Correspondence to:
Dario DiFrancesco, Università di Milano, Dipartimento di Fisiologia e Biochimica Generali, via Celoria 26, 20133 Milano, Italy. Fax:39-02-70632-884. E-mail:
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Abstract |
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Hyperpolarization-activated (pacemaker) channels are dually gated by negative voltage and intracellular cAMP. Kinetics of native cardiac f-channels are not compatible with HH gating, and require closed/open multistate models. We verified that members of the HCN channel family (mHCN1, hHCN2, hHCN4) also have properties not complying with HH gating, such as sigmoidal activation and deactivation, activation deviating from fixed power of an exponential, removal of activation "delay" by preconditioning hyperpolarization. Previous work on native channels has indicated that the shifting action of cAMP on the open probability (Po) curve can be accounted for by an allosteric model, whereby cAMP binds more favorably to open than closed channels. We therefore asked whether not only cAMP-dependent, but also voltage-dependent gating of hyperpolarization-activated channels could be explained by an allosteric model. We hypothesized that HCN channels are tetramers and that each subunit comprises a voltage sensor moving between "reluctant" and "willing" states, whereas voltage sensors are independently gated by voltage, channel closed/open transitions occur allosterically. These hypotheses led to a multistate scheme comprising five open and five closed channel states. We estimated model rate constants by fitting first activation delay curves and single exponential time constant curves, and then individual activation/deactivation traces. By simply using different sets of rate constants, the model accounts for qualitative and quantitative aspects of voltage gating of all three HCN isoforms investigated, and allows an interpretation of the different kinetic properties of different isoforms. For example, faster kinetics of HCN1 relative to HCN2/HCN4 are attributable to higher HCN1 voltage sensors' rates and looser voltage-independent interactions between subunits in closed/open transitions. It also accounts for experimental evidence that reduction of sensors' positive charge leads to negative voltage shifts of Po curve, with little change of curve slope. HCN voltage gating thus involves two processes: voltage sensor gating and allosteric opening/closing.
Key Words: HCN channels, pacemaker, hyperpolarization-activated channels, modeling, channel gating
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INTRODUCTION |
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Hyperpolarization-activated f/h channels, also referred to as "pacemaker" channels, have been described in a variety of cell types including cardiac and neuronal cells (
Four isoforms of hyperpolarization-activated, cyclic nucleotidegated (HCN)1 channels have been recently cloned in several organisms (
The availability of different isoforms of HCN channels allows a detailed analysis of isoform-specific gating and modulatory properties, and a comparison of these properties with those of native channels as a way to corroborate expression data. Further, the existence of HCN isoforms with different gating properties prompts the question whether a comprehensive model can be developed to describe the kinetic features of the HCN family.
In this work we have analyzed detailed properties of voltage gating of three HCN clones (mHCN1, hHCN2, and hHCN4) based on the following considerations. First, it is known that kinetic features of native f-channel activation/deactivation cannot be described satisfactorily by the use of Hodgkin-Huxley (HH) description (
Our starting point was based on a recent investigation in cardiac SAN cells, showing that basic kinetic and modulatory properties of native pacemaker f-channels can be explained by a hybrid HH-MWC model (
On the basis of the above considerations, we explored the possibility that, as well as by cAMP, gating of pacemaker channels is modulated allosterically by voltage, too. In this attempt, we extended the hypothesis made in the gating scheme proposed earlier (
We found that by simply selecting an appropriate set of rate constants for each isoform, the allosteric assumption accounts for quantitative features of voltage-dependent gating of different isoforms. The allosteric hypothesis provides an interpretation of the possible mechanisms responsible for the different gating properties of different isoforms, and may represent an unifying model for HCN channel gating.
