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The Journal of Neuroscience, May 1, 2002, 22(9):3392-3403
Functional Stoichiometry of Glutamate Receptor Desensitization
Derek Bowie1 and G. David Lange2 1 Department of Pharmacology, Emory University School of Medicine, Atlanta, Georgia 30322, and 2 Instrumentation and Computer Section, National Institute of Neurological Disorders and Stroke, National Institutes of Health, Bethesda, Maryland 20892
ABSTRACT
TOP
ABSTRACT
INTRODUCTION
Potassium (K+) channels and ionotropic glutamate receptors (iGluRs) fulfill divergent roles in vertebrate nervous systems. Despite this, however, recent work suggests that these ion channels are structurally homologous, sharing an ancestral protein, architectural design, and tetrameric subunit stoichiometry. Their gating mechanisms also are speculated to have overlapping features. Here we show that the mechanism of iGluR desensitization is unique. Unlike K+ channels, AMPA- and kainate-type iGluR subunits desensitize in several ordered conformational steps. AMPA receptors operate as dimers, whereas the functional stoichiometry of kainate receptor desensitization is dependent on external ions. Contrary to conventional understanding, kinetic models suggest that partially desensitized AMPA and kainate receptors conduct ions and are likely participants in synaptic signaling. Although sharing many structural correlates with K+ channels, iGluRs have evolved unique subunit-subunit interactions, tailoring their gating behavior to fulfill distinct roles in neuronal signaling.
Key words: glutamate; AMPA; kainate; desensitization; stoichiometry; gating
INTRODUCTION
Prokaryotes and eukaryotes use a myriad of membrane-bound ion channel proteins to detect and respond to numerous environmental stimuli, from the rapid "trap door" closure of carnivorous plants to the detection of high-frequency auditory signals in songbirds (Hille, 1992
). The most striking role of this plethora of ion channel proteins is to establish a complex array of signaling pathways in nervous systems of higher organisms. Although the molecular basis for the complexity of neuronal circuitry is still emerging, it is well established that ion channel proteins are essential components sculpting neuronal behavior via changes in developmental expression patterns, post-translational modification, and subcellular targeting. Two of the most important ion channels are K+ channels and ionotropic glutamate receptors (iGluRs). K+ channels detect changes in membrane potential and regulate neuronal firing properties (Yi and Jan, 2000
Despite their divergent roles, K+ channels and iGluRs share important structural features, including tetrameric subunit stoichiometry (MacKinnon, 1991
Consistent with this, K+ channel and iGluR activation pathways exhibit notable similarities. K+ channels (Chapman et al., 1997
In this study we have reexamined AMPA receptor desensitization in comparison with kainate receptors. Unlike K+ channels, both AMPA and kainate receptors desensitize in a series of sequential conformational steps. Statistical evaluation of different gating schemes suggests that AMPA receptors operate as dimers, whereas kainate receptor desensitization is more consistent with a tetramer arrangement. We also have identified the novel observation that kainate receptor desensitization is regulated strongly by external ions.
