it is necessary to understand the relationship between components and states. In this paper we resolve this problem for simple models.

To explore this relationship we examined the simple gating mechanism described by Scheme 2 for two closed and one open state: C₂-C₁-O₁. For this gating mechanism the dwell-time distribution of all closed intervals is described by the sum of fast E₁, and slow E₂ exponential components (Fig. 2). To relate exponential components to underlying states, the closed dwell-time distribution was divided into those intervals arising from single sojourns to C₁ in the gating sequence O₁-C₁-O₁, designated {C₁}, and into those intervals arising from all sojourns through the compound state C₁-C₂ from the gating sequence O1-C₁-(C₂-C₁)_n-O1 (where n has integer values from 1 to infinity, Table II), designated {C₁C₂}.

Our analysis shows that {C₁C₂} and E₂ superimpose at longer interval times when the number of {C1} intervals approaches 0 (3–5,⁴⁵, A–D). This indicates that E₂ at longer times is generated by and includes all intervals from {C₁C₂}. At shorter interval times, however, there are too few intervals in {C₁C₂} to account for E₂ (3–5,⁴⁵, A, B, and D). To complete E₂ at shorter times, intervals from {C₁} fill in the gap between {C₁C₂} and E₂, as these are the only other intervals available to do so (Eq. 9, 3–5,⁴⁵, C–E). The leftover intervals in {C₁} not used to fill in the gap then generate E₁ (Eq. 10, 3–5,⁴⁵, C–E). This same basic mechanism for the generation of E₁ and E₂ generally applies, independent of the rate constants in Scheme 2 (3–5,⁴⁵), and allows for a graphical/numerical solution for E₁ and E₂. Although such a procedure would not normally be used, it does illustrate the systematic manner in which the exponential components are generated from the closed states. E₂ is given by the projection of a straight line superimposed at long times on the decay of {C₁C₂} plotted on semilogarithmic coordinates (Fig. 3 B, dashed red line superimposed on blue line). {C₁C₂} is then subtracted from E₂ to determine the deficit of intervals required to fill in the gap between {C₁C₂} and E₂ at shorter times (Fig. 3 D, gray area). The intervals used to fill the gap, which come from {C₁}, are then subtracted from {C₁} to obtain E₁ (Fig. 3 C). E₁ is then plotted on semilogarithmic coordinates to define its magnitude and time constant (Fig. 3 B, dashed black line). Hence, E₂ arises from all intervals in {C₁C₂} plus selected intervals from {C₁} as needed to fill the gap, and E₁ arises from the leftover intervals in {C1}.

It is sometimes inferred that E₁ is comprised of all of the {C₁} intervals and that E₂ is comprised of all the {C₁C₂} intervals, so that E₁ is tightly linked to C₁ and E₂ is tightly linked to the compound state C₁C₂. Although the discussion in the previous section indicates that this assumption is not necessarily correct, it would be useful to know under what conditions such an assumption applies. Our analysis shows that there is negligible error associated with this assumption for Scheme 2 when the t_C2/t_C1 ratio is >100 (Figs. 6 and 7; Table III), and that the error remains negligible for 10⁶-fold changes in the transition probability ratio of P_C1-O/P_C1-C2 (Fig. 6). The errors associated with this assumption become progressively greater as the t_C2/t_C1 ratio decreases. For t_C2/t_C1 and P_C1-O/P_C1-C2 ratios of 1, 29% of the {C₁} intervals are in E₁ with the rest in E₂ (Table III). As the t_C2/t_C1 ratio becomes <1, the assumption that E₁ is comprised of all the {C₁} intervals and that E₂ is comprised of all the {C₁C₂} intervals becomes untenable, as the time constant of E₁ switches from tracking t_C1 to tracking t_C2, and the {C₁} intervals switch from mainly contributing to E₁ to mainly contributing to E₂ (3–7,⁴⁵⁶⁷).

