Chapman ML , VanDongen AM .

5R01 NS 031557-12 NINDS NIH HHS , R01 NS031557 NINDS NIH HHS

	Figure 1. Heteromeric pore conformations and subconductance levels. (A) When ion channels move from the closed to the open state, a conformational change occurs in the pore. K channels consist of four identical subunits (indicated by squares in the model) surrounding a central pore. Each subunit makes an identical contribution to the lining of the permeation pathway. Therefore, when a K channel opens, each subunit must change its conformation from “closed” (black) to “open” (white). Consequently, as the channel moves between the homomeric open and closed states, it must visit heteromeric states (H1–H3), in which some subunits are “open” and others are “closed.” Due to the tight packing of the subunits at the central pore region, it is likely that the transitions between the open and closed conformation are highly cooperative. Therefore, the heteromeric states H1–H3 are expected to be very short-lived. The permeation status of the heteromeric pore conformations is not immediately clear. It is conceivable that only the homomeric open state is capable of permeating ions. However, because subconductance levels (sublevels) were usually associated with transitions between the open and closed state in drk1, we have proposed that they arise from the heteromeric pore conformations H1–H3 (Chapman et al., 1997). Consistent with this idea, we have identified four distinct sublevels in drk1-L, a mutant with a large single channel conductance (Chapman et al., 1997). This subunit-subconductance hypothesis will be further tested in the remainder of this paper. (B) Single channel activity was simulated for the model shown in panel A using an approach described previously (VanDongen, 2004a). The rate constants leaving the fully open and closed states were set to 8,000 s−1, while all other rate constants (leaving sublevels) were set to 32,000 s−1. The amplitude levels for the subconductance states (H1–H3) were assigned values previously found for drk1-L (15, 37, 58, 82%). In this model, all gating transitions have a complex fine structure in which the channel visits at least three of the four subconductance levels. A representative opening transition is shown with high time resolution (and without noise or filtering), which follows the following itinerary: 0→15→58→37→15→58→82→100%. (C) The same transition as in B, on a slower (more realistic) timescale. Note the difference in scale bars. (D) The same transition as in C, after addition of Gaussian noise (standard deviation is 0.5 times the amplitude of the fully open state) and filtering (finite impulse response low-pass filter with a four-pole Bessel characteristic, cutoff frequency is 5 kHz). Note that the transition appears smooth. (E) An opening transitions taken from a single channel record of drk1 recorded in cell-attached mode at +40 mV. The record was low-pass filtered at 500 Hz. (F) The entire 50-ms record from which the transition in B–D was taken is shown without noise or filtering. (G) The same record as in F, with Gaussian noise added and after filtering (5 kHz). Note how most of the unsuccessful attempts to open or close that are seen in F do not survive the low-pass filter. Also, most of the complexity and fine-structure present in the opening and closing transitions is lost. However, occasionally a shoulder (arrow) can be observed, similar to those seen in wild-type drk1 (see Fig. 1 B in Chapman et al., 1997).
	Figure 2. Drk1-S and drk1-L have widely different activation thresholds. Drk1-L (for “Large”) is a previously described large-conductance mutant that has the same activation midpoint as drk1 (Chapman et al., 1997). Drk1-S (for “Shifted”) was identified in a screen of S4 mutations: mutation of residues 289–292 from RRV(V) to LLV(A) resulted in a shift of the activation midpoint by +70 mV. (A and B) Families of outward K currents for drk1-S (A) and drk1-L (B) were elicited by 400-ms step depolarizations from a holding potential of −80 mV to membrane potentials indicated. (C) Comparison of the normalized conductance–voltage relationships for the two mutants used in this paper. Normalized conductances were calculated from whole cell current–voltage relationships (A and B), as previously described (VanDongen et al., 1990).
	Figure 3. Single channel behavior of the parent channels drk1-S and drk1-L. Examples of single channel traces recorded in cell-attached patches for drk1-S and drk1-L, for step depolarizations to 0 and +40 mV. Traces for drk1-S at 0 mV were selected from several hundred records, in most of which the channel failed to open. Because of the very low open probability for drk1-S at 0 mV, it was not possible to estimate the number of channels in the patch. Due to the difference in single channel conductance, records for drk1-S (9 pS) are noisier than those for drk1-L (38 pS).