MATERIALS AND METHODS
Cell culture and expression of recombinant receptors. Human embryonic kidney 293 (HEK 293) cells (CRL 1573; American Type Culture Collection, Manassas, VA) and tsA201 cells (provided by Dr. R. Horn, Jefferson Medical College, Philadelphia, PA) were maintained at a confluency of 70-80% in MEM with Earle's salts, 2 mM glutamine, and 10% fetal bovine serum. After being plated at low density, the cells were transfected with cDNAs encoding GluR-A and GluR6 receptor subunits (supplied by Drs. M. L. Mayer and K. M. Partin at National Institutes of Health, Bethesda, MD, and Colorado State University, CO, respectively), using the calcium phosphate technique as described previously (Bowie et al., 1998
Electrophysiological recordings. All recordings were made with an Axopatch 200B amplifier (Axon Instruments, Foster City, CA) that used thin-walled borosilicate glass pipettes (2-5 M
) coated with dental wax to reduce electrical noise. Control and L-Glu (10 mM; 50 msec duration) solutions were applied rapidly to outside-out patches excised from HEK 293 or tsA201 cells expressing unedited GluR-A or GluR6 subunits as described previously (Bowie et al., 1998
External solutions contained 55, 150, or 405 mM NaCl, 5 mM HEPES, and 0.1 mM each of CaCl2 and MgCl2, pH 7.3. Internal solutions contained (in nM) 10, 115, or 360 NaCl, 10 NaF, 5 HEPES, 5 Na4BAPTA, 0.5 CaCl2, 1 MgCl2, and 10 Na2ATP, pH 7.3. Osmotic pressure was adjusted to 295 mOsm for 55 and 150 mM NaCl with sucrose and to 750 mOsm for 405 mM NaCl. The osmotic pressure did not influence channel gating. Current records were filtered at 5 kHz and digitized at 25-50 kHz; series resistances (3-10 M
Modeling AMPA and kainate receptor desensitization. Before performing fits, we normalized the amplitudes of all test responses in a given experiment to the peak conditioning response. Because the total channel number present on excised patches was variable, this normalization step permitted comparison of the fit parameters among different patches and ionic conditions.
Rates into and out of desensitization were estimated from the amplitude and decay kinetics of test GluR6 or GluR-A responses. Code was written that permitted the simultaneous fitting of all test responses with the use of a nonlinear steepest descent algorithm ("NonlinearFit," Mathematica, Wolfram Research; see Appendix). For each model the rates of entry into and out of desensitization were governed by the rate constants ki and ri, respectively. Gi represents the macroscopic conductance(s) in each model expressed in normalized units. Gi reflects the sum of the product of the unitary conductance(s) and open probability. There were no constraints imposed on the various fit parameters, nor was the outcome of the fits sensitive to initial guesses.
The four models we have tested form a hierarchy of functions. With such a hierarchy it is appropriate to measure the improvement in fit achieved by a more complex model compared with a less complex model as the ratio (F) of the relative decrease in error sum of squares over the relative decrease in error degrees of freedom (Horn, 1987
Assumptions of modeling strategy. To estimate the functional stoichiometry of desensitization, we made three assumptions. First, we assumed that AMPA and kainate receptors are tetramers composed of four identical subunits. Several studies suggest that AMPA receptors are tetramers (Rosenmund et al., 1998
The second assumption is that decay kinetics of test responses reflects entry into desensitization. This assumption remains valid even if the mechanism of desensitization includes coupling to channel opening or closure. Consistent with this, there is some evidence suggesting that deactivation and desensitization of AMPA receptors are coupled (Partin et al., 1996
Third, we have assumed that the agonist concurrently initiates channel activation and the onset of desensitization during each test response. In agreement with this, most studies suggest that AMPA and kainate receptors are not required to traverse the open state to enter desensitized states, because desensitization occurs with low agonist concentrations that fail to gate the channel (Dingledine et al., 1999
Our modeling of AMPA and kainate receptor desensitization is a simplified approach. An asset of this method is that analysis is more intuitive in nature. Ideally, transitions that are unlikely to impact on the outcome of analysis are neglected. The modeling described in this study does not account for agonist binding or unbinding steps. However, there is justification for this because our fitting strategy relies on fitting the decay of responses elicited by saturating agonist to which binding/unbinding events do not contribute significantly. Finally, the high concentrations of agonist suggest that our models describe mainly the behavior of different states of fully occupied receptors.
RESULTS
Kinetics of AMPA and kainate receptor desensitization
AMPA (Fig. 1) and kainate (Fig. 2) receptor desensitization was studied on outside-out patches excised from mammalian cells expressing GluR-A and GluR6 homotetramers, respectively (see Materials and Methods). Entry into and exit from desensitization was determined from the decay kinetics and amplitude of macroscopic responses elicited by paired agonist applications (10 mM L-Glu; 50 msec duration each) separated by varying time intervals (Figs. 1a, 2a). Agonists were applied to patches by an ultrafast solution exchange (10-90% rise times of 25-50 µsec) two orders of magnitude faster than desensitization kinetics. Our preliminary experiments (see Figs. 1-5) were recorded in 405 mM NaCl salt solutions to increase the amplitude of macroscopic currents in excised patches (see Materials and Methods). In subsequent experiments AMPA and kainate receptor behavior was examined in 55 and 150 mM NaCl solutions (see Figs. 6, 7).
