Voltageclamp experiments

The increase in the sodium permeability of the membrane during the spike that is predicted by the sodium hypothesis can be measured with radioactive tracers by the method illustrated in Fig. 4.4. But although this approach has the advantage of specificity, in that it provides unambiguous information about sodium movements and not those of any other ion, the time resolution of tracer experiments is rather poor, and the results refer only to the cumulative effect of a large number of impulses. In order to make a detailed study of the changes in membrane permeability in the course of a single action potential, it is necessary to resort to measurements of the electric current carried by the ions when they move across the membrane, which enable much greater sensitivity and much better time resolution to be achieved. However, the amount that can be learnt simply by recording the current that flows during the conducted action potential is very limited, because the permeability changes follow a fixed sequence determined by the nerve and not by the experimenter. To get around the difficulty, Hodgkin and Huxley exploited the approach originally due to Cole and Marmont in order to measure the ionic conductance of a nerve membrane whose potential was first 'clamped' at a chosen level and then subjected to a predetermined series of abrupt changes. This enabled them to explore in considerable detail the laws governing the voltage-sensitive behaviour of the sodium and potassium channels, and present-day knowledge of the permeability mechanisms that underlie not only excitation and conduction in nerve and muscle, but also synaptic transmission, is derived very largely from voltage-clamp studies.

A typical experimental set-up for voltage-clamping a squid giant axon is shown in Fig. 4.6. It requires the introduction of two internal electrodes, one for monitoring the potential at the centre of the stretch of axon to be clamped, and the other for passing current uniformly across the membrane over a somewhat greater length. In Hodgkin and Huxley's original apparatus, these electrodes were of the type seen in Fig. 2.2, and were constructed by winding two spirals of AgCl-coated silver wire on a fine glass rod. Nowadays, the potential is generally recorded by a 50 fxm micropipette filled with isotonic KCl (see Fig. 2.1), to which is glued a platinum wire 75 fxm in diameter whose terminal portion is left bare and platinized so that it will pass current without undue polarization. The external electrode consists of a platinized platinum

Fig. 4.6. Schematic diagram of the arrangement for measuring membrane current under a voltage-clamp in a squid giant axon.

sheet in three sections: the current flowing to the central section is amplified and recorded, while the two outer sections help to ensure the uniformity of clamping over the fully controlled region. After appropriate amplification, the internal potential is fed to a voltage comparator circuit along with the square wave signal to which it is to be clamped. The output from the comparator is applied to the internal current wire so as to increase or decrease the membrane current just enough to force the membrane potential to follow the square wave exactly. In electronic terms, this arrangement constitutes a negative feedback control system in which the potential across the membrane is determined by the externally generated command signal, and the resulting membrane current is measured. In order to voltage-clamp smaller non-myelinated nerve fibres, single nerve cells, muscle fibres, or the isolated node of Ranvier in a myelinated nerve fibre, various other electrode arrangements are called for, but the basic principle of the circuit remains the same.

The equivalent electrical circuit of the nerve membrane may be regarded, as shown in Fig. 4.7, as a capacitance Cm connected in parallel with three resistive ionic pathways each incorporating a resistance (RK, RNa and Rleak) in series with a battery. For a given ionic pathway, the driving forces acting on the ions are the membrane potential Em and the concentration gradient for that species of ion. As has been explained on p. 33, the concentration gradient may be equated with an electromotive force calculated from eqn (3.2) as the Nernst equilibrium potential, whence the appropriate values for the three battery potentials are EK, ENa and Eleak, and the net e.m.f. acting on each ion is the difference between Em and its Nernst potential. It follows from Ohm's Law that the ionic currents 1K, lNa and Ileak are given by

Em EK

Rk and so on. Although in accordance with electrical convention the ionic pathways are represented as resistances, it is often more convenient to think of