This paradoxical switch follows as a simple consequence of the mechanism by which E₁ and E₂ are generated. Because it is the t_C2/t_C1 ratio that determines the magnitude and shape of the {C₁C₂} distribution, it is the t_C2/t_C1 ratio that also determines the number of {C₁} intervals required to fill in the gap between {C₁C₂} and E₂ at shorter times to complete the E₂ exponential (3–5,⁴⁵, C and D). When t_C2 >> t_C1, the relative number of {C₁} intervals needed to fill in the gap is insignificant. Consequently, most {C₁} intervals go to generate E₁, and E₂ is comprised of mainly {C₁C₂} intervals (Figs. 4 and 6). In contrast, when t_C2 << t_C1, most of the {C₁} intervals are used to fill in the gap between the {C₁C₂} distribution and E₂, so there are few intervals available to generate E₁ (Figs. 5 and 6), and this is the case over six orders of magnitude change in the transition probability ratio of P_C1-O/P_C1-C2 (Fig. 6, D–F). E₁ has a very small amplitude and very fast time constant when t_C2 << t_C1 because essentially all the {C₁} intervals go to complete the E₂ exponential at shorter times so that there are few {C₁} intervals left to generate E₁ (Fig. 5, C and D; Fig. 6, D–F). Such a change in E₁ can have severe consequences on the interpretation of experimental data, as discussed in the following section.

Whereas it is relatively easy to detect slow exponential components of very small areas because of the high likelihood penalties that result if intervals of longer duration are not included in an exponential component (McManus and Magleby, 1988), it is much more difficult to detect fast exponential components of small area superimposed on slower components. For example, when t_C2 is fivefold less than t_C1 in Scheme 2, E₁ has a time constant of 0.18 ms and area of 0.01 (Fig. 5, Table III for k_C2-C1 of 5,000/s). It is unlikely that such a fast component with only 1% of the area would be detected in experimental data, leading to an incorrect conclusion of a single closed state with a lifetime of 2.22 ms, rather than two closed states with lifetimes of 1 ms (C₁) and 0.2 ms (C₂). It would be even more difficult to detect components arising from briefer duration closed states if there were additional intervening closed states before the open state, as is likely to be the case for data from real channels. Obtaining experimental data over a wide range of conditions that could lead to large changes in state lifetimes, together with simultaneous fitting of the data to gating mechanisms rather than with components could facilitate the detection of states.

The studies in this paper were performed for simple gating mechanisms and for data with perfect time resolution. With limited time resolution, brief duration intervals can go undetected, leading to the formation of compound states that include both open and closed states (Blatz and Magleby, 1986; Hawkes et al., 1992; Colquhoun and Hawkes, 1995b). Such compound states would need to be included when relating exponential components to states. Calculating the fractional change in exponential components for fractional changes in state lifetimes provides a method to examine the linkage between components and states (Eq. 11) for simple as well as highly complex models and also when time resolution is limited.

Understanding the relationship between components and states provides investigators with a physical interpretation for the exponential components in distributions of open and closed dwell times from single channels.

This section presents the analytical solution for the dwell-time distribution of closed intervals for Scheme 2 following Colquhoun and Hawkes (1981, 1982, 1994). The three rate constants that determine the distribution of closed intervals are designated as:

The distribution of all closed intervals, f(t), is given by the probability density function described by the sum of two exponential components:

where w₁ and w₂ are the magnitudes of the fast E₁ and slow E₂ exponential components, and τ_E1 and τ_E2 are the time constants. The time constants, magnitudes, and areas (a) of the exponential components are given by:

From the analytical solution it can be seen that the time constants, magnitudes, and areas of both E₁ and E₂ are determined by all of the rate constants that affect the lifetimes of both closed states. The relationship between components and states is not readily apparent from these equations, and see also Colquhoun and Hawkes (1981), Magleby and Pallotta (1983), and Jackson (1997) for analytical solutions of more complex gating mechanisms.

It can be shown by numerical substitution into Eq. A2 (or by setting k₊₁ to 0) that when k₊₁ << (β + k_-1)_, i.e., when t_C2 >> t_C1, that

indicating that τ_E1 approaches t_C1 when the t_C2/t_C1 ratio is >>1, as shown in the Results, and see Colquhoun and Hawkes (1994) for an alternative means to express limits for τ_E1.

It can also be shown by numerical substitution into Eq. A2 (or by setting k_-1 and β to 0) that when k₊₁ >> (β + k_-1), i.e., when t_C2 << t_C1, that

indicating that τ_E1 approaches t_C2 when the t_C2 << t_C1 ratio is <<1, as shown in the Results.

This work was supported in part by Lois Pope LIFE and American Heart Association, Puerto Rico/Florida Affiliate, Postdoctoral Fellowships to C. Shelley, and grants from the National Institutes of Health (AR32805) and the Muscular Dystrophy Associate to K.L. Magleby.

Olaf S. Andersen served as editor.