The Otto Loewi Minerva Center for Cellular and Molecular
Neurobiology and the Department of Neurobiology, the Hebrew University,
Jerusalem 91904, Israel
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INTRODUCTION |
One of the basic parameters characterizing
ligand-gated receptor channels is the number of agonist molecules that
must be bound to the receptor to transform it into an open channel. A number of theoretical-experimental approaches have been proposed for
evaluating this parameter. The most conventional and widespread method
is the Hill plot (see Stryer, 1981
, p. 68). This method derives the
number of binding sites from the maximal slope of a dose-response
curve obtained at low agonist concentrations. Using the Hill plot, the
number of binding sites for nicotinic acetylcholine receptors (N-AChR)
was found to be two (Land et al., 1984; Colquhoun and Sakmann, 1985;
Colquhoun and Ogden, 1988). For the quisqualate-type glutamate
receptor, (qGluR), also called the
-amino-3-hydroxy-5-methyl-4-isoxazole-propionic acid (AMPA) receptor
(Collingridge and Lester, 1989), the Hill coefficient values vary, with
findings of 2 (Johans and Sakmann, 1992; Hestrin, 1992; Tour et al.,
submitted for publication), 3 (Dudel et al., 1990a), and 5 (Dudel et
al., 1990b). These variations may be due in part to the low
signal-to-noise ratio at low agonist concentrations (the range from
which the Hill coefficient is extracted).
Additional approaches for calculating the number of binding sites have
been proposed by different investigators. Bates et al., (1990), used
the analysis of two-dimensional dwell-time histograms to calculate the
number of binding sites for qGluR and found four closed and four open
states for this receptor. However, differences in the calculated
maximum likelihood values in models assuming two, three, or four
binding sites were small. A similar method, based on analysis of
three-dimensional dwell-time histograms for discrimination of different
models, was proposed by Magleby and Weiss (1990). This method, however,
requires a very long simulation time. Clements and Westbrook (1991),
calculated the number of binding sites for
N-methyl-D-aspartate (NMDA) receptors. These researchers also used the maximum likelihood procedure and found that a
model with two binding sites for glutamate and two binding sites for
glycine best fits the initial phase of the NMDA receptor currents.
All the numerical methods described above demand preliminary
information concerning the kinetic scheme and rate constants of the
receptor under study. Without such information, convergence toward a
final model is unlikely. It seems, therefore, that an additional method
is still needed to resolve the fundamental question of how many agonist
molecules must be bound for a channel to open. In the present study, we
propose an experimental-theoretical procedure to evaluate this number.
According to this procedure, analysis of the initial phase of the
current provides the number of binding sites. Below, we show that,
unlike the Hill plot procedure, our approach yields the best results at
high agonist concentrations, which have a much better signal-to-noise
ratio. Using this method for qGluR, we found the number of sites that
must be bound for the receptor channel to open to be two.
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THEORY |
The kinetic model
Several models have been proposed to describe the time course of
ligand-gated receptor channels (e.g., del Castillo and Katz, 1957; Land
et al., 1984; McManus et al., 1988; Parnas et al., 1989; Dudel et al.,
1990b; Franke et al., 1991; Buchman and Parnas, 1992, 1994; Maconochie
et al., 1994; Colquhoun and Hawkes, 1995). Common to all these models
is the presumption that the process of channel opening occurs in
sequential steps. Accordingly, the receptor first binds the agonist
molecules, usually one-by-one in sequence. Then, when the required
number of agonist molecules are bound, the receptor undergoes a
conformational change resulting in channel opening.
The following is a generalized representation of such a sequential
kinetic scheme:
SCHEME 1
Ri denotes a receptor bound to
i agonist molecules (i is an integer ranging from
0 to n, where n is the number of sites that must
be bound for the channel to open). A stands for the agonist, and O for an open channel. ki and
k
i are forward and backward rate constants,
respectively, and ko and
kc are the channel opening and closing rate
constants, respectively. Such schemes have gained appreciable
experimental support and are widely accepted.
Variations on Scheme 1 have also been suggested (e.g., Colquhoun and
Sakmann, 1985; Bates et al., 1990). However, in many of these variant
models, the backbone of Scheme 1 is preserved, with one or more
bifurcating branches added. These added branches result in additional
openings. Usually, the rate constants associated with these branches
are smaller than those associated with the main backbone (Bates et al.,
1990; Colquhoun and Hawkes, 1995). Nevertheless, we will consider the
validity of our proposed procedure also for the case of more than one
open state.
Mathematical model to evaluate the number of binding sites,
n
The theoretical-experimental procedure proposed here evaluates
the number of binding sites from the rising phase of the current. To
highlight the essence of the procedure, we first examine the simplest
case
constant agonist concentration. By constant agonist concentration, we mean a situation in which, at t = 0,
agonist concentration is instantly stepped from zero to its assigned
level and is kept constant at this level. Such conditions can be
reproduced in experiments using outside-out patches with fast
application of the agonist (see Fig. 4, B, D, F).
