Active Behavior of the Cell Membrane
When a stimulus current pulse is arranged to depolarize the resting membrane of a cell to or beyond the threshold voltage, then the membrane will respond with an action impulse. An example of this is seen in Figure 2.8 in the action potential responses 3b and 4 to the transthreshold stimuli 3 and 4, respectively. The response is characterized by an initially rapidly rising transmembrane potential, which reaches a positive peak and then more slowly recovers to the resting voltage. This phasic behavior typifies what is meant by an action impulse.
A quantitative analysis of the action impulse was successfully undertaken by Alan L. Hodgkin and Andrew F. Huxley and colleagues in Cambridge (Hodgkin and Huxley, 1952abcd). Their work was made possible because of two important factors. The first was the selection of the giant axon of the squid, a nerve fiber whose diameter is around 0.5 mm, and consequently large enough to permit the insertion of the necessary two electrodes into the intracellular space. (Credit for discovering the applicability of the squid axon to electrophysiological studies is given to by J. Z. Young (1936).) The second was the development of a feedback control device called the voltage clamp, capable of holding the transmembrane voltage at any prescribed value.
This chapter describes the voltage clamp device, the experiments of Hodgkin and Huxley, the mathematical model into which their data were fitted, and the resulting simulation of a wide variety of recognized electrophysiological phenomena (activation, propagation, etc.). The voltage clamp procedure was developed in 1949 separately by K. S. Cole (1949) and G. Marmont (1949). Because of its importance, we first discuss the principle of the voltage clamp method in detail. The Hodgkin and Huxley work is important not only for its ability to describe quantitatively both the active and the passive membrane, but for its contribution to a deeper understanding of the membrane mechanisms that underlie its electrophysiological behavior.
A remarkable improvement in the research of membrane electrophysiology was made by Erwin Neher and Bert Sakmann, who in 1976 published a method for the measurement of ionic currents in a single ionic channel (Neher and Sakmann, 1976). This method, called patch clamp, is a further development of the voltage clamp technique. The patch clamp technique allows the researcher to investigate the operation of single ion channels and receptors and has a wide application, for instance, in the pharmaceutical research. By measuring the capacitance of the plasma membrane with the patch clamp technique, the researcher may also investigate the regulation of exocytosis of the cell.
The electric behavior of the axon membrane is, of course, described by the net ion flow through a great number of ion channels. The ion channels seem to behave "digitally" (as seen in the measurement result of the patch clamp experiment); however, because of the large number of ion channels, the electric currents of a large area of the axon membrane exhibit "analog" behavior, as seen in the measurement result obtained in a voltage clamp experiment.
Logically, discussion of the electric behavior of the membrane should begin by examining the behavior of single ion channels and then proceed to by explain the electric behavior of the membrane as the summation of the behavior of a large number of its constituent ionic channels. For historical reasons, however, membrane behavior and the voltage clamp method are discussed here first, before ionic channel behavior and the patch clamp method are explored.
4.2 VOLTAGE CLAMP METHOD
4.2.1 Goal of the voltage clamp measurement
In order to describe the activation mechanism quantitatively, one must be able to measure selectively the flow of each constituent ion of the total membrane current. In this section, we describe how this is accomplished by the voltage clamp measurement procedure.
The following current components arise when the axon is stimulated at one end and the membrane voltage as well as current of a propagating nerve impulse are measured distally:
Note that Io = -Ii.
Our particular goal is to measure selectively each individual ionic current, especially the sodium and potassium currents. Note that because we examine the ionic currents during the propagating nerve impulse, the membrane resistance (rm) is not constant; hence it is represented by a symbol indicating a variable resistance. Any measurement of membrane current with a propagating nerve impulse, however, will yield the sum of these currents.
The total membrane current (as illustrated in Figure 4.1) satisfies Equation 3.48, which can be rewritten in the form:
|where||im||= total transmembrane current per unit length [µA/cm axon length]|
|imI||= ionic component of the transmembrane current per unit length [µA/cm axon length]|
|cm||= membrane capacitance per unit length [µF/cm axon length]|
|Vm||= membrane voltage [mV]|
|t||= time [ms]|
|ri||= intracellular axial resistance per unit length of axon [k /cm axon length]|
|ro||= interstitial resistance per unit length [k /cm axon length]|
|x||= distance [cm]|
By measuring Vm(t) and the propagation velocity Θ, we could obtain Vm(t - x/Θ) and hence im from Equation 4.1. Although the determination of im is straightforward, the accuracy depends on the uniformity of the preparation as well as knowledge of the parameters ri, ro, and Θ. A more satisfactory procedure is based on the elimination of the axial currents.