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Figure 1. Entry into and exit from AMPA receptor desensitization. a, Typical recording showing 50 superimposed conditioning and test pulses (10 mM; 50 msec duration; Hp, 20 mV); patch number 010123p3. b, Schematic of concerted (left) and independent (right) models of desensitization highlights the behavior of individual subunits. c, Summary plot of GluR-A recovery in 405 mM NaCl solution (n = 6; mean ± SEM). Data were fit (solid line) by the expression: I(t) = Ipeak · [1
rec)], where I(t) is the response amplitude at any time, t, Ipeak is the peak test response, and
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Figure 2. Entry into and exit from kainate receptor desensitization. a, Typical recording showing 33 superimposed conditioning and test Glu pulses (10 mM; 50 msec duration; Hp,
During the first agonist application, or conditioning response, AMPA and kainate receptors were activated rapidly (rise times of 300 µsec) but subsequently were desensitized in the continued presence of agonist to an equilibrium level of 1.4 ± 0.2% (GluR-A, n = 6; Fig. 1a,e) and 0.8 ± 0.05% (GluR6, n = 7; Fig. 2a,d) of the peak response in 405 mM symmetrical NaCl. The rate of onset of AMPA receptor desensitization determined from conditioning responses was fit best by a double-exponential function with fast and slow time constants of 2.49 ± 0.27 and 8.04 ± 1.78 msec (20.7 ± 6.2% of peak; n = 6), respectively. The rate of onset of kainate receptor desensitization estimated from 50 msec conditioning responses to 10 mM Glu decayed with a single time constant of 6.03 ± 0.47 msec (n = 7). Note that for both receptor subtypes the membrane current observed at the end of the conditioning response relaxed to the zero current baseline more slowly than the decay of the peak response described above (Figs. 1e, 2d). The difference is more evident for AMPA (Fig. 1e) than for kainate receptors (Fig. 2d) and suggests that channels with different kinetic behavior mediate peak and equilibrium responses. Consequently, equilibrium desensitization does not reflect the recycling of AMPA or kainate receptors through the main open state as modeled previously (Partin et al., 1993
Recovery from desensitization was determined from the amplitude of a second agonist application, or test response, that reported the fraction of resensitized channels (Figs. 1a,c, 2a,b). AMPA receptors fully recovered from desensitization with interpulse intervals >1 sec (
The behavior of individual AMPA and kainate receptor subunits during desensitization is still not resolved fully. Most studies have proposed that entry into and exit from kainate receptor desensitization can be described reasonably well as first-order processes (Heckmann et al., 1996
In view of this, we have reinvestigated the behavior of individual AMPA receptor subunits during desensitization and, for comparison, examined the functional stoichiometry of kainate receptor desensitization also. Initially, we reviewed the concerted/independent models of desensitization that are analogous to K+ channel inactivation mechanisms (Fig. 1b, right, left). Although subunits behave differently in concerted and independent models, entry and exit rates from desensitization in each case display first-order kinetics. Furthermore, we examined the time course of recovery from desensitization first, because it is more than three orders of magnitude slower than the onset of desensitization. We anticipated that digressions from first-order behavior would be easier, at least initially, to identify experimentally during the recovery step rather than the onset of desensitization.