The situation may be different under physiological conditions. Results
reported by Clements (1996) and Dudel et al. (1999) suggest that,
during the rising phase of the miniature endplate postsynaptic current
(MEPSC), the transmitter concentration rises almost linearly with time
before it eventually declines. Therefore, we will also expand the
proposed theoretical-experimental procedure to include a case in which
the agonist concentration is not constant but rather rises linearly
with time.
We begin with some intuitive considerations. The number of binding
sites in Scheme 1 is n, and the total number of steps
leading to channel opening is n + 1. Consequently, at
t 
0, the open probability Po is
proportional to tn+1, or, in a semi-formal way,
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(1)
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The first derivative of Po,
P'o under the conditions of Eq. 1, is
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(2)
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Dividing Eq. 1 by Eq. 2 gives,
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(3)
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Deriving Eq. 3, with respect to t, at t = 0, we obtain
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(4)
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Equations 2-4 demonstrate the procedure for extracting
n + 1. Specifically, from the derivative of the ratio
(Po/P'o) at
t = 0, we can extract n + 1.
Using the Taylor expansion of the function
(Po/P'o)(t),
we proved (see Appendix A, Constant agonist concentration) that, for
constant agonist concentration, the proportionality in Eq. 4 can be
replaced by equality,
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(5)
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For linearly increasing agonist concentration, Eq. 5 should be
modified (see Appendix A, Linearly increasing agonist concentration) as
follows:
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(6)
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MATERIALS AND METHODS |
Theoretical methods
Implementation of Eqs. 5 and 6
Direct experimental implementation of Eqs. 5 and 6 is impossible
for two reasons. First, solution of these equations demands an exact
definition of the beginning of the current, a point that is difficult
to resolve from experimental data. Second, the value of
(Po/P'o)' at
t = 0, which is necessary to evaluate n,
cannot be measured using only one experimental point at t = 0. These two difficulties can be circumvented if the rationales
underlying Eqs. 5 and 6 are retained while the exact procedure of
solution is varied somewhat.
Concerning the first difficulty, we notice that, if the reciprocal
function,
t(Po/P'o)
(existence and properties of the reciprocal function are shown in
Appendix B), is used instead of (Po/P'o)(t)
(Eqs. 5 and 6) the need to pinpoint t = 0 is replaced by a need to pinpoint Po = 0. This is
easier to achieve. The second problem can be circumvented as follows.
If we expand the range of measurements from exactly t = 0 to a finite range of t in the vicinity of
t = 0, we can approximate the reciprocal function t(Po/P'o) by a
polynomial function. The first derivative of this polynomial function
can be used instead of the derivative of
t(Po/P'o) for
calculation of n + 1 (Eq. 5) or 2n + 1
(Eq. 6).
The polynomial function is obtained by a Taylor expansion of
t(Po/P'o)
(Courant, 1937b),
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(7)
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Here, bj stands for a constant
coefficient of a jth polynomial term (j is an
integer ranging from 1 to 
).
To extract n from the polynomial function (Eq. 7), we repeat
the procedure used for the direct function. We thus derive Eq. 7 at
Po/P'o = 0,
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(8)
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Here, b1 is the first polynomial
coefficient. Also, according to the theorem of the reciprocal function
(Courant, 1937a), the left side of Eq. 8 equals
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(9)
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Combining Eqs. 8 and 9, we obtain
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(10)
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Incorporating Eq. 5, for constant agonist concentration, into
Eq. 10 we obtain
![b<SUB>1</SUB>=<FR><NU>1</NU><DE>[1/(n+1)]</DE></FR>=n+1.](/content/vol78/issue2/fulltext/731/img012.gif)
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(11)
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Incorporating Eq. 6, for linearly increasing agonist
concentration, into Eq. 10 we obtain
![b<SUB>1</SUB>=<FR><NU>1</NU><DE>[1/(2n+1)]</DE></FR>=2n+1.](/content/vol78/issue2/fulltext/731/img013.gif)
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(12)
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Actual evaluation of b1
Fitting the infinite-order polynomial (Eq. 7) to the actual
experimental data is impossible. Therefore, we must truncate Eq. 7 to
finite-order polynomials. For later use, we notice that the first-order
polynomial is given by
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(13)
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the second-order polynomial is given by
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(14)
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and the third-order polynomial is given by
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(15)
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Assessment of b1 from the finite-order
polynomials (Eqs. 13-15) will yield a smaller value than expected,
based on the infinite-order polynomial (Eqs. 11 and 12). Hence, we must
seek the maximal value of Po (denoted as
Pv) that still guarantees a valid value of
b1; i.e., will satisfy the condition of
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(16)
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for constant agonist concentration, and the condition of
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(17)
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for linearly increasing agonist concentration.
Even though we are interested in b1 only,
b2 (Eq. 14 and 15) and b3
(Eq. 15) must satisfy certain conditions for the evaluation of
b1 to be correct. In Appendix C, we show that,
for both constant and linearly increasing agonist concentrations, these
conditions are
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(18)
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The theoretical approach presented above is general for any
number (n) of binding sites and does not depend on the value of the rate constants involved. Neither does it depend on experimental conditions such as temperature or agonist concentration (excluding the
requirement of constant agonist concentration). By contrast, Pv can be strongly affected by all the above
parameters. Therefore, Pv must be evaluated
under various conditions, such as agonist concentration and temperature.