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MATERIALS AND METHODS |
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Molecular Biology
Functional Expression of HCN Clones in Phoenix Cells
Mouse HCN1 cDNA (mHCN1) (construct pSD64TF, provided by Dr. B. Santoro, Columbia University, New York), was cut with EcoRV and NotI restriction enzymes, filled blunt and ligated into the eukaryotic expression vector pIRES-EGFP (CLONTECH Laboratories, Inc.) predigested with EcoRV. Human HCN2 minigene (hHCN2) was subcloned into the eukaryotic expression vector pcDNA3.1 (Invitrogen) as previously described (
Electrophysiology
Experiments were performed on Phoenix cells incubated after transfection at 37°C in 5% CO2 for 1 5 d. Before electrophysiological measurements, cells were dispersed by trypsinization and plated onto 35-mm plastic Petri dishes at low density, where they were allowed to settle for 34 h. Petri dishes were transferred under the stage of an inverted microscope, and single cells were patch-clamped in the whole-cell configuration according to standard methods (
Whole-cell pipets were filled with an intracellular-like solution containing (in mM): 10 NaCl, 130 KCl, 1.0 EGTA, 5 HEPES-KOH, 0.5 MgCl2, 2 ATP (sodium salt), 0.1 GTP (sodium salt), and 5 phosphocreatine, pH 7.2. We added 1 mM BaCl2, 2 mM MnCl2, 100 µM NiCl2, and 20 µM nifedipine to perfusing solutions (in mM: 110 NaCl, 30 KCl, 1.8 CaCl2, 0.5 MgCl2, and 5 HEPES-NaOH, pH 7.4) to reduce interference from overlapping Ca2+ and K+ components in the measurement of HCN currents.
Data Analysis
Activation curves were obtained by standard two-step voltage-clamp protocols previously used (
Curve Fitting
For all HCN isoforms investigated, activation/deactivation traces were fitted by single exponential curves after an initial delay; delay, time-constant, and current amplitude were free to vary in the best-fitting routine.
Numerical Computations
Analytical treatment of the allosteric scheme proposed (see Fig 2) is given in the Appendix. Calculations were performed by integration of the vectorial differential Equation A2:
where the probability vector p, the vector k, and the matrix A are defined in the Appendix. The integration was performed by a fourth order Runge-Kutta approximation method, with an integration step set to 1/1,500th of the total integration time. The channel open probability at any time t was calculated as Po(t) = p6(t) + p7(t) + p8(t) + p9(t) + p10(t). In computations reproducing experimental current kinetics, the predicted current time course at a given voltage V was calculated as I(V,t) = Imax(V) · Po(V,t), where Imax(V) = gmax · (V - Vrev) was the fully activated current extrapolated from experimental data.
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To reproduce individual experimental traces (Fig 4 Fig 5 Fig 6) we used a semi-empirical fitting routine. Computations were initially performed by assigning first-guess values to the selected parameters, chosen by previous identification of general criteria for restricting parameter values (see RESULTS) and then by trial-and-error approximation, until satisfactory simulations were obtained.
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RESULTS |
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HCN Channels Have Kinetic Properties That Are Not Compatible with HH Models
Early investigation of the kinetics of native hyperpolarization-activated channels (f-channels of cardiac Purkinje fibres) has identified features that are not compatible with HH gating models. These include a sigmoidal activation time course that cannot be described by a fixed power of an exponential at all voltages, sigmoidal deactivation and removal of activation "delay" by conditioning prehyperpolarizing steps (
To probe whether HCN channels also have features not complying with HH kinetics, we analyzed the time course of current activation in cells expressing HCN1, HCN2, or HCN4, and checked if it could be described by a fixed power of an exponential. In Fig 1 A, HCN1, HCN2, and HCN4 current activation traces recorded on hyperpolarization at voltages in the range -75 to -115 mV are shown. Plots of the function F(t) = ln(1- (I(t)/I)1/n) should result in straight lines with zero y-intercepts and slope = -1/
if the current activates according to the nth power of an exponential. To verify this, we plotted in Fig 1 B the function F(t) for each current record with n = 1, 2 and 3, and in Fig 1 C, we plotted the y-intercepts of fits to the linear part of F(t) plots.