To study the recovery process carefully, we mapped out the entire time course of GluR6 and GluR-A recovery from the peak amplitudes of 33 and 50 test pulses, respectively, which subsequently were fit with a single-exponential function (Figs. 1c, 2b). Contrary to many previous studies (Lomeli et al., 1994
The concerted/independent models also predict that entry rates into desensitization are independent of the fraction of resensitized channels. To examine this, the decay kinetics of test responses at different interpulse intervals was compared (Figs. 1e, 2d). Contrary to the concerted/independent models, decay rates were different between test pulses. For example in Figures 1e and 2d, the first GluR-A and GluR6 test responses decayed with time constants of 10.1 ± 1.9 and 20.8 ± 3.2 msec, respectively, whereas the last (fourth or sixth) test responses shown were 5.7 ± 0.4 and 11.6 ± 0.9 msec, respectively. At time intervals >840 msec (11 ± 3% desensitized;
Determining the functional stoichiometry of iGluR desensitization
Alternatively, AMPA and kainate receptors may desensitize in several sequential steps. Because individual subunits contain a binding site for a single agonist molecule, desensitization may result from a number of possibilities including combined or independent subunit conformations. To determine the number of conformational steps (or gates) involved, we developed a fitting program to examine four models of desensitization: (1) concerted/independent (one gate; Fig. 1b), (2) dimer with independence or cooperativity (two gates; Fig. 3a), and (3) tetramer (four gates; Fig. 4a). The program fits experimental records, with each model providing information on transition rates into (kN) and out of (rN) desensitization as well as the macroscopic conductance of each state (GN). In the independent dimer, paired subunits are equivalent and independent; therefore, rate constants are proportional to the number of open or desensitized subunit pairs. In the cooperative dimer, paired subunits are not independent; therefore, their contribution is not proportional to the number of subunit pairs. The F ratio test was used to evaluate statistically the goodness of fit among models with different degrees of freedom. A complete treatise of the statistical methods and mathematical formulae used for fitting is described elsewhere (see Materials and Methods, Appendix, and the web address).
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Figure 3. Determining the functional stoichiometry of AMPA receptor desensitization. a, Schematic of the cooperative dimer model. b, GluR-A experimental records (dots; patch number 010123p1) fit with the concerted/independent models (left, solid line) and the cooperative dimer model (right, solid line). The first 25 msec of four test responses [time after conditioning response (tc) was 8, 40, 60, and 105 msec] were superimposed for comparison. The dashed line indicates zero current level. c, Plot profiling the distribution of each state in the cooperative dimer model at a range of interpulse intervals. d, Summary plots showing how fits of experimental data with different models of desensitization were compared. Fits were compared pairwise, using the F ratio test. Top plot shows the independent dimer compared with the concerted/independent models. Middle plot compares the cooperative dimer with concerted/independent models and independent dimer model. Bottom plot compares the tetramer model with all other models of desensitization. Each bar indicates the confidence level distinguishing between two models fit to the same experimental data. For every comparison six bars are shown that represent the results from six separate patch recordings. The dashed line in each plot denotes the 95% confidence level.
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Figure 4. Determining the functional stoichiometry of kainate receptor desensitization. a, Schematic of the tetramer model. b, GluR6 experimental records (dots; patch number 000720p2) fit with the concerted/independent (left, solid line) and tetramer (right, solid line) models. Entire 50 msec of three test responses (tc = 15, 135, and 285 msec) are superimposed for comparison. The dashed line indicates zero current level. c, Plot profiling the distribution of each state in the tetramer model at a range of interpulse intervals. d, Summary plots showing how fits of experimental data with different models of desensitization were compared. Fits were compared pairwise, using the F ratio test. Each bar indicates the confidence level distinguishing between two models fit to the same experimental data. For every comparison seven bars are shown that represent the results from seven separate patch recordings. The dashed line in each plot denotes the 95% confidence level.