By convention Vm, the transmembrane voltage, is taken as the intracellular potential, Φi, relative to the extracellular potential, Φo. That is, Vm = Φi - Φo. Further, the positive direction of transmembrane current is chosen as outward (from the intracellular to the extracellular space). These conventions were adopted in the mid-1950s so that in reading earlier papers one should be alert to encountering an opposite choice. The aforementioned conventions are reflected in Equation 4.1. Also, to maintain consistency with the tradition of drawing electronic circuits, in the equivalent circuits of the cell membrane, the reference terminal, that is the outside of the cell, is selected to be at the bottom and the terminal representing the measured signal, that is the inside of the cell, is at the top. In those figures, where it is appropriate to illustrate the membrane in the vertical direction, the inside of the membrane is located on the left-hand side and the outside on the right-hand side of the membrane.
Fig. 4.1. The principle of membrane current measurement with a propagating nerve impulse.
(A) It is assumed that a propagating wave is initiated at the left and has a uniform velocity at the site where the voltage is measured. To obtain the transmembrane current, Equation 4.1 can be used; implementation will require the measurement of the velocity of propagation so that ²Vm/x² = (1/Θ²)²Vm/t² can be evaluated.
(B) A portion of the linear core conductor model (assuming the extracellular medium to be bounded) which reflects the physical model above. (Note that because we examine the ionic currents during the propagating nerve impulse, the membrane resistance rm is not constant; hence it is represented by a symbol indicating a variable resistance. To the extent that the ion concentrations may change with time then Em can also be time-varying.) The symbols are explained in the text.
|where||im||= the total current per unit length [µA/cm axon length]|
|imI||= the ionic current per unit length [µA/cm axon length]|
|cm||= the capacitance of the preparation per unit length [µF/cm axon length]|
Because the apparatus ensures axial uniformity, it is described as space clamped. The electric model of the space clamped measurement is illustrated in Figure 4.2.
|im = imI||(4.3)|
and the membrane current is composed solely of ionic currents. (In the moment following the onset of the voltage step, a very brief current pulse arises owing to the capacitance of the membrane. It disappears quickly and does not affect the measurement of the ensuing activation currents.)
The voltage clamp procedure is illustrated in the space-clamp device shown in Figure 4.3. A desired voltage step is switched between the inner and outer electrodes, and the current flowing between these electrodes (i.e., the transmembrane current) is measured.
The actual voltage clamp measurement circuit is somewhat more complicated than the one described above and is shown in Figure 4.4. Separate electrodes are used for current application (a, e) and voltage sensing (b, c) to avoid voltage errors due to the electrode-electrolyte interface and the resistance of the thin current electrode wires. Figure 4.4 illustrates the principle of the measurement circuit used by Hodgkin, Huxley, and Katz (1952). The circuit includes a unity gain amplifier (having high input impedance), which detects the membrane voltage Vm between a wire inside the axon (b) and outside the axon (c). The output is sent to an adder, where the difference between the clamp voltage (Vc) and the measured membrane voltage (Vm) is detected and amplified. This output, K(Vc - Vm), drives the current generator. The current generator feeds the current to the electrode system (a, e) and hence across the membrane. The current is detected through measurement of the voltage across a calibrated resistance, Rc. The direction of the controlled current is arranged so that Vm is caused to approach Vc, whereupon the feedback signal is reduced toward zero. If K is large, equilibrium will be established with Vm essentially equal to Vc and held at that value. The principle is that of negative feedback and proportional control.
The measurements were performed with the giant axon of a squid. The thickness of the diameter of this axon - approximately 0.5 mm - makes it possible to insert the two internal electrodes described in Figure 4.4 into the axon. (These were actually fabricated as interleaved helices on an insulating mandrel.).