Outside o

Inside

Fig. 4.7. The equivalent electrical circuit of the nerve membrane according to Hodgkin and Huxley (1952). RNa and RK vary with membrane potential and time; the other components are constant.

them as the reciprocal conductances gK,gNa andgleak. These represent the ease with which that particular ion can pass across the membrane, and are thus directly comparable with the permeability coefficients that appear in the constant field equation (see p. 33), though they are measured in different units. In

the equivalent circuit, RK —— and RN —— are indicated as being variable,

and the object of voltage-clamp experiments is to investigate their dependence on membrane potential and time. Rleak, to which the main contributing ion is Cl~, is constant. In the absence of externally applied current, the electrical model predicts that the value of Em will be determined by the relative sizes of the ionic conductances. If gK is, as in the resting condition, much larger than gNa, Em will lie close to EK; but when the sodium channels are opened and gNa rises, Em will move towards ENa.

When the potential at which the membrane is clamped is suddenly altered, the current flowing across the membrane will consist of the capacity current

Fig. 4.8. Membrane currents for large depolarizing voltage-clamp pulses; outward current upwards. The figures on the right show the change in internal potential in mV. Temperature 3.5°C. From Hodgkin (1958) after Hodgkin, Huxley and Katz (1952).

required to charge or discharge Cm plus the ionic current that is to be measured. Hence the total current I will be given by dE

where I is the sum of the current flowing through all three ionic pathways.

With a well-designed voltage-clamp system,-should have fallen to zero and dt the flow of capacity current should therefore have ceased after no more than about 20 ¡s, so that all subsequent changes in the recorded current can be attributed to alterations in the sodium and potassium conductances operative at the new membrane potential. Fig. 4.8 shows a typical family of superimposed current records for a squid giant axon subjected to increasingly large voltage steps in the depolarizing direction. The initial capacity transients were too fast to be photographed, and what is seen is purely the ionic current. It is evident that there is an early phase of ionic current which flows inwards for small depolarizations and outwards for large ones, and a late phase which is always outwards. This is consistent with the postulates of the sodium hypothesis, and next we have to consider how the contributions of L, and I„ can be

separated from one another.

The method adopted by Hodgkin and Huxley for the analysis of their voltage-clamp records was to suppress the inward sodium current by substituting choline for sodium in the external medium. This procedure yielded

Hodgkin Huxley Experimental Set
Fig. 4.9. Membrane currents associated with depolarization of 65 mV in presence and absence of external sodium. Outward current and internal potential shown upward. Temperature 11 °C. From Hodgkin (1958) after Hodgkin and Huxley (1952).

records of the type shown in Fig. 4.9, from which it is apparent that the removal of external sodium converts the initial hump of inward current into an outward one, but has no effect on the late current, confirming that they are carried by Na+ and K+ ions respectively. To eliminate the sodium current completely, it was necessary to leave some sodium in the external medium and to take Em exactly to ENa, where by definition I = 0. In the experiment of Fig. 4.10 this was achieved by reducing [Na]o to one tenth and depolarizing by 56 mV; trace b shows the resulting record of the potassium current by itself. Subtraction of trace b from trace a, which was recorded in normal sea water, then yielded trace c as the time course of the sodium current. The currents thus measured were finally converted into conductances by taking gK = V(Em " EK) and ¿Na = 7Na/(Em " £Na)' A plot °f ¿K

and gNa against time (Fig. 4.11) shows that, as explained on p. 48, gNa rises internal potential

56 mV

56 mV

Fig. 4.10. Analysis of the ionic current changes in a squid giant axon during a voltage-clamp pulse that depolarized it by 56 mV. Trace a(= /Na + /K) shows the response with the axon in sea water containing 460 mM-Na. Trace b (= /K) is the response with the axon in a solution made up of 10% sea water and 90% isotonic choline chloride solution. Trace c(= /Na) is the difference between traces a and b. Temperature 8.5°C. From Hodgkin (1958) after Hodgkin and Huxley (1952).

quickly and is then inactivated, while gK rises with a definite lag and is not inactivated.