Because we cannot find the value of Pv
analytically, we turn to computer simulations. The appropriate
Pv depends, as mentioned above, on n
itself. This leads to the question of the n to be used in
simulations. In determining the value of n for the
simulation, we take the following factors into consideration. If
Pv is selected to determine n, say,
of two, and the actual n of the receptor under study is
higher than two, we will not be able to extract this higher
n. However, if Pv is selected to
yield a high n, it will still be absolutely suitable for
yielding a lower n. Therefore, at least for the nicotinic
receptor, a good practice is to assign a value of 3 to n in
the simulations to determine the appropriate Pv.
For an unknown receptor, it is also advisable to select
Pv suitable for n = 3 and change
this only if the analysis suggests a higher n.
Three agonist concentrations were considered: low (0.01 mM), medium
(0.1 mM), and high (1 mM). The values of the rate constants were taken
to be those corresponding to the AChR in the frog neuromuscular junction. For the high temperature, we took the parameters' values to
be those provided by Franke et al. (1991) (Table
1) and modified them slightly for a model
containing three binding sites. The effect of the temperature was
modeled by lowering the values of all rate constants.
Because the influence of the experimental noise on the approximated
polynomial increases with the order of the polynomial, we decided to
obtain Pv only for the first-, second-, and
third-order polynomials. We begin with the first-order approximation
(linear approximation) for constant agonist concentration (Eq. 13). The procedure for finding Pv in this case is
exemplified in Fig. 1, A and
B. We simulated the model with n = 3, and
transformed the obtained function
Po(t) (shown on Fig. 1 A)
into the function t(Po/P'o) (Fig.
1 B, empty squares). Next, the linear function was
fitted to function
t(Po/P'o) on a
wide range of Po (dashed line). Then
the range of Po was sequentially decreased until
the slope of the linear fit, b1, reached the
correct value of b1 > 3 (solid
line, see Eq. 16). The maximal value of Pv,
required by the first-order polynomial, was obtained at high agonist
concentration and low temperature. This very small current of
Pv = 0.004 · Pmax cannot be measured accurately. Therefore, we must
increase the order of the polynomial and use Eqs. 14 and 15 instead of
Eq. 12. Eqs. 14 and 15 allow us to increase the range of
Pv.
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FIGURE 1
Results of simulations of a three-binding-sites model
(n = 3) used for extracting Pv
at constant agonist concentration. A standard slow set of parameters
was used for simulations (Table 1). (A) Time course
(solid line) of the open probability at 1 mM of agonist. The
magnified initial phase is shown in the insert. (B) Results
of the first-order approximation of the function
t(Po/P'o) (Eq. 13). Empty squares, values of the function; striped
line, linear fit at t = 0.03 ms
(b1 = 2.62, suggesting two binding sites
instead of three, see Eq. 17); solid line, linear fit at
t = 0.025 ms (b1 = 3.02,
suggesting the correct number, 3, of binding sites). In this case,
Pv = 0.004 · Pmax. (C) Second- (Eq. 14,
striped line) and third- (Eq. 15, solid line)
order polynomial approximation of
t(Po/P'o)
(empty squares). For the second-order polynomial
approximation, Pv was found to be
0.08 · Pmax (b1 = 3.09). For the third-order polynomial approximation,
Pv was found to be 0.25 · Pmax (b1 = 3.2).
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Fig. 1 C shows an example using Eqs. 14 and 15. Second-
(striped line) and third- (solid line) order
polynomials were fitted to the simulation results (open
squares). For the second-order polynomial,
Pv = 0.08 · Pmax. For the third-order polynomial, Pv = 0.25 · Pmax.
The same procedure was performed for the linearly increasing agonist
concentration (Fig. 2, A and
B). With the exception of agonist concentration, which rose
from 0 to 1 mM over 0.3 ms, the parameters for the simulations were the
same as those used for Fig. 1, A, B, and C. For
the second-order polynomial (Fig. 2 B, striped line),
Pv = 0.16 · Pmax. For the third-order polynomial (Fig. 2 B,
solid line), Pv = 0.39 · Pmax. These values are even larger
than those obtained for constant agonist concentration, rendering the
estimation of n easier.
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FIGURE 2
Results of simulations of a three-binding-sites model
(n = 3) used for extracting Pv
with linearly increased agonist concentration. A standard slow set of
parameters was used for simulations (Table 1). (A) The time
course (solid line) of the open probability. Agonist
concentration was increased linearly from 0 to 1 mM for 0.3 ms and then
decreased exponentially. The insert shows the magnified initial phase.
(B) Second- (Eq. 14, striped line) and third-
(Eq. 15, solid line) order polynomial approximation of
t(Po/P'o)
(empty squares). For the second-order polynomial
approximation, Pv was found to be
0.16 · Pmax (b1 = 5.23, n = 3, see Eq. 17). For the third-order polynomial
approximation, Pv = 0.39 · Pmax (b1 = 5.43, n = 3).