Fig. 4.4. Realistic voltage clamp measurement circuit. Current is applied through electrodes (a) and (e), while the transmembrane voltage, Vm, is measured with electrodes (b) and (c). The current source is controlled to maintain the membrane voltage at some preselected value Vc.
Fig. 4.7. Selective measurement of sodium and potassium current: The extracellular sodium ions are replaced with an inactive cation to reduce the sodium Nernst potential so that it corresponds to the clamp voltage value.
Our object here is to find equations which describe the conductances with reasonable accuracy and are sufficiently simple for theoretical calculation of the action potential and refractory period. For sake of illustration we shall try to provide a physical basis for the equations, but must emphasize that the interpretation given is unlikely to provide a correct picture of the membrane. (Hodgkin and Huxley, 1952d, p. 506)
In spite of its simple form, the model explains with remarkable accuracy many nerve membrane properties. It was the first model to describe the ionic basis of excitation correctly. For their work, Hodgkin and Huxley received the Nobel Prize in 1963. Although we now know many specific imperfections in the Hodgkin-Huxley model, it is nevertheless essential to discuss it in detail to understand subsequent work on the behavior of voltage-sensitive ionic channels.
The reader should be aware that the original Hodgkin and Huxley papers were written at a time when the definition of Vm was chosen opposite to the convention adopted in the mid-1950s. In the work described here, we have used the present convention: Vm equals the intracellular minus extracellular potential.
In this model, each of these four current components is assumed to utilize its own (i.e., independent) path or channel. To follow the modern sign notation, the positive direction of membrane current and Nernst voltage is chosen to be from inside to outside.
The model is constructed by using the basic electric circuit components of voltage source, resistance, and capacitance as shown in Figure 4.10. The ion permeability of the membrane for sodium, potassium, and other ions (introduced in Equation 3.34) is taken into account through the specification of a sodium, potassium, and leakage conductance per unit area (based on Ohm's law) as follows:
|where||GNa, GK, GL||= membrane conductance per unit area for sodium, potassium, and other ions - referred to as the leakage conductance [S/cm²]|
|INa, IK, IL||= the electric current carried by sodium, potassium and other ions (leakage current) per unit area [mA/cm²]|
|VNa, VK, VL||= Nernst voltage for sodium, potassium and other ions (leakage voltage) [mV]|
|Vm||= membrane voltage [mV]|
The above-mentioned Nernst voltages are defined by the Nernst equation, Equation 3.21, namely:
where the subscripts "i" and "o" denote the ion concentrations inside and outside the cell membrane, respectively. Other symbols are the same as in Equation 3.21 and z = 1 for Na and K but z = -1 for Cl.
In Figure 4.10 the polarities of the voltage sources are shown as having the same polarity which corresponds to the positive value. We may now insert the Nernst voltages of sodium, potassium, and chloride, calculated from the equations 4.7 ... 4.9 to the corresponding voltage sources so that a calculated positive Nernst voltage is directed in the direction of the voltage source polarity and a calculated negative Nernst voltage is directed in the opposite direction. With the sodium, potassium, and chloride concentration ratios existing in nerve and muscle cells the voltage sources of Figure 4.10 in practice achieve the polarities of those shown in Figure 3.4.
Because the internal concentration of chloride is very low small movements of chloride ion have a large effect on the chloride concentration ratio. As a result, a small chloride ion flux brings it into equilibrium and chloride does not play an important role in the evaluation of membrane potential (Hodgkin and Horowicz, 1959). Consequently Equation 4.9 was generalized to include not only chloride ion flux but that due to any non-specific ion. The latter flux arises under experimental conditions since in preparing an axon for study small branches are cut leaving small membrane holes through which small amounts of ion diffusion can take place. The conductance GL was assumed constant while VL was chosen so that the sum of all ion currents adds to zero at the resting membrane potential.
When Vm = VNa, the sodium ion is in equilibrium and there is no sodium current. Consequently, the deviation of Vm from VNa (i.e., Vm - VNa) is a measure of the driving voltage causing sodium current. The coefficient that relates the driving force (Vm - VNa) to the sodium current density INa is the sodium conductance, GNa - that is, INa = GNa(Vm - VNa), consistent with Ohm's law. A rearrangement of this leads to Equation 4.4. Equations 4.5 and 4.6 can be formed in the same way.