Separation of the two components of the ionic current can now be achieved more easily by recording the sodium current after completely blocking the potassium channels, and vice versa. A good method of abolishing IK is through the introduction of caesium into the axon by perfusion or dialysis: the Cs+ ions enter the mouths of the potassium channels from the inside, and block them very effectively. Fig. 4.12 shows a typical family of IN records for voltage-clamp pulses of different sizes applied to a squid giant axon dialysed with caesium fluoride. For the sodium channels, a blocking agent is now available which acts externally at extremely low concentrations, this being the Japanese puffer-fish poison tetrodotoxin, usually abbreviated to TTX, whose affinity constant for the sodium sites is no more than about 3 nanomolar. Fig. 4.13 shows a family of IK records for a squid axon dialysed with a potassium

56 mV

internal potential

Fig. 4.11. Time courses of the ionic conductance changes during a voltage-clamp pulse calculated from the current records shown in Fig. 4.10. The dashed lines show the effect of repolarization after 0.6 or 6.3 ms. From Hodgkin (1958) after Hodgkin and Huxley (1952).

sodium conductance potassium conductance

Fig. 4.11. Time courses of the ionic conductance changes during a voltage-clamp pulse calculated from the current records shown in Fig. 4.10. The dashed lines show the effect of repolarization after 0.6 or 6.3 ms. From Hodgkin (1958) after Hodgkin and Huxley (1952).

fluoride solution and bathed in a sodium-free solution containing 1 jxm-TTX. A quantitative analysis of such records gives results identical with those obtained by Hodgkin and Huxley, and not only confirms their conclusions in every respect but also provides convincing evidence for the validity of the assumption that the sodium and potassium channels are entirely separate entities between which the only connection is a strong dependence on a potential gradient common to both of them.

Hodgkin and Huxley next proceeded to devise a set of mathematical equations which would provide an empirical description of the behaviour of the sodium and potassium conductances as a function of membrane potential and time. Thus the sodium conductance was found to obey the relationship where ¿Na is a constant representing the peak conductance attainable, m is a dimensionless activation parameter which varies between 0 and 1, and h is a similar inactivation parameter which varies between 1 and 0. The corresponding equation for the potassium conductance was

0.40

0.40

Fig. 4.12. Superimposed traces of the sodium current in a voltage-clamped squid axon whose potassium channels had been blocked by internal dialysis with 330 mM CsF + 20 mM NaF and which was bathed in a K-free artificial sea water containing 103 mM-NaCl and 421 mM-Tris buffer. The membrane was held at -70 mV, and pulses were applied taking the potential to levels varying between -40 and +80 mV in steps of 10 mV. Current scales mA/cm2. For the smaller test pulses, the current flowed inward (downward), but above about +50 mV its direction reversed. For the largest pulses, inactivation was no longer complete, and the channels ended up in the non-inactivating open state. Temperature 5°C. Computer recording made by R. D. Keynes, N. G. Greeff, I. C. Forster and J. M. Bekkers.

Fig. 4.12. Superimposed traces of the sodium current in a voltage-clamped squid axon whose potassium channels had been blocked by internal dialysis with 330 mM CsF + 20 mM NaF and which was bathed in a K-free artificial sea water containing 103 mM-NaCl and 421 mM-Tris buffer. The membrane was held at -70 mV, and pulses were applied taking the potential to levels varying between -40 and +80 mV in steps of 10 mV. Current scales mA/cm2. For the smaller test pulses, the current flowed inward (downward), but above about +50 mV its direction reversed. For the largest pulses, inactivation was no longer complete, and the channels ended up in the non-inactivating open state. Temperature 5°C. Computer recording made by R. D. Keynes, N. G. Greeff, I. C. Forster and J. M. Bekkers.