Figure 3 summarizes our results from fits of AMPA receptor data. To provide information on rates into and out of AMPA receptor desensitization, our program simultaneously fit 50 test responses as typified by the patch recording shown in Figure 1a. For convenience, we have not illustrated the fits to all 50 test responses but have selected, instead, several responses that occur at the beginning of the recovery process because they exhibit different decay kinetics (Fig. 3b). Moreover, to compare the goodness of fit of all four models, we have summarized the results of statistical comparisons in a bar graph (Fig. 3d). Figure 3b shows several AMPA receptor test responses fit with the concerted/independent models (Fig. 1b) and cooperative dimer model (Fig. 3a). As expected, the concerted/independent models did not account for experimental observations, particularly the test responses observed at brief interpulse intervals (Fig. 3b, left). In contrast, AMPA receptor desensitization was described well by the cooperative dimer model, accounting for the different decay kinetics observed with test responses at the beginning of the recovery process (Fig. 3b, right). Although not shown, the cooperative dimer model fit all 50 test responses well in all patch recordings. The goodness of fit for the cooperative dimer compared with the other models is described below. Because the fitting program provides information on the rate constants governing recovery from desensitization, the fractional occupancy of the various states at different time points during recovery can be calculated. For the cooperative dimer model (Fig. 3a) the distribution of the three states at a range of interpulse intervals is shown in Figure 3c.
To evaluate statistically which of the four models of desensitization fit best our experimental data, we have used the F ratio test (see Materials and Methods). The F ratio test examines whether the decrease in the sum of squares, often found when the data are fit to more complicated models, is merited by the loss of degrees of freedom (i.e., additional variables). The results of evaluating the goodness of fit of each model in these pairwise comparisons are summarized in Figure 3d. Figure 3d shows three plots that illustrate paired comparisons between the independent dimer model and concerted/independent models (Fig. 3d, top plot), the cooperative dimer with the independent dimer or concerted/independent models (Fig. 3d, middle plot), and, finally, the tetramer model with all three other models (Fig. 3d, bottom plot). The value denoted by each bar in each of the three plots represents the confidence level observed when two models were compared with data from a single patch recording. For example, the comparison between the tetramer and concerted/independent models (Fig. 3d, bottom plot, left column) shows six bars or six comparisons, which indicates that data from six different patch recordings were used. By noting the position of the bar in a data set (e.g., third bar in a group of six), it is possible to compare how different models fit the data from an individual experiment.
As expected, both dimer (independent and cooperative) and tetramer models fit better the experimental data (>95% confidence level; six of six patches) than the concerted/independent models (Fig. 3d, left column). Moreover, five of six and four of six patches favored (>95% confidence level; Fig. 3d, dotted line) the cooperative dimer and tetramer models, respectively, over the independent dimer model (Fig. 3d, middle column). However, the goodness of fit of the cooperative dimer model was favored over the tetramer for all patches that were tested (Fig. 3d, bottom, right column; n = 6), suggesting that AMPA receptors assembled from identical subunits operate as dimers of dimers. The rate constants and macroscopic conductances estimated from fits of the dimer model to AMPA receptor data are summarized in Table 1. It is worth noting that the results of our fits suggest that intermediate desensitized conductance states contribute to membrane conductance. As described below, intermediate desensitized states contribute more significantly to the membrane conductance elicited by kainate receptors.
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Table 1. Summary of the rate constants and macroscopic conductance values obtained from model fits Figure 4 summarizes our results from fits of kainate receptor data. Consistent with AMPA receptor data, GluR6 test responses also were not fit well by the concerted/independent models. Figure 4b shows three test responses observed during the initial phase of recovery and fit with the concerted/independent models (Fig. 4b, left). As expected, the concerted/independent model overestimated the peak amplitude of test responses (Fig. 4b, left). In contrast, the tetramer model fit well the GluR6 responses even at brief interpulse intervals (Fig. 4b, right). The distribution of each state of the tetramer model at different interpulse intervals was calculated and is profiled in Figure 4c. As mentioned previously, we have shown only test responses at brief interpulse intervals in Figures 3b and 4b for convenience. However, a comparison of fits of different models to the entire range of GluR6 responses, including early and later phases of the recovery process, can be reviewed in Figure 5 (see also the web address in Materials and Methods).