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Computer simulations and data analysis
Differential equations were solved using the fourth-order
Runge-Cutta method. Stochastic behavior of the channels was modeled as
a series of single-channel simulations that used the Monte-Carlo method. All simulations were performed on SGI (Indy, Unix Irix 5.3, SGI
Inc., Mountain View, CA) workstations using BIOQ, a program developed
in our laboratory (Ashkenazi and Parnas, 1997). Data analyses were
performed on a PC using Microsoft Excel.
Experimental methods
Preparations and experimental conditions
Frog. Miniature endplate postsynaptic currents
(MEPSCs) were recorded from the cutaneous pectoris neuromuscular
junction (NMJ) of a frog (Rana ridibunda). The preparation
was isolated using the technique elaborated by Bloch et al. (1968) and
was bathed in a standard solution (Dudel, 1989). Temperature was kept
at 8 ± 1°C.
Crayfish. MEPSCs were recorded from the opener
neuromuscular system of a crayfish, Procambus clarkii (Dudel
and Kuffler, 1961), and single-channel currents were recorded from the
crayfish deep abdominal extensor muscles (Parnas and Atwood, 1966). The
preparations were bathed in standard van Harreveld solutions (Franke et
al., 1987), at a temperature of 12 ± 1°C. Patch electrodes for
outside-out recordings were filled with intracellular low
Cl solution (Franke et al., 1987).
MEPSC recordings and averaging
For both the frog and the crayfish, the standard macropatch
technique (Dudel, 1983) was used to stimulate the preparation and
record the currents. Each preparation was stimulated 5-10 thousand
times at 2-3 Hz. Quantal content was in the range of 0.2. MEPSCs from
each experiment were recorded separately at a sampling rate of 100 kHz,
using the LabView interface (AT-MIO-16F-5, NI-DAQ 4.9.0 driver
software, National Instrument Corporation, Austin, TX). A preliminary
selection of traces was made by excluding all traces with zero or
multiple MEPSCs. The remaining MEPSCs were then aligned (using our
own programs developed for this purpose) as follows: To measure the
baseline (Fig. 3 A, striped
line) and the peak value (marked as 1), we subjected the raw data
(solid line) to a 10-kHz filter (dashed line).
The rise time (tr, 10-90% of peak current) was
then measured and averaged. All traces in which
tr differed from the average
tr by more than 10% were excluded. Because it
is impossible to pinpoint the exact beginning of the current, we
defined an effective beginning as 10% of the peak value (marked as 2).
All traces were aligned to this point. Traces in which the noise level
in the vicinity of this point exceeded 10% of the peak current were
excluded from further processing.
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FIGURE 3
MEPSCs: recordings and averaging. (A) Single
MEPSC recorded from frog neuromuscular junction (N-AChR receptor).
Solid line, one example of raw data; dashed line,
the same MEPSC filtered at 10 kHz. Baseline (striped line)
was calculated by averaging 50 points before the rising phase of the
current. The peak value of the current (1) is calculated from the
filtered current. The point of alignment (2) is chosen as the point at
which the value of the nonfiltered current is equal to 0.1 of the peak
value. Here, and in all the following figures showing experimental
results, time zero (abscissa) was chosen to be this alignment point.
(B) Rising phase (points) of an average current of 98 synchronized MEPSCs recorded from frog neuromuscular junction. The
baseline and the synchronization point (2) were determined as in
A. The current in the range between points 3 and 4 is used
for analyzing the number of agonist binding sites. (C) The
same as B, recorded from crayfish neuromuscular
junction (qGluR receptor).
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Following the steps described, 50-200 traces remained suitable
for averaging. These aligned nonfiltered traces were averaged (Fig.
3, B and C).
Single channel recording and averaging
The fast application system used for applying pulses of
glutamate (L-glutamic acid sodium salt, BDH, Poole, UK) to
the outside-out patches has been described in detail (Franke et al.,
1987; Dudel et al., 1990b). Briefly, a well-defined thin liquid
filament was produced by using pressure to eject a solution containing
the agonist from a polyethylene tube glued to a steel needle. The tube
was moved rapidly by a piezo crystal so that the liquid filament washed
the tip of a patch pipette. The speed of solution exchange at the
pipette tip depends primarily on the velocity of the ejected solution
(Maconochie and Knight, 1989). Speed of solution exchange at the tip of
an open patch pipette was tested as follows (Maconochie et al., 1989).
The solution was diluted in the liquid filament by 20% to produce a
junction potential so that currents (osmotic currents) could be
recorded when switching from the control solution to a diluted one. In
some experiments, fast application of the agonist was needed, although,
in others, slow application was required. The ejected solution
velocities used for such applications were 300 and 100 mm/msec,
respectively. The shapes of the osmotic current profiles for both cases
were recorded with an open pipette. The 10-90% rise time for the slow
application was 310 µs (Fig. 4 A). For the fast
application, it was 160 µs (Fig. 4 B).