Now the four currents discussed above can be evaluated for a particular membrane voltage, Vm. The corresponding circuits are formed by:
(Regarding these circuit elements Hodgkin and Huxley had experimental justification for assuming linearly ohmic conductances in series with each of the emfs. They observed that the current changed linearly with voltage when a sudden change of membrane voltage was imposed. These conductances are, however, not included in the equivalent circuit in Figure 4.10. (Huxley, 1993))
On the basis of their voltage clamp studies, Hodgkin and Huxley determined that the membrane conductance for sodium and potassium are functions of transmembrane voltage and time. In contrast, the leakage conductance is constant. Under subthreshold stimulation, the membrane resistance and capacitance may also be considered constant.
One should recall that when the sodium and potassium conductances are evaluated during a particular voltage clamp, their dependence on voltage is eliminated because the voltage during the measurement is constant. The voltage nevertheless is a parameter, as may be seen when one compares the behavior at different voltages. For a voltage clamp measurement the only variable in the measurement is time. Note also that the capacitive current is zero, because dV/dt = 0.
For the Hodgkin-Huxley model, the expression for the total transmembrane current density is the sum of the capacitive and ionic components. The latter consist of sodium, potassium, and leakage terms and are given by rearranging Equations 4.4 through 4.6. Thus
|where||Im||= membrane current per unit area [mA/cm²]|
|Cm||= membrane capacitance per unit area [F/cm²]|
|Vm||= membrane voltage [mV]|
|VNa, VK, VL||= Nernst voltage for sodium, potassium and leakage ions [mV]|
|GNa, GK, GL||= sodium, potassium, and leakage conductance per unit area [S/cm²]|
As noted before, in Figure 4.10 the polarities of the voltage sources are shown in a universal and mathematically correct way to reflect the Hodgkin-Huxley equation (Equation 4.10). With the sodium, potassium, and chloride concentration ratios existing in nerve and muscle cells the voltage sources of Figure 4.10 in practice achieve the polarities of those shown in Figure 3.4.
Note that in Equation 4.10, the sum of the current components for the space clamp action impulse is necessarily zero, since the axon is stimulated simultaneously along the whole length and since after the stimulus the circuit is open. There can be no axial current since there is no potential gradient in the axial direction at any instant of time. On the other hand, there can be no radial current (i.e., Im = 0) because in this direction there is an open circuit. In the voltage clamp experiment the membrane current in Equation 4.10 is not zero because the voltage clamp circuit permits a current flow (necessary to maintain the clamp voltage).
[it] depends on the distribution of charged particles which do not act as carriers in the usual sense, but which allow the ions to pass through the membrane when they occupy particular sites in the membrane. On this view the rate of movement of the activating particles determines the rate at which the sodium and potassium conductances approach their maximum but has little effect on the (maximum) magnitude of the conductance. (Hodgkin and Huxley, 1952d, p. 502)
Hodgkin and Huxley did not make any assumptions regarding the nature of these particles in chemical or anatomical terms. Because the only role of the particles is to identify the fraction of channels in the open state, this could be accomplished by introducing corresponding abstract random variables that are measures of the probabilities that the configurations are open ones. In this section, however, we describe the Hodgkin-Huxley model and thus follow their original idea of charged particles moving in the membrane and controlling the conductance. (These are summarized later in Figure 4.13.)
The time course of the potassium conductance (GK) associated with a voltage clamp is described in Figure 4.11 and is seen to be continuous and monotonic. (The curves in Figure 4.11 are actually calculated from the Hodgkin-Huxley equations. For each curve the individual values of the coefficients, listed in Table 1 of Hodgkin and Huxley (1952d), are used; therefore, they follow closely the measured data.) Hodgkin and Huxley noted that this variation could be fitted by a first-order equation toward the end of the record, but required a third- or fourth-order equation in the beginning. This character is, in fact, demonstrated by its sigmoidal shape, which can be achieved by supposing GK to be proportional to the fourth power of a variable, which in turn satisfies a first-order equation. Hodgkin and Huxley gave this mathematical description a physical basis with the following assumptions.