Clearly, in none of the three examples presented can a fixed power of an exponential adequately describe the time course at all voltages. This is especially evident for HCN2. The voltage dependence is less marked for HCN4 and absent for HCN1, but a delay is apparent also for these two isoforms (see also Fig 3). The analysis of Fig 1 shows that, for all isoforms investigated, the activation time course can be described as a single exponential after a delay. The delay is small in HCN1, larger in HCN4 and most marked in HCN2 records.
Other gating properties of native f-channels in Purkinje fibres that cannot be reconciled with HH kinetics include sigmoidal deactivation and the removal of activation delay by conditioning prehyperpolarizing steps (
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Description of Kinetic Properties of HCN Channels by Allosteric Voltage-dependent Gating
In Fig 2, the reaction scheme used to describe allosteric voltage-dependent gating of HCN channels is shown. Channels are thought to be composed of four subunits, each possessing one voltage sensor that can assume two configurations (one "willing" and one "reluctant" to open channels) and that is gated independently by voltage. With the allosteric hypothesis, all subunits undergo simultaneous, concerted transitions between closed/open states.
The analytical treatment, and the methods for numerical computations of the kinetics predicted by the model in Fig 2 are described in Appendix. Under the assumptions made, the kinetic behavior of the allosteric model is fully characterized by knowledge of the parameter a and the eight parameters o, ßo,
o,
o, z
, zß, z
, and z
describing the voltage dependence of rate constants
, ß,
,
according to Equation A9. A further simplification is obtained by setting zß = -z
, z
= -z
(see Appendix). This reduces the overall free parameters to n = 7.
HCN Channels Activation and Deactivation Kinetics: Selection of First-guess Fitting Parameters
Since the number of free parameters needed to simulate channel kinetics (n = 7) was too large to allow adequate fitting of data, we searched for criteria to restrict parameters values. The parameter "a" represents the multiplying factor of equilibrium constants of closed/open transitions upon movement of one voltage sensor from the reluctant to the willing state.
The model assumes that the transition of a voltage sensor to the willing state increases the open-to-closed channel ratio by a fixed factor, regardless of the channel subunit the voltage sensor belongs to. This implies that the parameter a is a constant in the range 0 < a < 1. Values of "a" close to 1 determine the condition L L1
L2
L3
L4, corresponding to practically ineffective voltage sensors; this would tend to abolish delays in activation and deactivation. Small values of "a" (a << 1), on the other hand, determine the condition L >>L1 >> L2 >> L3 >> L4. This would imply that most openings occur when all four voltage sensors are in the willing state (C4
O4), and that activation kinetics could be approximated by the linear scheme C
C1
C2
C3
C4
O4, with well separated "delaying" (C
C4) and "opening" (C4
O4) processes. Because of the symmetry of the cyclic allosteric scheme proposed, small values of "a" would also favor closing to occur mainly when all voltage sensors are in the reluctant state (O
C).
To test the predictive ability of the model, we tentatively assumed a = 0.2 and ran pilot computations. We found that, indeed, as expected for small values of a, the single exponential time course (after a delay) of computed activation traces approximated that of C4O4 transitions (time constant
4L; see Equation A10). We also noticed that plots of the delay displayed a voltage dependence similar to that of the time constant of voltage sensor transitions (
K(V); see Equation A10), reflecting the fact that, in the model of Fig 2, the process responsible for delaying channel opening is the displacement of voltage sensors from reluctant to willing states (data not shown).
These preliminary observations provided a method for a rough estimate of some of the required parameters. We fitted experimental traces by single exponentials after an initial delay. Plots of time constant and delay data, thus, measured for HCN1, HCN2, and HCN4 channels are shown in Fig 3. We best-fitted the time constant data with the equation for 4L(V) (see Equation A10; Fig 3 A); this yielded the functions
4(V) and ß4(V), from which first-guess parameters
o, ßo, and zß (= -z
) could be calculated as reported in the legend to Fig 3.
Delay data (Fig 3 B) were also best-fitted with the equation for K(V) (see Equation A10); this generated the functions
(V) and
(V), from which the parameters
o,
o, and z
(= -z
) were calculated as reported in Fig 3 legend. Table 1 lists the fitting parameters that were used as an initial set for predicting the kinetic behavior of the three HCN isoforms investigated.