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FIGURE 4
Outside-out patch recordings from glutamate channels
activated by slow (A, C, E) and fast (B, D, F)
application of glutamate. (A) The time course of the
solution exchange at an open patch pipette was tested using osmotic
currents. The velocity of the ejected solution was ~100 µm/msec
(slow application). The exchange of the solution, expressed as the rise
time of the current (10-90%) took 310 µsec. (B) Same as
A but for a case where the velocity of the ejected solution
was ~300 µm/msec (fast application). Here, the exchange in
solutions (10-90%) took 160 µsec. (C) One example
(thin solid line) of current elicited by slow application of
10 mM glutamate to an outside-out patch with a holding potential of
 70 mV. The outcome of averaging 1000 such currents is the heavy solid
line. The base line (striped line) was established by
averaging 1000 applications of glutamate-free solution on the same
patch. (D) Same as C but for the fast application
of glutamate. (E) and (F) are the rising phases
of the averaged currents depicted in C and D,
respectively, shown in more detail.
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An example of one recorded trace of channel activity in response to a
slow application of 10 mM glutamate is shown in Fig. 4 C
(thin solid line). The heavy solid line is the outcome
of averaging 1000 such traces. Electrical artifacts and leak currents were eliminated from the glutamate-activated ensemble currents by
subtracting average currents recorded from the same patch where glutamate was absent from the applied solution (Fig. 4 C,
striped line). The magnified initial phase of the resulting
current is shown in Fig. 4 E. Each patch was then analyzed
separately. Currents activated by fast glutamate application were
analyzed and averaged in the same manner (Fig. 4, D and
F).
Averaged data is given as the mean ± SD.
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RESULTS |
The total current through the channels is proportional to the open
probability. We can therefore use the function t(I/I') instead of
t(Po/P'o) in Eqs.
14 and 15.
Evaluation of n for the nicotinic receptor from
MEPSCs
Because an n of two is well established for N-AChR
(Rang, 1974; Prinz and Maelicke, 1983; Unwin, 1993; Hucho et al., 1996; see also references in Introduction), we began our study with this
receptor to test the validity of our theory. The MEPSCs were recorded,
aligned, and averaged as described in Materials and Methods (Fig. 3,
A and B). Then, the rising phases of the aligned and averaged currents were analyzed as described in Eqs. 14 and 15.
Analysis for one experiment is shown in Fig.
5 A. A second-order polynomial (Eq. 14, dashed line) was successfully fitted to
the experimental data points (empty squares) of
t(I/I') at the range of 0.025 · Ipeak-0.22 · Ipeak. The
first derivative of this polynomial, b1 (Eqs. 12
and 16 are used, as the ACh concentration in the synaptic cleft
increases linearly with time), at (I/I') = 0, is 4.41, implying n = 2 (Eq. 17). The third-order
polynomial (solid line) was fitted to the range of
0.025 · Ipeak-0.37 · Ipeak, with b1 = 3.92, also implying n = 2. This experiment
was repeated 14 times. Second-order polynomials were successfully
fitted for 12 of 14 experiments. In 2 of 12 cases,
b1 was smaller than 3 (suggesting one binding site); in 10 of 12 cases, b1 was greater than 3 and smaller than 5 (suggesting two binding sites). The second-order
polynomials fitted to the remaining two patches did not meet the
requirements set in Eq. 18. The average value of
b1 for the second-order polynomials was
3.75 ± 0.62. Third-order polynomials were successfully fitted to
11 of 14 cases. In 2 of 11 cases, b1 was smaller
than 3 (suggesting one binding site); in 9 of 11 cases
b1 was greater than 3 and smaller than 5 (suggesting two binding sites). For the remaining 3 patches, a
third-order polynomial did not meet the requirements set in Eq. 18. The
average value of b1 for the third-order
polynomials was 3.74 ± 0.62. These results are summarized in
Table 2.
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FIGURE 5
Examples of the polynomial fits of the function
t(I/I') calculated for the different types of experiments.
Empty squares, calculated values of t(I/I');
striped line, second-order polynomial approximation (Eq. 14); and solid line, third-order polynomial approximation
(Eq. 15). The value of n is calculated using Eq. 17 for
(A), (B), and (C), and using Eq. 16 for
(D). (A) The analysis of averaged MEPSC recorded from N-AChR
(Fig. 3 B). The fitting range for the second-order
polynomial is from 0.025 · Ipeak to
0.22 · Ipeak. The obtained value of
b1 = 4.41; therefore n = 2.
The fitting range for the third-order polynomial is from
0.025 · Ipeak to 0.37 · Ipeak. The obtained value of b1 = 3.92, therefore n = 2. (B) Analysis of averaged
MEPSC (qGluR, Fig. 3 C). The fitting range for the
second-order polynomial is from 0.02 · Ipeak to 0.18 · Ipeak.
The obtained value of b1 = 3.56; therefore
n = 2. The fitting range for the third-order polynomial
is from 0.02 · Ipeak to 0.34
· Ipeak. The obtained value of
b1 = 3.56; therefore n = 2. (C) Analysis of the averaged currents recorded using the patch
clamp technique with slow agonist application (Fig. 4, A, C
and E) recorded for qGluR. The fitting range for the
second-order polynomial is from 0.02 · Ipeak to 0.2 · Ipeak.
