Hons Neuroscience Professor R.R. Ribchester QUANTAL ANALYSIS AT THE NEUROMUSCULAR JUNCTION Our present understanding of the fundamental physiological mechanism of transmitter release at synapses is mainly due to the work of B.Katz and his colleagues, based on their studies of transmission at the frog neuromuscular junction made during the 1950 s-1970 s at University College London. (Katz was awarded a Nobel Prize for his work in 1970). They showed that neurotransmitter is released in multi-molecular packets ( quanta ). Electron microscopy established that these corresponded with synaptic vesicles in the motor nerve terminals. The evidence for quantized release of transmitter was based on Katz s statistical analysis of electrophysiological recordings made from neuromuscular junctions during stimulation and at rest. Refinements of this quantal analysis are still widely used in cellular electrophysiology to establish the amount of transmitter released at synapses during activity, in a variety of tissues: including the neuromuscular junction, autonomic ganglia, the hippocampus, and the cerebral cortex. At the resting neuromuscular junction, miniature end-plate potentials (MEPPs) are generated spontaneously at the endplate. An evoked end-plate potential (EPP) is produced in the muscle fibre following nerve stimulation. Normally this response triggers a muscle fibre action potential and contraction. When transmission is weakened either by blocking receptors with a nicotinic antagonist (e.g. tubocurarine) or by suppressing transmitter release with solutions containing reduced Ca 2+ ions, the EPP becomes too small to trigger an active response: the EPP is said to be subthreshold. The essence of Katz s quantal hypothesis of synaptic transmission was threefold: 1. The quantum of transmitter underlying the smallest nerve-evoked EPP and the spontaneous MEPP are one and the same; 2. The release of each quantum of neurotransmitter is independent of the release of other quanta and occurs with a very low statistical probability (i.e. random); 3. The evoked EPP is caused by the synchronous release of several quanta, due to a transient and large increase in the probability of release of individual quanta. Evidence supporting the hypothesis was obtained by recording intracellular EPPs and MEPPs and ascertaining the relationship between their amplitudes. In particular it was noted that EPPs are variable in amplitude from stimulus to stimulus, whereas the MEPPs are roughly constant in amplitude. The variability could be accounted for on the basis of point (2) above, by showing that the distribution of EPP amplitudes conformed to a binomial distribution, which simplified under conditions of low release probability to a poisson distribution. Binomial model of synaptic transmission Consider a nerve terminal containing a number (n) of quanta /synaptic vesicles. Suppose each has a small chance (p) of fusing with the plasma membrane and releasing transmitter across the synapse. If the synapse is stimulated repetitively, say 100 times, then the mean number (m) of quanta released will be : m = n.p (1) By analogy, imagine tossing a coin 100 times. The probability of each toss coming up heads is 0.5. The average number of times the coin will come up heads is therefore 100 times 0.5: i.e., 50. 1
In matters of transmitter release, however, the probability of a vesicle fusing with the presynaptic membrane is normally considerably less than 0.5, but for the sake of argument let s suppose that n=3 and p=0.1. On average, a stimulus will evoke 0.3 quanta. In other words, some of the time there will be no release (a failure ). On the other occasions, release will consist of 1, 2 or 3 quanta. Thus the quantal content of the EPPs will vary between zero and three. How are the quantal amplitudes distributed? How often would one or two quanta be released in response to a stimulus? And how often would no quantal release occur? We make the assumption that all the quanta released following a stimulus are recycled, so that the number available on each occasion remains constant. Under this condition, the overall probability that all 3 quanta will be released (P) is simply the product of their individual release probabilities: P(3) = p.p.p = p 3 (= 0.001) Similarly, the probability that no quanta are released can be stated formally. By the rules of probability, either something or nothing must happen and certainty has the value of 1.0. So the probability for each vesicle not being released (q) is 1-p. Therefore the overall probability of a stimulus failing to release any of the three quanta in our imaginary synaptic terminal is: P(0) = (1-p).(1-p).(1- p) = q.q.q = q 3 (= 0.729) How about the overall probability of release of one quantum? By similar reasoning, for each quantum in the store this is p.q.q. The rules of probability require us to apply this condition to each of the three quanta in the store, thus: P(1) = p.q.q + p.q.q + p.q.q = 3 p.q 2 (= 0.243) Likewise, P(2) = p.p.q + p.p.q + p.p.q = 3 p 2.q (=0.027) 2