	Figure 4. Ensemble averages for drk1-L and drk1-S. Probability of being open following a step depolarization to 0 mV in patches containing N channels (NPO), where N = 1 for drk1-L, and N ≥ 1 for drk1-S (see Fig. 3). NPO was obtained from ensemble averages after records were idealized by the TRANSIT algorithm (VanDongen, 1996, 2004a). Ensemble averages (thin line) were fit with the following model modified after Hodgkin and Huxley (1952): PO (t) = ninf * [1 − exp(−t/τn)]Q. ninf is the steady-state open probability, τn is the activation time constant, and (Q) is the exponent. The thick line illustrates the optimal fit. Optimized parameters were as follows: for drk1-L: τn = 6.29 ms, ninf = 0.641, Q = 1.05, and for drk1-S: τn = 112.0 ms, ninf = 0.024, Q = 1.00.
	Figure 5. Subunit-subconductance model for voltage-gated K channels. The open↔close behavior illustrated in Fig. 1 G reflects the intrinsic kinetic properties of the gate, as described by the model in Fig. 1 A. In voltage-gated channels, this intrinsic gating behavior is allosterically controlled by membrane potential. A subunit-based kinetic scheme for K channels assembled from two S and two L subunits is constructed as follows. (A) The drk1-S and drk1-L subunits are modeled separately, since they differ in their voltage dependence. As before, each subunit is described by a linear three-state model that directly couples voltage sensor movement to channel opening. Closed states are represented by squares, open states by circles. Open squares indicate a resting voltage sensor; filled squares/circles imply the sensor has moved. The direction of the arrows specifies the position of the equilibrium and the width of the arrow the relative magnitude of the rate constants. The direction and width of the arrows are reflective of a membrane potential of 0 mV. Open arrows are voltage dependent (and involve sensor movements), closed arrows are voltage independent (indicating open–close gating transitions). (B–D) Combining two S and two L subunits in an S-L-S-L configuration results in a complex three-dimensional 36-state model. However, at intermediate membrane potentials, S-L channels will preferentially visit a subset of states, because the L subunits are likely to activate before the S subunits. The preferred itinerary is shown by the 15-state triangular model in B, while C and D illustrate all possible activation and opening transitions, respectively. The 15-state model contains three functionally homomeric states, at the corners of the triangle. Voltage sensor movements are represented by vertical transitions, open–close transitions by horizontal arrows. Rows labeled P1–P3 indicate partially activated channels, Columns H1–H3 represent the heteromeric pore conformations that are proposed to produce subconductance levels (Fig. 1 A). Highly unstable states are shown on a gray background: these are the heteromeric pore states that are connected by two wide filled arrows to neighboring states. Heteromeric pore states labeled Sub1, Sub2, and Sub3 are not as unstable as the remaining states in H1–H3, because channels in these states can only exit to the left. Sub1 and Sub2 are boxed to indicate that they are expected to predominate at membrane potentials below the activation threshold of the S subunits. At a negative holding potential, channels accumulate in the homomeric resting state. Following a step depolarization to 0 mV, they are expected to migrate downward and to the right. Because the two S subunits only activate slowly and reluctantly at this membrane potential (Fig. 4), channels will spend a relatively long time at the row labeled P2, before moving slowly to the bottom row.
	Figure 6. Single channel characteristics of the channel formed by the S-L dimer. (A and B) Single channel records of K channels expressed from the S-L dimer were recorded in cell-attached patches. Channels were activated by 750-ms step depolarizations to 0 and +40 mV. At 0 mV, early openings are dominated by visits to two sublevels, indicated by dotted lines. At +40 mV, sublevel visits become much more rare, although occasionally (e.g., record 2) persistent sublevel behavior can still be observed. The amplitude levels of Sub1 and Sub2 (see Fig. 7) are indicated by dotted lines.
	Figure 7. The S-L dimer channel has two subconductance levels. (A) Raw amplitude histograms were constructed and fitted with sums of Gaussian components as previously described (Chapman et al., 1997). Current levels arising from transitions (as judged by their slope) were omitted. At all potentials studied, four Gaussians were required; in addition to the fully open and closed level there were two well-resolved sublevels (Sub1 and Sub2). (B) Conductances were estimated by plotting the single channel amplitudes versus membrane potential, resulting in values of 4 pS, 9 pS, and 15 pS for the Sub1, Sub2, and the fully open state, respectively. The 4 pS and 9 pS values are similar to the 15% and 37% subconductance levels reported previously for drk1-L at threshold (Chapman et al., 1997).