The obtained value of b1 = 3.45; therefore
n = 2. The fitting range for the third-order polynomial
is from 0.02 · Ipeak to 0.38
· Ipeak. The obtained value of
b1 = 3.18; therefore n = 2. (D) Analysis of the averaged currents recorded using the Patch
clamp technique with fast application of the agonist. A second-order
polynomial was fitted to the range from 0.02 · Ipeak to 0.2 · Ipeak.
The obtained value of b1 = 2.94; therefore,
n = 2. A third-order polynomial was not fitted in this
case, because it does not meet the condition of Eq. 18.
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Evaluation of n for the quisqualate receptor
from MEPSCs
The procedure described above was repeated for the glutamate
receptor, qGluR. Analysis for one experiment is shown in Fig. 5 B. The function t(I/I') was approximated by
second- (dashed line) and third- (solid line)
order polynomials. The second-order polynomial was fitted to the range
from 0.02 · Ipeak to 0.18 · Ipeak, and the third-order polynomial was fitted to
the range from 0.02 · Ipeak to
0.34 · Ipeak. The values of
b1 were 3.56 in both cases, suggesting
n = 2 (Eq. 17). The experiment was repeated 21 times.
The average values of b1 were 3.75 ± 0.43 for the second-order polynomial and 3.84 ± 0.51 for third-order
polynomial. The results are summarized in Table 2.
Evaluation of n for the quisqualate receptor from
outside-out recordings (slow application of glutamate)
A solution of 10 mM glutamate was slowly applied (see Materials
and Methods) 1000 times to a patch, containing qGluRs. The recorded
currents were averaged as described in Materials and Methods. In this
case as well, n was evaluated from Eqs. 12 and 17, because
the slow application of agonist causes the agonist concentration to
rise almost linearly with time (see Fig. 4 A). The results
of one experiment are shown in Fig. 5 C. The data points
(empty squares) of t(I/I') were approximated by
second- (dashed line) and third- (solid line)
order polynomials. The fit range for the second-order polynomial was
from 0.02 · Ipeak to 0.23 · Ipeak. For the third-order polynomial, it was from
0.02 · Ipeak to 0.38 · Ipeak, with a b1 of 3.45 and
3.18, respectively, both suggesting n = 2 (Eq. 17).
Eleven such experiments were conducted, and the results are summarized
in Table 2.
Evaluation of n for the quisqualate receptor from
outside-out recordings (fast application of glutamate)
In this type of experiment, we tried to create the conditions that
would satisfy Eqs. 11 and 16. To do so, a fast application procedure
was used (see Materials and Methods and Fig. 4 B). The results of one experiment are shown in Fig. 5 D. A
second-order polynomial was fitted to the function t(I/I')
at a range of 0.02 · Ipeak to
0.2 · Ipeak. We selected a wider
range than required by Eq. 14 to decrease the influence of the noise
near the point (I/I') = 0. As will be discussed, the
widening of the fitting range is partially compensated by the
noninstantaneous application of glutamate. The value of
b1 is 2.94, reconfirming two binding sites
(n = 2) for qGluR (Eq. 16). We did not repeat the
analysis with a third-order polynomial (Eq. 15), because, in this
case, the requirements set by Eq. 18 could not be met. This experiment was repeated 16 times. In all 16 cases, a second-order polynomial was
successfully fitted, although a third-order polynomial could fit only 2 out of 16 experiments. The results are summarized in Table 2.
The robustness of the proposed method
To check for the robustness of the proposed
theoretical-experimental method, we turn to an examination of factors
that may affect the precision of the evaluation of n.
Stochastic noise
The real experimental data includes stochastic noise that results
from the stochastic behavior of the channels. To check the effect of
the stochastic noise on our results, we replaced the deterministic
model used previously (Figs. 1 and 2) with a Monte-Carlo model. Figure
6, A and B shows
the results of one simulation with n = 3 and linearly
increasing agonist concentration. The resulting current (Fig.
6 A, filled diamonds) is an average of 10,000 single-channel currents (representing the average of 100 MEPSCs when
each one consists of 100 single-channel currents). Transformation of
this current in Fig. 6 A (in range marked by the dashed
lines) to the function
t(Po/P'o) is shown
in Fig. 6 B (empty squares). As shown, a third-order
polynomial (Eq. 15, solid line) was successfully fitted to
these data points. The value of b1 = 5.86
was obtained. Eleven such simulations were performed. A third-order
polynomial was successfully fitted to nine of them (the other two
simulations did not meet the requirements set in Eq. 18) with the
following results: b1 values were 5 < b1 < 7 (suggesting three binding sites) in
five simulations, 7 < b1 < 9
(suggesting four binding sites) in one simulation, and 3 < b1 < 5 (suggesting two binding sites) in three
simulations. The average value of b1 = 5.39 ± 1.31, suggests the correct number, of n = 3.
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FIGURE 6
Example of the polynomial fit of the simulation results
obtained using Monte-Carlo method with linearly increasing agonist
concentration. Standard slow set of parameters was used for simulations
(Table 1). (A) The initial phase of the simulated current.
Average current from 10,000 single channels is shown as filled squares,
and the current that obtained for the same model using the numerical
solution of the differential equation is shown as a solid line. The
region between the dashed lines is then transformed to the function
t(Po/P'o).