As is known, the potassium ions cross the membrane only through channels that are specific for potassium. Hodgkin and Huxley supposed that the opening and closing of these channels are controlled by electrically charged particles called n-particles. These may stay in a permissive (i.e., open) position (for instance inside the membrane) or in a nonpermissive (i.e., closed) position (for instance outside the membrane), and they move between these states (or positions) with first-order kinetics. The probability of an n-particle being in the open position is described by the parameter n, and in the closed position by (1 - n), where 0 n 1. Thus, when the membrane potential is changed, the changing distribution of the n-particles is described by the probability of n relaxing exponentially toward a new value.
Fig. 4.11. Behavior of potassium conductance as a function of time in a voltage clamp experiment. The displacement of transmembrane voltage from the resting value [in mV] is shown (all are depolarizations). These theoretical curves correspond closely to the measured values.
In mathematical form, the voltage- and time-dependent transitions of the n-particles between the open and closed positions are described by the changes in the parameter n with the voltage-dependent transfer rate coefficients αn and βn. This follows a first-order reaction given by :
|where||αn||= the transfer rate coefficient for n-particles from closed to open state [1/s]|
|βn||= the transfer rate coefficient for n-particles from open to closed state [1/s]|
|n||= the fraction of n-particles in the open state|
|1 - n||= the fraction of n-particles in the closed state|
If the initial value of the probability n is known, subsequent values can be calculated by solving the differential equation
Thus, the rate of increase in the fraction of n-particles in the open state dn/dt depends on their fraction in the closed state (1 - n), and their fraction in the open state n, and on the transfer rate coefficients αn and βn. Because the n-particles are electrically charged, the transfer rate coefficients are voltage-dependent (but do not depend on time). Figure 4.12A shows the variations of the transfer rate coefficients with membrane voltage. Expressions for determining their numerical values are given at the end of this section.
Furthermore Hodgkin and Huxley supposed that the potassium channel will be open only if four n-particles exist in the permissive position (inside the membrane) within a certain region. It is assumed that the probability of any one of the four n-particles being in the permissive position does not depend on the positions of the other three. Then the probability of the channel being open equals the joint probability of these four n-particles being at such a site and, hence, proportional to n4. (These ideas appear to be well supported by studies on the acetylcholine receptor, which consists of five particles surrounding an aqueous channel, and where a small cooperative movement of all particles can literally close or open the channel (Unwin and Zampighi, 1980).)
The potassium conductance per unit area is then the conductance of a single channel times the number of open channels. Alternatively, if GK max is the conductance per unit area when all channels are open (i.e., its maximum value), then if only the fraction n4 are open, we require that
where GK max = maximum value of potassium conductance [mS/cm²], and n obeys Equation 4.12.
Equations 4.12 and 4.13 are among the basic expressions in the Hodgkin and Huxley formulation.
|where||= steady-state value of n|
|= time constant [s]|
We see that the voltage step initiates an exponential change in n from its initial value of n0 (the value of n at t = 0) toward the steady-state value of n (the value of n at t = ). Figure 4.12B shows the variation of n and n4 with membrane voltage.
Fig. 4.12. (A) Variation of transfer rate coefficients αn and βn as functions of membrane voltage.
(B) Variation of n and n4 as functions of membrane voltage (GK n4 ).
Fig. 4.13. In the Hodgkin-Huxley model, the process determining the variation of potassium conductance with depolarization and repolarization with voltage clamp.
(A) Movement of n-particles as a response to sudden depolarization. Initially, αn is small and βn is large, as indicated by the thickness of the arrows. Therefore, the fraction n of n-particles in the permissive state (inside the membrane) is small. Depolarization increases αn and decreases βn. Thus n rises exponentially to a larger value. When four n-particles occupy the site around the channel inside the membrane, the channel opens.
(B) The response of the transfer rate coefficients αn and βn to sudden depolarization and repolarization.
(C) The response of n and n4 to a sudden depolarization and repolarization (GK n4 )
|where||αm||= the transfer rate coefficient for m-particles from closed to open state [1/s]|
|βm||= the transfer rate coefficient for m-particles from open to closed state [1/s]|
|m||= the fraction of m-particles in the open state|
|1 - m||= the fraction of m-particles in the closed state|
An equation for the behavior of sodium activation may be written in the same manner as for the potassium, namely that m satisfies a first-order process:
The transfer rate coefficients αm and βm are voltage-dependent but do not depend on time..