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To verify if the parameters in Table 1 could represent a starting set for subsequent fitting of kinetic data, we selected representative experimental current traces recorded during activation/deactivation voltage-clamp protocols, and used the parameter values in Table 1 to run model computations with the same protocols. We found that despite the elementary criteria used to select rate parameters, the computations appeared to succeed in reproducing qualitatively the main kinetic features of the three HCN isoforms, including the voltage dependence of rates of activation and deactivation and the different degrees of activation and deactivation delay for different isoforms, although, not surprisingly, they failed to reproduce accurately the time course of individual traces. To verify how the parameter a affected computed channel kinetics, prediction tests were also performed using a = 0.1, 0.3, and 0.5, and were less satisfactory (data not shown). Therefore, we selected a = 0.2.
Fitting of Activation and Deactivation Time Course for HCN1, HCN2, and HCN4 Channels
To verify if the model was able to more accurately mimic the kinetics of current activation and deactivation of HCN channels in specific cases, we used semi-empirical fitting routines on individual sets of traces as shown in Fig 4 Fig 5 Fig 6 for HCN1, HCN2, and HCN4 channels, respectively.
For each trace, we initially used Table 1 to assign starting values to selected parameters (y4, z, 4L, and
K) and calculated the rate constants
, ß,
, and
according to Equation A10 and Equation A6 of Appendix. These values (with a = 0.2) allowed calculation of the matrix A and of vector k (Equation A7b and Equation A8), and thus of the current time course. The process was repeated by trial-and-error approximation, until a satisfactory simulation was obtained.
For all isoforms, the model was able to reproduce the detailed kinetic features of activation and deactivation in the whole voltage range investigated. The data points selected with the above procedure for y4, z, 4L, and
K deviated from the first-guess values assumed on the basis of the preliminary analysis of Fig 3 (compare closed circles with broken lines in Fig 4 Fig 5 Fig 6, panels CF). They were best-fitted using the equations for y4, z,
4L, and
K (see Equation A9, and Equation A10) and the resulting rate parameters are reported in Table 2. These can be directly compared with those of Table 1.
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Steady-state Open Probability
Steady-state solution of the allosteric model in Fig 2 yields the open probability-voltage relation Po(V) expressed by Equation A13, which clearly does not represent a Boltzmann type of voltage dependence. Nonetheless, to verify if it can describe the steady-state open probability curve of HCN channels, we used it to fit in Fig 7 the mean activation curves for HCN1, HCN2, and HCN4.
For all three isoforms, the allosteric Po relation was able to fit adequately the steady-state activation curves. Po values obtained independently by fitting activation/deactivation kinetics in Fig 4 Fig 5 Fig 6 (open circles) fell near the Po curves (full lines) obtained with Equation A13, indicating that the two procedures led to comparable predictions.
Removal of the Delay in Current Activation by Conditioning Prehyperpolarization
In cardiac Purkinje fibres, activation delay can be removed by conditioning prehyperpolarizations (
To verify the ability of the allosteric model to mimic this behavior, we fitted records at -110 mV and at -150 mV with the procedure used above (Fig 6), and then ran two-step model computations reproducing experimental protocols (Fig 8 C). The model accurately predicted a progressive current acceleration and removal of delay with increasing prepulse durations. With sufficiently long prepulses, the computed time course became a single exponential, as apparent from the semilog plot of Fig 8 D. This is expected if the action of prehyperpolarizing steps is to "prime" the channel opening process by pushing the majority of voltage sensors to their willing state, such that at -110 mV, the current development reflects mostly C4O4 transitions. The computed traces at -110 mV, however, did not perfectly overlap experimental ones particularly after the longest prepulses. This may be due to the fact that, especially with long prepulses, the experimental current at -110 mV underwent an early acceleration compared with a single exponential (also observed in native f-channel;
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DISCUSSION |
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Early investigation of native pacemaker f-channels has shown that description of detailed kinetic properties requires a complex multistate, non-HH model (
HCN channels belong to the superfamily of Kv and CNG channels. The amino acid sequence comprises six transmembrane (TM) domains with a pore region between S5 and S6 and a positively charged domain S4, which acts as the primary voltage sensor (
A gating model taking into account the molecular information available and the structural similarities with Kv and CNG channels should, therefore, regard HCN channels as tetramers whose closed/open transitions are regulated by voltage sensors. Two additional observations help clarify the features of this model: first, previous work on native f-channels has shown that the cAMP-induced modulation can be described by an allosteric model (
Allosteric Model of HCN Channels Voltage Gating and Comparison with Previous Models
Based on the above considerations, we asked whether the voltage gating of HCN channels could be described as an allosteric process, which led to a 10-state cyclic reaction scheme (Fig 2). A major problem that we faced in using the model was to restrict the choice of free model parameters in a way that would allow the selection of parameter values by data fitting, since clearly too many free parameters would imply the need of arbitrary selection of some of the values.