(B) Third-order polynomial fit of
t(Po/P'o).
Empty squares, the values of the function; solid
line, polynomial fit. The obtained value of
b1, 5.86, suggests the correct number
n = 3 (Eq. 17).
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More than one open state
Colquhoun and Sakmann (1985) reported that N-AChR exhibits two
open states: one from a single-liganded receptor and the main open
state from a double-liganded receptor. For this receptor, we found that
n = 2 (Fig. 5 A). How strong, then, must
the opening from the singly bound receptor be for the observed
n to be one and not two? To answer this question, we
conducted a simulation with the model shown in Scheme 2. This
simulation captured the findings of Colquhoun and Sakmann.
Colquhoun and Sakmann (1985) fixed the values of the various rate
constants to fit a membrane potential of 130 mV, whereas the rate
constants that we used (Table 1) were established for resting
potential. For Scheme 2, therefore, we also used the rate constants of
Table 1 and supplemented them with the values of ko1, kc1,
k3, and k3. The values
of ko1 and kc1 were
chosen to conserve the relationships with ko2
and kc2, respectively, as given by Colquhoun and
Sakmann (1985). We simulated the model shown in Scheme 2 with a
linearly increasing agonist concentration and transformed the obtained
Po into the reciprocal function
t(Po/P'o) (Fig.
7 A, empty squares).
The third-order polynomial (solid line) was successfully
fitted. The obtained value of b1 = 4.28
suggests two binding sites (Eqs. 12 and 17). This result can be
explained by the fact that ko1 is four orders of
magnitude smaller than ko2, and, therefore, the
channel's probability of being in state O1 is
much smaller than its probability of being in state
O2. To determine conditions for which the
presented method will yield n = 1, we increased the
ratio ko1/ko2 to 2 and
analyzed the results of the simulations. Figure 7 B shows
the simulation results for this case. As shown, only when the ratio
ko1/ko2 
1 (four
orders of magnitude higher than in the real case described by Colquhoun and Sakmann, 1985) does the obtained value of b1
suggest n = 1. Below this ratio, in spite of an opening
from a single-bound receptor, the evaluated n will be two.
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FIGURE 7
Analysis of opening from the partially liganded state.
The open probability, Po, in this case, is equal
to Po1 + Po2, where
Po1 and Po2 are the
channel's probabilities of being in states O1
and O2, respectively. (A) Calculation
of n for the model shown on Scheme 2. The value of
k3 was assigned to be equal to
k2, and the value of k3
was chosen to conserve energy. ko1 = 0.0001 · ko2,
kc1 = 10 · kc2 (Colquhoun and Sakmann, 1985; see Table 1
for the values of the other rate constants). The obtained value of
b1 = 4.28 suggests two binding sites (Eq. 17). (B) The obtained value of b1
(empty circles) as a function of the ratio
ko1/ko2. When this ratio
is less than 1, the obtained value of b1 is
greater, than 3 (dashed line), and, therefore, suggests that
n = 2. When this ratio is greater than or equal to
1, the obtained value is less than 3, and suggests n = 1 (Eq. 17).
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DISCUSSION |
In the present study, we propose a theoretical-experimental
approach for calculating the number of agonist molecules, n,
that must be bound for a receptor channel to open. This method is based on analysis of the initial phase of the MEPSC or current produced by
agonist application to outside-out patches. Unlike the methods used in
previous studies (see references in Introduction), our method is based
on analysis of an analytical function
t(Po/P'o) rather
than on numerical reconstruction of the current or dwell-time histograms. The advantage of this method is that it requires neither an
initial model nor knowledge of the rate constant values.
Let us examine factors that may affect the accuracy of this method. The
factors to be considered relate to experimental conditions and the
recording system. To maximize precision in measuring the initial phase
of the current, the signal-to-noise ratio must be increased as much as
possible. To achieve this, a high concentration of agonist must be
used. Indeed, in our experiments, a saturate concentration of
glutamate, 10 mM (Tour et al., 1995) was used. An additional way to
increase the signal-to-noise ratio is to record and average a
sufficiently large number of currents. Conducting the experiments at
low temperature also reduces the total noise by reducing the white
noise. Low temperature also decreases the reaction rates, thereby
causing a slower rise time. This permits sampling more data points
during the initial rising phase of the current. For all the above
reasons, the experiments presented here were conducted at low temperature.
Another important factor concerns filtration of the data. High
filtration causes smoothing of the current, and consequently, reduces
P'o, increasing the value of
Po/P'o. Ideally,
the currents should be recorded with no filtration at all, but
practically, this is impossible, given that the recording system has
built-in filtration (approximately 100 kHz; Ogden, 1994). For the same reason, the frequency of the analog-to-digital conversion of the recorded currents must be as high as possible. We recorded the data of
our experiments without additional filtration and sampled the data at
0.01 ms per point.
For the case of constant agonist concentration, the time required to
increase agonist concentration from zero to its assigned value must be
as short as possible. We managed to reduce this time to 0.16 ms.