The overall distribution of transmitter release is obtained by adding together all four probability terms, and these must all add up to one (i.e. there are no other possibilities): P(0) + P(1) + P(2) + P(3) = p 3 + 3p 2 q + 3 pq 2 + q 3 = 1 This simplifies to : (p+q) 3 = 1 The above is a binomial expression : it contains two terms, p and q. Mathematical theory shows that in general we can predict from such an expression that for any number of quanta n with release probability p, that a particular nerve stimulus will release x quanta (x n) from the formula: P(x) = n!. p x. q (n-x) The Binomial Distribution (2) (n-x)!x! Try this out on the example we have used above with n=3 and p=0.1 ( m=0.3): P(0) = P(1) = P(2) = P(3) = The Poisson model The problem with applying binomial analysis to real synapses is that there is rarely any independent way of estimating n or p. We can only estimate the mean quantal content, m, by dividing the mean EPP amplitude by the mean MEPP amplitude. It turns out that we can still nonetheless predict the distribution of amplitudes if we assume that n is very large (n>>p) and p is very small (p<<1). Under these conditions x<<n. Based on these assumptions we can make a number of simplifications to the binomial distribution (equation 2, above). For example, we may write: n! n x (e.g. try this with n=10,000 and x=3) (n-x)! and q (n-x) q n recalling that q = (1-p), we can now substitute these terms in the binomial distribution (equation 2): P(x) = n x. p x. (1-p) n x! 3
since m = n.p (equation 1) this immediately simplifies to : P(x) = m x. (1-p) n (3) x! A little mathematical tinkering further simplifies the expression (1-p) n. First we apply natural logarithms in the identity: Ln (1-p) n = n. Ln (1-p) A mathematical formulation called McLaurin s theorem can be used to express Ln (1-p): Ln (1-p) = -p - p 2 - p 3 - p 4 -... - p 2! 3! 4! But since p<<1 by our assumption in the present analysis, then all the terms after the first one in the McLaurin series must be very small and we can ignore them. Thus: Ln(1-p) -p and therefore n.ln (1-p) -n.p Taking the antilogarithm of both sides: (1-p) n = exp (-n.p) = exp (-m) [Note: Theory of logarithms - if y=ln(x), then x=exp(y)] Substituting back in equation (3) we obtain: P(x) = m x. exp(-m) The Poisson Distribution (4) x! Once again, calculate what the distribution of probability of occurrence of EPPs containing 0,1,2,3 quanta are when the mean quantal content is 0.3 P(0) = P(1) = P(2) = P(3) = 4
Application of the Poisson Distribution to estimating quantal contents Experimentally, what does P(x) mean? It is simply the fraction of occasions on which the evoked postsynaptic potential (EPP, or in the case of CNS synapses, the EPSP) has a quantal content of x. To evaluate the quantal hypothesis and to use the Poisson distribution, we must compare the predicted variability in the amplitudes of EPPs with the actual variability observed experimentally. One of the most elegant demonstrations of the coincidence of the model and the data was obtained in a study by Boyd & Martin in a study of synaptic transmission in cat muscle: The data in the figure below were obtained from intracellular microelectrode recordings at a single neuromuscular junction in an isolated preparation in which neuromuscular transmission was depressed using a low Ca ion-high Mg ion bathing medium. The upper right of the figure shows the histogram of MEPP amplitudes as a bar chart and the superimposed graph is a fit of a normal (gaussian) distribution to the amplitudes. The lower graph shows the distribution of EPP amplitudes as a bar chart (including failures ) and a fit of the Poisson distribution, taking account of the gaussian variation in MEPP amplitudes. Note that the number of failures is accurately predicted, as well as the distribution of the peaks. The mean and variance of each peak is a unit multiple of the first, which has the same mean and variance as the MEPP amplitude distribution. Data such as these provide confirmation of the quantal hypothesis. 5