	Figure 8. Activation kinetics of sublevels and the fully open state in the S-L dimer. (A and B) Probabilities of being in the fully open state or a subconductance level as a function of time after the step depolarization at both 0 and +40 mV were estimated by ensemble averages following idealization and interpretation by TRANSIT (VanDongen, 1996). The Po curves were fitted with a sum of three exponentials plus a constant (a0), shown by the solid line. Po (t) = a0 + a1 * exp (−t/τ1) + a2 * exp (−t/τ2) + a3 * exp (−t/τ3). The optimal values in the format a0, τ1 (a1), τ2 (a2), τ3 (a3) were as follows: for sublevels at 0 mV, 21.8, 367 (13,3), 8.48 (−3813), 8.40 (3777); for the open state at 0 mV, 45, 11.3 (7.4), 81.4 (16.7), 84.4 (−69.8); for sublevels at +40 mV, 13.1, 365 (8.6), 5.40 (−3812), 5.36 (3790); for the open state at +40 mV, 63.8, 9.3 (15), 58.7 (1.0), 60.7 (−79.8). Time constants (τ1–τ3) are in ms, their amplitudes (a1–a3) in percentages.
	Figure 9. Activation kinetics of the three conductance levels in the S-L dimer. (A) Probabilities of being in each of the three conducting states (Sub1, Sub2, and Open) were calculated from ensemble averages of idealized and “interpreted” single channel data (see MATERIALS AND METHODS). Shown here are the probabilities of being in each state as a function of time following a step depolarization to 0 mV. These curves were fit with the following model: Psub1 (t) = nL (t), Psub2 (t) = nLQ (t) and Popen (t) = nLQ (t) * nSR (t), where nLQ (t) = ninf, L [1 − exp(−t / τL)]Q and nSR (t) = ninf, S [1 − exp(−t/τS)]R. The model is derived on the Hodgkin and Huxley equations describing the change in normalized K conductance as a function of time following a step depolarization: n(t) = ninf [1 − exp(−t/τ)]4, where ninf is the steady-state value and τ the activation time constant. The HH model assumes that four identical and independent charged particles must move before the channel can open, resulting in an exponent of 4. Our model assumes that activation of Sub1, Sub2, and Open requires movement of 1 L sensor, 2 L sensors, and 2 L + 2 S sensors, respectively. If the sensors move independently and do not display cooperativity, then both exponents Q and R should have a value of 2. Cooperativity between the sensor movements would result in deviations from these integer values. By optimizing the exponents (Q) and (R), our model allows for cooperativity between the movements of the voltage sensors. The optimal parameters values were: τL = 10.3 ms, τS = 101.6 ms, ninf, L = 0.22, ninf, S = 0.47, (Q) = 2.47, and R = 1.0.
	Figure 10. Cumulative first latency distributions for the S-L dimer K channel. (A) The waiting times until the first visits to Sub1, Sub2, and Open (first latencies) were determined from the idealized and interpreted single channel data, and a cumulative distribution (FL) was constructed. The curves were fitted with the following equation: FL (t) = [1 − exp(−t/τ1)] * [1 − exp(−t/τ2)]Q. The optimal values for were as follows. For Sub1, τ1 = 14.2, τ2 = 108.9, Q = 1.3; for Sub2, τ1 = 8.6, τ2 = 95.8, Q = 9.2; for Open, τ1 = 38.9, τ2 = 167.7, Q = 1.3. Time constants τ1 and τ2 are in ms. The exponent describing the latency for sub2 is very large (9.2), indicating that visits to this sublevel are subject to a pronounced delay. This seems consistent with the model of Fig. 5 B, where a minimum of four transitions are required for an initial visit to sub2: movement of both L sensors and two opening transitions. (B) This panel shows the same data on an expanded timescale to highlight the kinetic differences between the three latencies.
	Figure 11. Conditional open probabilities for the S-L dimer K channel. COPs at a membrane potential of 0 mV were estimated following idealization and interpretation by TRANSIT (VanDongen, 1996). COPs were estimated by ensemble averages that were constructed after the waiting time until the first opening (to Sub1, Sub2, or Open) was removed. The COP curves were fitted with a sum of two exponentials plus a constant: COP (t) = a0 + a1 * exp (−t/τ1) + a2 * exp (−t/τ2). The optimal fit is shown by a solid line. The optimized values were as follows: for sub1, τ1 = 4.1, τ2 = 44.6, a0 = 0.067, a1 = 0.308, a2 = 0.079; for Sub2, τ1 = 30.1, τ2 = 1151, a0 = −0.290, a1 = 0.194, a2 = 0.568; for Open, τ1 = 33.6, τ2 = 200, a0 = 0.563, a1 = −0.289, a2 = −0.131. Time constants (τ1, τ2) are in ms.