where is the peak potassium conductance and n is another dimensionless activation parameter. The quantities m, h and n described the variation of the conductances with potential and time, and were determined by the differential equations dm

dh dt

where the as and /s are voltage-dependent rate constants whose dimensions are time-1. The precise details of the voltage-dependence of the six

Fig. 4.13. Superimposed traces of the potassium current in a voltage-clamped squid axon whose sodium channels had been blocked by bathing it in artificial sea water containing 1 ^m-TTX, and which was dialysed internally with 350 mM-KF. The membrane was held at -70 mV, and pulses were applied taking the potential to levels varying between -60 and +40 mV steps. Outward current is upward. Temperature 4°C. Computer recording made by R. D. Keynes, J. E. Kimura and N. G. Greeff.

Fig. 4.13. Superimposed traces of the potassium current in a voltage-clamped squid axon whose sodium channels had been blocked by bathing it in artificial sea water containing 1 ^m-TTX, and which was dialysed internally with 350 mM-KF. The membrane was held at -70 mV, and pulses were applied taking the potential to levels varying between -60 and +40 mV steps. Outward current is upward. Temperature 4°C. Computer recording made by R. D. Keynes, J. E. Kimura and N. G. Greeff.

Fig. 4.14. Comparison of computed (a,b) and experimentally recorded (c,d) action potentials propagated in a squid giant axon at 18.5°C, plotted on fast and slow time scales. The calculated conduction velocity was 18.8 m/s, and that actually observed was 21.2 m/s. From Hodgkin and Huxley (1952).

rate constants need not concern us further, since eqns (4.3) to (4.7) have mainly been cited in order to help the mathematically-minded reader to follow the steps that were necessary for the achievement of Hodgkin and Huxley's primary objective of testing the correctness of their description of the permeability system by calculating from their equations the shape of the propagated action potential.

Fig. 4.15. The time courses of the propagated action potential and underlying ionic conductance changes computed by Hodgkin and Huxley from their voltage-clamp data. The constants used were appropriate to a temperature of 18.5°C. The calculated net entry of Na+ was 4.33 pmole/cm2, and the net exit of K+ was 4.26 pmole/cm2. Conduction velocity = 18.8 m/s. From Hodgkin and Huxley (1952).

Fig. 4.15. The time courses of the propagated action potential and underlying ionic conductance changes computed by Hodgkin and Huxley from their voltage-clamp data. The constants used were appropriate to a temperature of 18.5°C. The calculated net entry of Na+ was 4.33 pmole/cm2, and the net exit of K+ was 4.26 pmole/cm2. Conduction velocity = 18.8 m/s. From Hodgkin and Huxley (1952).

The final step in Hodgkin and Huxley's arguments was thus the computation from data obtained under voltage-clamp conditions of the way in which the conducted action potential would be expected to behave. Fig. 4.14 shows an example of the excellent agreement between the predicted time course of the propagated action potential at 18.5°C and what was observed experimentally at this temperature. The velocity of conduction of the impulse could also be computed, and again the theoretical and observed values were satisfactorily close to one another. Lastly, the net exchange of sodium and potassium could be predicted from the calculated extents and degree of overlap of the changes ingNa andgK during the spike that are illustrated in Figs. 4.15 and 4.16. The total entry of sodium and the exit of potassium in a single impulse each worked out to be about 4.3 pmole/cm2 membrane, which fits very well with the results of the analytical and tracer experiments discussed on p. 43. No more could have been asked of the sodium hypothesis than that it should have yielded from purely electrical

ms 0 1 23456 7

Fig. 4.16. Time relations of the events during the conducted impulse. a, membrane potential. b, ionic movements. c, membrane permeability. d, local circuit current flow. e, membrane impedance.

ms 0 1 23456 7

Fig. 4.16. Time relations of the events during the conducted impulse. a, membrane potential. b, ionic movements. c, membrane permeability. d, local circuit current flow. e, membrane impedance.

measurements figures which checked so nicely with those based on a chemical approach.

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