ESTIMATING QUANTAL CONTENT There are three principal methods. Other methods are based on complex analysis of EPP amplitudes. 1. Direct method: Under favourable conditions, both MEPPs and EPPs can be recorded in sufficient numbers to allow cross checking of the quantal content of EPPs, comparing Poisson statistics with direct estimation of the mean quantal content.m = (mean EPP amplitude) / (mean MEPP amplitude) This methos is not always possible for various technical reasons, e.g the MEPPs might be very infrequent; or they may be too small, buried in the noise of the recording system; or the mean quantal content may be large, resulting in non-linear summation of EPPs (see below). Applying the Poisson equation alone is sometimes sufficient in such cases. There are two methods of estimating quantal content based on the Poisson distribution: the Method of Failures and the Variance Method. 2. Method of Failures If the mean quantal content is low enough (as in the examples above), a significant fraction of stimuli will fail to evoke a response. This represents the P(0) expression in the Poisson distribution: P(0) = exp (-m). m 0 /0! since, by mathematical definintion, both m 0 and 0! are equal to 1 : P(0) = exp (-m) Taking natural logartithms : Ln (P0) = -m Substituting for P(0)=(Number of Failures)/(Number of Stimuli) and rearranging: m = Ln (Stimuli/Failures) 3. Variance method Another property of the Poisson distribution is that its variance equals its mean. From this it can be derived that: m = (mean EPP amplitude) 2 (variance of EPP amplitudes) This is often expressed in terms of the coefficient of variation (standard deviation/mean = σ/µ) : m = C.V. -2 6
Non-linear summation of synaptic potentials As the amount of transmitter released onto the postsynaptic membrane increases - e.g. as the quantal content increases - the effectiveness of each quantal packet declines. This is because transmitter molecules at excitatory synapses like the neuromuscular junction act on receptors coupled to ion channels. When the channels open, positive ionic current flows into the postsynaptic cell. The electromotive driving force is determined by the ion gradient and the membrane potential. For example, at neuromuscular junctions the receptor/ion channel is the nicotinic ACh receptor which gates permeability to Na and K ions about equally. The reversal potential is about -5 mv. This means that as the membrane potential approaches -5 mv, the ionic current flowing through the open channels becomes vanishingly small; and if membrane potential becomes more negative than -5 mv, an outward ionic current is produced instead and the EPP reverses in sign. Each quantal component of the EPP depolarises the membrane potential towards the reversal potential by a small amount, but as the amount of overall depolarisation becomes greater each successive quantal component has a weaker and weaker effect on further depolarisation. For instance, if the mean MEPP amplitude is 1 mv, then an EPP comprising 5 quanta may well produce a depolarisation of 5 mv. But an EPP comprising 20 quanta may only produce about 15 mv of depolarisation. This is called non-linear summation of synaptic potentials. It means that under normal conditions of synaptic transmission when quantal contents can be quite high, the direct method of quantal analysis will underestimate mean quantal content and the variance method will overestimate mean quantal content. (Note: The failures method cannot usually be applied when mean quantal content is greater than about 5 because there are so few failures; P(0)=exp(-5) = 0.007; i.e. less than 1 in 100 stimuli would be expected to result in failure of transmission). The relationship between membrane potential, synaptic current and synaptic potential were investigated by McLachlan & Martin (1981), by alternately voltage- and current-clamping of the endplate. They showed that the relationship between the observed amplitude of the EPP, and the amplitude which would be obtained if transmitter quanta produced linear summation is: V = V / (1-f.V/E) where V is the predicted amplitude, V the observed amplitude; E is the difference between the resting membrane potential and the reversal potential and f - critically - is a factor which varies from muscle fibre to muscle fibre depending on its length, diameter and specific membrane and cytoplasmic electrical resistance. Normally it is not possible to measure this fudge factor directly for every muscle fibre (it requires alternate voltage and current clamping of the endplate to do this). But there are rules of thumb: long muscle fibres mostly have an f-factor of 0.8; short muscle fibres have factors of about 0.3. Applying the correction for non linear summation to each EPP before calculating mean quantal content results in more accurate estimates by either the direct or variance methods. References Katz,B.(1969) The release of neural transmitter substances. Liverpool University Press. McLachlan EM. Martin AR. (1081) Non-linear summation of end-plate potentials in the frog and mouse. Journal of Physiology. 311:307-24. Katz,B.(1996) Neural transmitter release: from quantal secretion to exocytosis and beyond. The Fenn Lecture. Journal of Neurocytology. 25,677-86. JH Byrne & JL Roberts (2009) From molecules to networks. 2 nd edn. Sinauer (Chapter 8) 7