Jeff Gauthier
Channels I, Week 5 (November 28, 2003)

"Voltage Clamp Analasys of Acetylcholine Produced End-Plate Current Fluctuations at Frog Neuromuscular Junction"
C.R. Anderson and C.F. Stevens
Journal of Physiology 1973, 235, pp 655-691

	In the early 1970s, physiologists knew that endplate transmission from nerve to muscle relied on acetylcholine (ACh), and that acetylcholine acted by increasing ionic conductance at the endplate.  No one knew how acetylcholine determined the time course of conductance or what biophysical properties of the endplate allowed conductance to change.  A popular hypothesis was that ion channels, proteinacious pores, were imbedded in the membrane.
	Anderson and Stevens wanted to corroborate the existence of ion channels by estimating their parameters, such as the conductance of a single ion channel.  The first step, though, was to demonstrate that the muscle membrane was the site of interesting conductance changes, rather than the interstitial space.

First Easy Experiment: Conductance Time Constant Depends on Voltage

	During an evoked end plate current, the end plate conductance increases transiently and then returns to its initial level.  Explaining the rise was straightforward: ACh is released into the cleft and binds to sites on the muscle. The decrease in conductance, on the other hand, could be due to several mechanisms: 1) ACh might unbind and diffuse away, 2) enzymes might alter ACh, or 3) the binding sites might desensitize while ACh is still bound.  Of these three hypotheses, only the third predicts that the rate of conductance decrease would depend on muscle membrane potential.
	Anderson and Stevens measured the time course of conductance while varying the muscle voltage, and found that the rate of conductance decrease did change with voltage (Fig 9. A, B).  These data suggested that ACh stays bound but the end plate sites become less sensitive to it.  Of course, there could be some truth to all three hypotheses, but the large effect of voltage suggested that desensitization was the primary factor.

	The first easy experiment verified that the interesting part of endplate transmission is found on the muscle rather than the narrow space between muscle and nerve.  This left Andersen and Stevens to further elaborate their ion channel hypothesis and test its predictions.  They characterized the model by three parameters which could be estimated from data.  If independent estimates of the parameters were found to agree, it would be strong evidence for their model of ion channels.
	The model was generated by analogy to the recently discovered kinetics of enzymes in solution.  Andersen and Stevens proposed proteinacious ion channels which open and close at fixed rates upon the binding of ACh.

¨

The parameters in this model were the rates of closing and opening, alpha and beta, and the conductance of a single channel, gamma.

Second Easy Experiment: Estmating the Conductance of a Single Channel

	For an evoked end plate potential the amount of released ACh varies, so it's hard to interpret the exact magnitude of the end plate conductance.  Andersen and Stevens got around this difficulty by exposing the end plate to a constant concentration of ACh via an electrophoretic pipette.  Under steady state conditions, the end plate conductance represents a well-controlled quantity.  Even if the concentration of ACh wasn't known precisely, at least the concentration wasn't changing over time.
	If one assumes that the ACh binding sites are discrete and in one-to-one correspondence with discrete units of conductance, then the magnitude of a single conductance can be measured.  Anderson and Stevens assumed ion channels with individual sites for ACh, so these conditions held in their case.  The statistical analysis required to make this measurement only depends on elementary probability theory.
	For independent ion channels, the number of open channels at any given moment will follow a binomial distribution.  If we further assume that only a few channels are open at once, the distribution can be approximated to be poisson.  Let # be the number of open channels.  Then mean (#) = var (#) because # is poisson.  If the conductance of a single channel is g, the total conductance is g = g * #.  A little bit of algebra shows that var (g) = g * mean (g).  The quantities # and g could not be measured directly, but g could.  Andersen and Stevens recorded the total conductance in response to a constant concentration of ACh, and plotted variance as a function of mean (figure 3).  The results clearly showed this was a straight line, consistent with # being a poisson process.  The slope of the best fit line was used to generate their first estimate of g.
	The linearity of the plot was encouraging for another reason.  For random variables drawn from any distribution, the standard deviation is proportional to the mean if and only if the variables are independent.  Since the standard deviation seemed proportional to the mean in this case, the individual conductances were probably independent of one another.  Independence was a (tacit) prediction of the ion channel model, so the model was further strengthened.

	Andersens and Stevens, though, weren't satisfied with these straightforward measurements.  The mean and standard deviation plot was encouraging since it showed a linear relationship, and the voltage dependence of the time constant suggested interaction at the membrane, but those were no no guarantee of ion channels.  Independent measurements would have to be made and closely compared to provide a more robust corroboration of the channel kinetics theory: Andersen and Stevens applied a more sophisticated analysis to the constant concentration ACh data to get another estimate of g and indirectly measure the rate constants of channel closing and opening, alpha and beta.

Hard Analysis: Corroborating the Ion Channel Model with Consilient Estimates

	To power their analysis, Andersen and Stevens looked to the fluctuation-dissipation theorem from statistical mechanics (published sixteen years earlier in 1957 by R. Kubo in the Journal of the Physical Society of Japan).  In a 1972 paper by Magleby and Stevens, the theorem was used to derive estimates of alpha, beta, and gamma from the predicted constant concentration data.  If the ion channel model were accurate, these complicated estimates would agree with the simple estimates discussed above.
	For ACh at a constant concentration, the conductance values over time were a special kind of random process known as a "stationary process".  For a stationary process, a transformation called the spectrum can reveal information not easily identified in the raw data.  In this case, the theory of ion channel kinetics made several predictions about the spectrum.  The "hard analysis" consists of comparing the model's predictions to the observed data (the hard part has been left out).
	These were the model's predictions:

	1) The shape of the spectrum will be Lorentzian (i.e. the shape in figure 4).
	Inspection verifies the first prediction, but the Lorentzian might be predicted from many other theories as well.

	2) The frequency at the cutoff of the Lorentzian will be equal to alpha/(2*pi).  These estimages will therefore depend exponentially on voltage, and will be comparable to values of alpha obtained from evoked potentials (see p. 669, second paragraph for definition of "cutoff frequency").
	The paper's most impressive figure, Figure 9C, elegantly demonstrates this agreement.  The three estimates of alpha, independently obtained from quantal release and constant concentration of ACh, align perfectly.

	3) The plateau amplitude of the Lorentzian will be proportional to gamma (for a known proportionality constant) and this estimate will be comparable to the estimate of gamma which was based on the mean and variance.
	When the proportionality constant was divided out, the resulting estimates of gamma fell very near to the estimates from the "second easy experiment" above (Figure 12).

	Since the estimates generated by the hard analysis closely approximated the estimates from the easy analysis, Andersen and Stevens had given strong evidence for a model of ion channels.  This theory explained what ACh binds to (individual channels) and why evoked end plate potentials decay in exponential fashion (spontaneous channel closure).

"The Selective Inhibition of Delayed Potassium Currents in Nerve by Tetraethylammonium Ion"
Bertil Hille
Journal of Physiology 1967, 50, pp 1287-1302