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Figure 5. Kainate receptor desensitization fit with concerted/independent and tetramer models. a, b, Four superimposed test responses (tc = 15, 150, 225, and 300 msec; dots; patch number 000720p2) recorded during the early phase of recovery from GluR6 desensitization and fit with the concerted/independent model (a) or tetramer model (b). The dashed line indicates zero current level. Note that the tetramer model fits the data better than the concerted/independent model. c, d, Ten superimposed test responses from the same patch recording (tc = 0.040, 0.84, 1.24, 1.64, 2.04, 2.44, 2.84, 3.64, 7.95, and 15.95 sec; patch number 000720p2) showing both early and late phases of recovery from GluR6 desensitization and fit with the concerted/independent model (c) or the tetramer model (d). Note that the tetramer model fits the data better than the concerted/independent model. For clarity, the total number of data points on each test response sweep has been reduced (eightfold). The dashed line indicates zero current level.
Similar to our analysis of AMPA receptors, we also compared statistically the goodness of fit between pairs of models (Fig. 4d). As noted for AMPA receptors, dimer (independent and cooperative) and tetramer models accounted better for the experimental data than the concerted/independent models in all patches that were tested (Fig. 4d, left column; seven of seven; >95% confidence level). However, unlike AMPA receptors, the tetramer model fit the data with GluR6 receptors better than the independent or cooperative dimer models in all cases (Fig. 4d, bottom, middle and right columns; six of seven patches; >95% confidence level), suggesting that the functional stoichiometry of kainate receptor desensitization is most consistent with a tetramer arrangement. The rate constants and macroscopic conductances for the tetramer model of GluR6 are summarized in Table 1. It is worth noting that, when compared with AMPA receptors, the intermediate desensitized states (e.g., G3 and G4) of the kainate receptor make a significant contribution to the total membrane conductance (see Table 1), the physiological significance of which is discussed below. Although fits of GluR6 data consistently favored the tetramer model, the experiments described below revealed that the apparent functional stoichiometry of kainate receptor desensitization was dependent on the external ion concentration.
External ions regulate the gating behavior of kainate, but not AMPA, receptors
External ions modulate C-type inactivation of K+ channels (Baukrowitz and Yellen, 1995
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Figure 6. Kainate receptor desensitization is modulated by external ions. a, b, Plots showing typical GluR6 or GluR-A receptor responses in symmetrical solutions of 55, 150, and 405 mM NaCl. Bottom traces show junction currents recorded with an open electrode tip. c, d, Summary plots illustrating the effect of changing the NaCl concentration on the time course of GluR6 and GluR-A receptor desensitization. Data are expressed as mean ± SEM.
In view of their ion sensitivity, we determined whether kainate receptors still operated as functional tetramers in 55 and 150 mM NaCl. Figure 7a shows the first 20 test responses typically observed in recordings with GluR6 receptors in 55 mM NaCl. In contrast to responses in 405 mM NaCl, the decay kinetics for all test responses was very similar (Fig. 7b). For example, the conditioning response (
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Figure 7. External ions regulate the functional stoichiometry of kainate receptor desensitization. a, Typical GluR6 test responses recorded in 55 mM NaCl solutions exhibited similar decay kinetics (patch number 000724p2). b, When the first six test responses from a were aligned for comparison, the decay kinetics was revealed to be almost identical. c, Recovery from GluR6 desensitization in 55 (filled circles) and 150 (open circles) mM NaCl solutions. GluR6 recovery in 55 mM NaCl was monoexponential, where
GluR6 responses in 150 mM NaCl exhibited intermediate behavior. Recovery was not monoexponential in nature (Fig. 7c, arrow, open symbols), deviating clearly from the concerted/independent models during the early recovery phase (Fig. 7d). However, the monoexponential function fit the data better than GluR6 responses observed in 405 mM NaCl solutions (Fig. 2b). Statistical comparison of the goodness of fit between model pairs revealed that the experimental data were best fit by the cooperative dimer model (Fig. 7e, open bars). In four of four patches that were tested, the dimer cooperative model was favored (>95% confidence level) over the concerted/independent, independent dimer, and tetramer models (Fig. 7e, open bars).