One assumption of the model at one time intuitive and useful to the effect of limiting free parameters, was to consider that the action on closed/open channel ratio of any voltage sensor's displacement is always the same, regardless of the distribution of voltage sensors (i.e., the number of sensors in the willing state versus those in the reluctant state), or to which channel subunit the voltage sensor belongs. This meant that the parameter a could be considered as a constant (in the range 01). This assumption, together with the assumption of balanced rate constant changes (i.e., when the equilibrium constant is multiplied by a, then the opening rate constant is divided by a, and the closing rate constant is multiplied by
a), was very effective in that it reduced the set of 10 originally free "vertical" rate constants in Fig 2 (
, ß,
1, ß1,
2, ß2,
3, ß3,
4, and ß4) to just three free rate constants (the parameter a and
, ß or, equivalently,
4 and ß4).
On this basis, after a preliminary selection of model parameters (Table 1), we used individual curve fitting to estimate rate constants (Fig 4 Fig 5 Fig 6 and Table 2). In all three HCN isoforms investigated, the allosteric scheme of Fig 2 could reproduce the observed kinetic behavior, including sigmoidal activation and deactivation (Fig 4 Fig 5 Fig 6) and removal of activation delay by prehyperpolarizations (Fig 8). Also, the model accounted for the position and shape of the open probability curve for the three isoforms (Fig 7).
The allosteric scheme examined here is analogous to that proposed by
In our model of HCN channels, closed/open transitions are indeed severalfold slower than sensor transitions for all isoforms (see Fig 10); however, fitting of the time constant curves in Fig 3 A and curve reconstruction in Fig 4 Fig 5 Fig 6 indicate that closed/open transitions have a substantial voltage dependence (zß = -z = 0.94, 1.07, and 0.80 equivalent electronic charges for HCN1, HCN2, and HCN4 isoform, respectively, Table 2).
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It is important to stress that the values of z and zß reported in Table 2 were not assumed, but resulted from data fitting, and, therefore, cannot be reduced arbitrarily. In other words, the assumption that each voltage sensor contributes equally to favor channel opening, together with the other model assumptions, leads to experimental data fitting indicating the presence of a substantial voltage dependence for closed/open transitions for all isoforms.
Investigating a model where closed/open transitions are weakly voltage-dependent would imply a different set of assumptions, since most of the voltage dependence of kinetics would derive from voltage sensors' state changes. The operating framework of the model would be likely to change significantly. We would need to release the assumptions concerning the constancy of a and/or the balanced rate constant changes, and let most of the 10 vertical rate constants , ß,
1, ß1,
2, ß2,
3, ß3,
4, and ß4 free to vary. In fact, the situation could be even more complex since releasing the condition a = constant would also affect the horizontal rate constants.
We did not adopt this solution because it would have involved too much arbitrariness in the selection of parameter values. More importantly, we did not have any real indication that closed/open transitions are weakly voltage-dependent or voltage-independent for HCN channels, since we did not have evidence for regions of shallow slope at negative voltages in time constant curves of any of the HCN isoforms investigated (Fig 3 A). However, the model scheme of Fig 2 is not incompatible with a situation where closed/open transitions are nearly voltage-independent (i.e., with a low z) and most of the voltage dependence of channel kinetics is attributable to sensor movements. Although we lack experimental evidence, we cannot rule out this possibility entirely, since, due to experimental constraints, the voltage range where recordings could be made was limited (positive to -135 mV; Fig 3 A).