	Figure 12. Subunit-subconductance models. To investigate whether the subunit-subconductance model can accurately describe the single channel behavior of the S-L dimer, Markov models were generated that represent specific instances of the general model shown in Fig. 5 B. The six extremely short-lived sublevels, which have a gray background in Fig. 5 B, were omitted from the Markov models, since the majority of visits to these levels will go undetected due to the low pass filtering. This results in nine resolvable states: five closed states (C1–C5), three sublevels (S1–S3), and the open state (O). Six variations were used (models 1–6) for which the number of free (optimized) parameters increases with the model, by reducing the number of constraints. Each Markov scheme defines a set of nine differential equations that were integrated to generate models of Po(t), FL(t), and COP(t) for Sub1, Sub2, and Open, at 0 and +40 mV (a total of 18 curves). Activation rate constants (vertical transitions) depend on membrane potential; rate constants describing channel opening (horizontal transitions) are voltage independent. For the Po(t) and FL(t) curves, it was assumed that all channels are in C1 at t = 0. The initial conditions for the COP(t) curves are not known and were therefore estimated. There are nine Markov states for three COP curves (Sub1, Sub2, Open) at two membrane potentials, resulting in 54 initial conditions to be estimated. In addition, rate constants and allosteric factors were estimated for each model by minimizing the sum of squared residuals (RSS) between the data and the model using the Solver in Microsoft Excel. Optimized values are provided for each model. The natural log of the residual sum of squares (RSS) is shown for each model as a relative measure of goodness-of-fit. Model 1 is the most severely constrained model and it makes two assumptions: (1) that there are four identical and independent voltage sensors, and (2) that opening transitions do not depend on the number of activated subunits. These constraints result in a model with only six free rate constants. The model provided a very poor description of the data, as illustrated by the Po(t) curve fits below. Model 2 takes into account that the S-L dimer contains subunits with distinct voltage sensors. Activation is described by two fast L sensors and two slower S sensors. Voltage sensors still move independently, and opening transitions do not depend on the number of activated subunits. This model has 10 free rate constants. Model 3 improves upon model 2 by introducing an allosteric factor (f) that allows opening efficiency to depend on the number of activated subunits. Although the number of parameters increases by only 1, the goodness-of-fit increases significantly, as shown by the Po(t) curve fits below. Model 4 allows the opening transitions to be different for the L and S subunit, while they share a single allosteric factor. This results in a small increase in the goodness-of-fit. Model 5 removes all constraints from the opening (horizontal) transitions and incrementally improves the fit. In model 6, all constraints are removed, resulting in 24 free rate constants. This last model has the smallest value of ln(RSS), consistent with it having the least number of constraints. However, the improvements made by models 4–6 over model 3 are only incremental (1%), compared with the substantial reductions in ln(RSS) made by models 2 and 3, (23% and 11%, respectively).
	Figure 13. Model fit of PO, first latency, and conditional PO at 0 mV. Model 3 in Fig. 12 was used to simultaneously fit 18 curves describing the behavior of the S-L dimer (Po, FL, and COP for three states at two membrane potentials). This figure illustrates the results for the data obtained at a membrane potential of 0 mV. Fig. 14 shows the results for a membrane potential of +40 mV.
	Figure 14. Model fit of PO, first latency, and conditional PO at +40 mV. See legend to Fig. 13.
	Figure 15. A model requiring all sensors to move before channel opening. (A) Diagram of a Markov model in which four sensor movements (in two S and two L subunits) are required before channels are allowed to open. Channel opening occurs via sojourn into Sub1 and Sub2. This model was not able to properly describe the 18 curves, as illustrated for the open probabilities at 0 mV (B) and +40 mV (C).
	Figure 16. Allosteric models for voltage-gated K channels. (A) It is assumed that voltage-dependent K channels can undergo two distinct conformational changes: voltage sensor movement (S4: resting↔active) and channel opening (pore: closed↔open). These two processes are allosterically coupled, indicated by the filled arrow, such that activation promotes channel opening. It is further assumed that these conformational changes occur in individual subunits. Consequently, each of the four subunits that make up a K channel may exist in one of four possible states, indicated by open and filled squares and circles (R = resting, A = active, C = closed, O = open). (B) The model in Fig. 5 assumes that a subunit can only move to the open conformation if its sensor has activated (linear three-state model). A more general model would allow subunits to visit the resting–open conformation (four-state model). (C) The general allosteric model for a protein with four subunits and two distinct conformational changes. The model in Fig. 5 B corresponds to the lower triangle of this general model (indicated with a gray background), which is a direct consequence of disallowing opening for resting subunits (B). Partially activated channels are indicated by rows P1–P3. Heteromeric pore conformations are indicated by columns H1–H3, which correspond to sublevels under the hypothesis tested in this paper. (D) Allosteric model previously proposed for voltage-gated K channels, which corresponds to columns labeled C and O in panel C. This model assumes that all opening transitions are highly cooperative, resulting in the heteromeric pore conformations in columns H1–H3 being unobservable transition states.

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