DISCUSSION
Until recently, it was not possible to speculate on the most basic properties of evolutionary precursors to eukaryotic K+ channels and iGluRs (Hille, 1992
Comparison with previous studies
To our knowledge, a direct comparison between AMPA and kainate receptor desensitization has not been examined previously. Many studies, however, have studied desensitization for each receptor subtype, and, in most cases, the recovery process has been described as first order (Lomeli et al., 1994
Some investigations have described more complex recovery behavior (Trussell and Fischbach, 1989
Raman and Trussell (1995)
Do AMPA and kainate receptors share common or disparate gating mechanisms?
Although AMPA and kainate receptors have distinct pharmacological profiles (Dingledine et al., 1999
External ions regulate kainate, but not AMPA, receptor gating behavior
We report that external ions distinguish between AMPA and kainate receptor desensitization mechanisms. Recent experiments suggest that both external anions and cations regulate GluR6 responses via a novel allosteric mechanism (Bowie, 2002
The physiological role of intermediate desensitized states
Our modeling suggests that AMPA and particularly kainate receptors containing desensitized subunits contribute to membrane conductance. This result is contrary to previous models of AMPA and kainate receptors in which desensitized states are designated as nonconducting (Raman and Trussell, 1992
Finally, the possibility that intermediate desensitized states contribute to membrane conductance may provide insight into the paradoxical observation that decay kinetics of recombinant kainate receptors in rapid perfusion systems are two orders of magnitude faster (
As proposed from anatomical work, neuronal kainate receptors may be juxtaposed to transmitter release sites but would reside mainly in intermediate desensitized states because of their slow recovery from desensitization. For example, if postsynaptic GluR6 receptors were activated at 1 Hz, synaptic responses would be attenuated by >80% (Fig. 7c; 150 mM NaCl), consistent with the small synaptic events observed from in vitro slice recordings (Cossart et al., 1998
FOOTNOTES Received Oct. 30, 2001; revised Feb. 4, 2002; accepted Feb. 13, 2002.
This work was supported by National Institutes of Health (NIH) Grants RO1 MH62144 (to D.B.) and RO1 NS36654 (to S. F. Traynelis) and by an NIH intramural research program (to G.D.L.). We thank Dr. J. R. Howe for sharing results before publication, Dr. P. Seeburg for permission to use GluR-A and GluR6 cDNAs, Drs. K. Partin and M. L. Mayer for providing cDNAs, Dr. R. Horn for the tsA201 cell line, and Dr. H. J. Motulsky for suggesting the use of AIC method. We are indebted to Dr. S. F. Traynelis for support and Dr. R. Dingledine for the piezoelectric stack. Preliminary experiments identifying GluR6 ion effects were performed in Dr. Mayer's laboratory. We thank Drs. R. W. Aldrich, C. Deutsch, K. Swartz, and G. Yellen for discussions and R. Horn, J. W. Johnson, and D. Weiss for critically reading a previous version of this manuscript.
Correspondence should be addressed to Dr. Derek Bowie, Department of Pharmacology, Emory University School of Medicine, Rollins Research Center, 1510 Clifton Road, Atlanta, GA 30322. E-mail: dbowie{at}pharm.emory.edu.
APPENDIX
Deriving a Mathematical Function for Fitting
For each model, two sets of first order linear differential equations are formulated. The first is concerned with recovery from desensitization. The independent variable (x) is the time between agonist pulses. The state functions (Pi) represent the probability of open channels of a given macroscopic conductance current (Gi). The transitions between states are governed by the recovery rate constants (ri). The second set of rate constants are concerned with entry into desensitization. The independent variable (t) is the time from the beginning of the second agonist pulse. The state functions (pi) again represent the probability of opening of channels. The transitions between states are governed by the entry rate constants (ki). The solutions of the first set of equations provide the initial conditions for the second set of equations. Finally, an equation defining the total conductance is achieved by multiplying each state function by its associated conductance (Gi), substituting in the initial conditions and summing over i. The total conductance (GT) is then a function of both x and t, that is, of both the time between agonist pulses and the time since the beginning of the second pulse.
(a) Concerted Model
Forward differential equations (recovery from desensitization),
Solution,
Reverse differential equations (entry into desensitization),
Solution,