Implications of Allosteric Design
Recent data have shown that replacement of basic residues in the S4 segment of HCN2 with neutral ones leads to negative shifts of the activation curve voltage dependence (
In Fig 9 A, the sensor distribution parameter z (reluctant/willing configurations) was assumed to undergo changes due to reduction of the equivalent charge number (z) by 1/6, 2/6, 3/6, and 4/6 of the control value (labeled 14). As expected, reductions of z
led to proportional reductions in the slope of the z(V) curve. However, in Fig 9 B, the Po(V) relations calculated with the different z(V) curves in Fig 9 A were shifted to more negative voltages (by up to 46.9 mV with a four-sixths reduction) with only a minor change in slope. Thus, a possible explanation for the lack of modification of the slope of the activation curve in experiments where charged residues are mutated is that the activation process can be separated, as in the allosteric model proposed here, into two distinct steps (i.e., redistribution of voltage sensors and closed/open transitions). A direct consequence is that the slope of the Po curve does not necessarily reflect the slope of gating charge voltage dependence. Since the model provides a molecular interpretation of gating properties of HCN channels, it will be interesting to use it to simulate expectations of experiments in which structural elements of HCN channels are modified (i.e., mutations and/or coexpression of different isoforms).
Model Interpretation of the Different Kinetic Features of HCN Isoforms
The kinetics and cAMP dependence of different HCN isoforms differ quantitatively. For example, HCN1 subunits form channels with fast voltage-dependent gating and reduced cAMP sensitivity, whereas HCN2 and HCN4 channels respond more efficiently to cAMP and have slower kinetics (
The comparison led to the following observations: (1) the y4(V) curves of all three isoforms are almost superimposable; (2) the voltage dependence of 4L(V) curves is similar for all isoforms, the main difference being a scaling-up factor when going from HCN1 to HCN2 and HCN4 curves; (3) the z(V) curves of HCN1 and HCN4 are little different, whereas that of HCN2 is some 40 mV more negative; and (4) a similarly large voltage shift is present for the
K(V) curve of HCN2 relative to HCN1 and HCN4 curves; the peak
K increases when going from HCN1 to HCN2 and to HCN4.
The above observations allow a few tentative speculations on the mechanisms responsible for the kinetics of HCN channels and their differences. First, a scaling-up of time constant curve that leaves unaltered the distribution curve implies in a Boltzmann mechanism that on- and off-rates are divided by the same factor (>1), equivalent to adding a fixed positive energy term to the energy changes associated to on- and off-reactions. The data in Fig 10 (A and B), thus, would be compatible with the assumption that closed/open transitions of HCN channels involve two mechanisms: (1) a voltage-dependent one, similar for all isoforms and involving movement of an equivalent electric charge of 1; and (2) a voltage-independent "mechanical" one perhaps associated to intersubunit interactions that are loose in HCN1, tighter in HCN2, and tighter still in HCN4 channels. Assuming simple scaling-up of time constant curves, the
4L curve of HCN1 is
5 and 15 times smaller than those of HCN2 and HCN4, respectively. Compared with HCN1, the extra energies required by HCN2 and HCN4 closed/open transitions, thus, would be RT ln(5) = 0.956 kcal/mol and RT ln(15) = 1.609 kcal/mol (see Equation A9 and Equation A10).
Second, according to the data in Fig 10, faster activation of HCN1 relative to HCN2 and HCN4 is attributable to both faster voltage sensor activation and channel opening, whereas the more negative position of the HCN2 activation curve relative to that of HCN1 or HCN4 (Fig 7) essentially depends on the voltage dependence of voltage sensors. Third, transition rates of voltage sensors are several times faster than closed/open transition rates, implying that for all isoforms, the activation process involves early sensor movements (from reluctant to willing states) as a preliminary requirement for efficient channel opening. Finally, lack of major changes in the equivalent charge associated to movement of voltage sensors and closed/open transitions suggests common structural rearrangements responsible for voltage-dependent state changes in the isoforms studied.