Precise definition of the longest permissible time is beyond the scope
of the present study. However, preliminary analysis using computer
simulations shows this value to be less than 0.11 ms (analysis not
shown). It follows that the time interval during which the data may be
collected for evaluation of the initial rise must be expanded. However,
enlarging the interval of the approximation above the permissible
value, Pv, causes the obtained value of
n to decrease (see Theoretical Methods). As a result of the
various sources of inaccuracy, n cannot be determined based on constant agonist concentration experiments alone. Conclusions regarding the true value of n, 2, were also confirmed by
results of experiments in which agonist concentration rises linearly
(Fig. 5, B and C). This result is in accord with
the value of n estimated for the same preparation from
measurements of the Hill coefficient (Tour et al., 1999). Our value of
2 differs, however, from results obtained by Dudel et al., 1990a,b for
the same preparation.
The obtained value also agrees with estimates obtained in other
systems: for AMPA subtype glutamate receptors in pyramidal cells of the
rat hippocampus (Johans and Sakmann, 1992), for glutamate-activated channels in the visual cortex (Hestrin, 1992), and for AMPA receptors from cultured hippocampal neurons (Clements et al., 1998). The first
two estimates were obtained using measurements of the Hill coefficient.
Clements et al. (1998) estimated the value of n from the
best fit of these models to the initial phase of the current at low
concentration of agonist. Clements et al. confirmed their estimations
by analysis of the antagonist displacement experiments.
Rosenmund et al. (1998), also using antagonist displacement
experiments, suggested that a conformational change (achieved by
binding agonist) in two subunits of the tetrameric receptor is
sufficient to open the channel to a certain extent. Conformational changes in additional subunits open the pore further. We believe that
our theory accounts for this case also, and the actual value of the
observed n will yield the minimal number of agonist
molecules needed to open the channel (see Fig. 7).
The Hill plot is by far the most common method used to evaluate
n. The method used here offers several advantages over the Hill plot. First and most important, it does not require lengthy and
complex experiments such as establishing a dose-response curve. Rather, it uses MEPSCs, which are obtained as a result of conducting routine experiments associated, for example, with release of
neurotransmitter. Second, it does not require measurements at low
agonist concentrations, measurements that are experimentally difficult
to perform and result in noisy data. Third, the "run down"
phenomenon, (Hestrin, 1992; Johans and Sakmann, 1992; Tour et al.,
1995), which greatly affects the Hill plot, has no effect whatsoever on
the precision of our method. Finally, because, in the proposed method
we analyze only the initial phase of the current, desensitization,
which affects the Hill plot, does not affect the analysis presented here.
The main disadvantage of our method is that its precision depends on
the time resolution of the recording system; hence very fast processes,
caused, for example, by positive cooperativity, can be overlooked.
However, the precision of this method can be increased in the future.
Several investigators have recently reported application systems with a
stabilization time shorter than 0.1 ms (Maconochie et al., 1994;
Heckmann et al., 1996).
We developed our method assuming a single opening from the fully
liganded receptora widely accepted property of ligand-gated receptors. Existence of an additional opening from the partially liganded receptor, suggested, for example, by Colquhoun and Sakmann (1985) for the nicotinic receptor, can also affect the accuracy of the
proposed method. We show that, for the nicotinic receptor, to see the
minimal number of sites that must be bound for the receptor to open,
the opening rate from the partially liganded state should be the same
as the opening rate from the fully liganded state, (that is, the number
of bindings that causes the opening from the partially liganded state).
This disadvantage is shared by the Hill plot method.
To conclude, this new method can be a powerful instrument for
determining the number of binding sites of ligand-gated receptor channels. In addition, the proposed method can be used to calculate the
number of steps preceding the opening of voltage-dependent channels.
Because the membrane potential can be changed experimentally in several
microseconds, the equations for the case of constant agonist
concentration are suitable. It appears that the analytical method
presented can be a promising addition to the existing methods in this
field (e.g., Hodgkin and Huxley, 1952; Mika and Palti, 1994).
The following system of differential equations describes the
kinetic model of Scheme 1.
To adapt the model to a linearly increasing agonist
concentration, we supplement Eqs. A1 and A2 with
TOP
ABSTRACT
INTRODUCTION
i stands for its decay
constant. Notice that ![<FR><NU>P<SUB><UP>o</UP></SUB></NU><DE>P′<SUB><UP>o</UP></SUB></DE></FR>(t)=<FR><NU>P<SUB><UP>o</UP></SUB>(0)+<LIM><OP>∑</OP><LL><UP>j=1</UP></LL><UL><UP>∞</UP></UL></LIM> [P<SUP>(<UP>j</UP>)</SUP><SUB><UP>o</UP></SUB>(0)/j!] · t<SUP><UP>j</UP></SUP></NU><DE>P′<SUB><UP>o</UP></SUB>(0)+<LIM><OP>∑</OP><LL><UP>j=1</UP></LL><UL><UP>∞</UP></UL></LIM> [P<SUP>(<UP>j+1</UP>)</SUP><SUB><UP>o</UP></SUB>(0)/j!] · t<SUP><UP>j</UP></SUP></DE></FR>.](/content/vol78/issue2/fulltext/731/img025.gif)