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Footnotes |
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1 Abbreviations used in this paper: CNG, cyclic nucleotidegated; HCN, hyperpolarization-activated, cyclic nucleotidegated; HH, Hodgkin-Huxley; MWC, Monod-Whyman-Changeaux; Po, open probability; SAN, sinoatrial node; TM, transmembrane.
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Acknowledgements |
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We thank Antonio Malgaroli, Richard B. Robinson and Arnaldo Ferroni for discussion, and Monica Beltrame for assistance in development of the molecular biology of HCN clones. We also wish to thank B. Santoro and S. Siegelbaum (Columbia University, New York, NY) for supplying mHCN1 cDNA, and U.B. Kaupp of the University of Jülich for supplying hHCN4 cDNA.
This work was supported by the MURST (Cofin 1999 to D. DiFrancesco) and by Telethon (Grant no 971 to D. DiFrancesco).
Submitted: 19 December 2000
Revised: 9 April 2001
Accepted: 13 April 2001
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Appendix |
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Kinetics of Allosteric Model
A general set of equations governing the kinetics of the scheme in Fig 2 is
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(A1) |
where pi is the probability of state i, ki,j is the rate constant for the transition i to j, and n is the number of possible states (pi = 1 and expressed in matrix notation as
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(A2) |
where the vectors p and k and the matrix A are defined as:
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(A3) |
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(A4) |
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(A5) |
To apply the general Equation A5 to the scheme of Fig 2, it is useful to first define the rate constants ki,j. As illustrated in the legend to Fig 2, the following relations hold for equilibrium constants and rate constants in the reaction scheme (note that a = 1/f2):
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(A6) |
Using the numbering of states 110 as labeled in Fig 2 and the relations A6, the rate constants ki,j can be tabulated as in Table A1.
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We can now explicitly write the elements of Equation A2 as follows:
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(A7a) |
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(A7b) |
Although integration of Equation A2 is possible (, ß,
and
are known. The voltage dependence of rate constants can be described by Boltzmann functions:
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(A8) |
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(A9) |
where o, ßo,
o, and
o are rate constants at zero voltage, z
, zß, z
, and z
are the equivalent units of electronic charge moving in the electric field during corresponding state transitions, and r = RT/F = 25.85 mV (T = 300°K). It is assumed that zß = -z
and z
= -z
. The former assumption is based on the experimental observation that activation and deactivation rate constants of native pacemaker channels have a nearly symmetrical dependence on voltage (
o, ßo,
o,
o, zß, and z
and of the parameter a.
An equivalent description, useful to a more intuitive interpretation of the meaning of parameters, can be obtained by using fractional distribution parameters and time constants. One such set of parameters equivalent to Equation A9 is for example:
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(A10) |
(note that 4 =
/a2 and ß4 = ß a2 according to A6). Here y4 and z represent the fractional distribution between channel states O4 and C4 and between sensor willing and reluctant states, and
4L and
K represent the corresponding time constants of equilibrium changes.
An explicit solution of A2 describing the probability of open state can be obtained by the MWC formulation of cyclic allosteric reactions (
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(A11) |
Summing over open and closed states and rearranging yields total open ([Ot]) and total closed ([Ct]) concentrations:
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(A12) |
from which the probability of open state is derived as
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(A13) |
The voltage dependence of Po results from the voltage dependence of rate constants according to Equation A9 and the definition of equilibrium constants L and K in the scheme of Fig 2:
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(A14) |
where
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(A15) |
are the midpoint voltages of the fractional distribution parameters y = 1/(1+L) (distribution between O/C states) and z = 1/(1+K) (distribution between willing/reluctant sensor states). As mentioned above (Equation A10) an equivalent description can be obtained by referring to C4O4, rather than C
O transitions. In this case, the variables used are (see Equation A6):
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(A16) |
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(A17) |
and from A13
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(A18) |
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