Fig. 4.14. Behavior of sodium conductance in voltage clamp experiments. The clamp voltage is expressed as a change from the resting value (in [mV]). Note that the change in sodium conductance is small for subthreshold depolarizations but increases greatly for transthreshold depolarization ( Vm = 26 mV).
On the basis of the behavior of the early part of the sodium conductance curve, Hodgkin and Huxley supposed that the sodium channel is open only if three m-particles are in the permissive position (inside the membrane). Then the probability of the channel being open equals the joint probability that three m-particles in the permissive position; hence the initial increase of sodium conductance is proportional to m3.
The main difference between the behavior of sodium and potassium conductance is that the rise in sodium conductance, produced by membrane depolarization, is not maintained. Hodgkin and Huxley described the falling conductance to result from an inactivation process and included it by introducing an inactivating h-particle. The parameter h represents the probability that an h-particle is in the non-inactivating (i.e., open) state - for instance, outside the membrane. Thus (1 - h) represents the number of the h-particles in the inactivating (i.e., closed) state - for instance, inside the membrane. The movement of these particles is also governed by first-order kinetics:
|where||αh||= the transfer rate coefficient for h-particles from inactivating to non-inactivating state [1/s]|
|βh||= the transfer rate coefficient for h-particles from non-inactivating to inactivating state [1/s]|
|h||= the fraction of h-particles in the non-inactivating state|
|1 - h||= the fraction of h-particles in the inactivating state|
and satisfies a similar equation to that obeyed by m and n, namely:
Again, because the h-particles are electrically charged, the transfer rate coefficients αh and βh are voltage-dependent but do not depend on time.
The sodium conductance is assumed to be proportional to the number of sites inside the membrane that are occupied simultaneously by three activating m-particles and not blocked by an inactivating h-particle. Consequently, the behavior of sodium conductance is proportional to m3h, and
|where||GNa max||= maximum value of sodium conductance [mS/cm²], and|
|m||= obeys Equation (4.16), and|
|h||= obeys Equation (4.18), and|
Following a depolarizing voltage step (voltage clamp), m will rise with time (from m0 to m ) according to an expression similar to Equation 4.14 (but with m replacing n). The behavior of h is just the opposite since in this case it will be found that h0 h and an exponential decrease results from the depolarization. Thus the overall response to a depolarizing voltage ste includes an exponential rise in m (and thus a sigmoidal rise in m3 ) and an exponential decay in h so that GNa, as evaluated in Equation 4.19, will first increase and then decrease. This behavior is just exactly that needed to fit the data described in Figure 4.14. In addition, it turns out that the normal resting values of m are close to zero, whereas h is around 0.6. For an initial hyperpolarization, the effect is to decrease m; however, since it is already very small, little additional diminution can occur. As for h, its value can be increased to unity, and the effect on a subsequent depolarization can be quite marked. This effect fits experimental observations closely. The time constant for changes in h is considerably longer than for m and n, a fact that can lead to such phenomena as "anode break," discussed later in this chapter. Figure 4.15A shows variations in the transfer rate coefficients αm, βm, αh, and βh with membrane voltage. Figure 4.15B shows the variations in m, h, and m3 h with membrane voltage.
Fig. 4.15. Variation in (A) αm and βm, (B) αh and βh, (C) m and h, and (D) m3h as a function of membrane voltage. Note that the value of m3h is so small that the steady-state sodium conductance is practically zero.
Fig. 4.16. The process, in the Hodgkin-Huxley model, determining the variation of sodium conductance with depolarization and repolarization with voltage clamp.
(A) Movement of m- and h-particles as a response to sudden depolarization. Initially, αm is small and βm is large, as indicated by the thickness of the arrows. Therefore, the fraction of particles of type m in the permissive state (inside the membrane) is small. Initially also the value of αh is large and βh is small. Thus the h-particles are in the non-inactivating position, outside the membrane. Depolarization increases αm and βh and decreases βm and αh. Thus the number of m-particles inside the membrane, m, rises exponentially toward unity, and the number of h-particles outside the membrane, h, decreases exponentially toward zero.
(B) The response of transfer rate coefficients αm, βm, αh, and βh to sudden depolarization and repolarization. (C) The response of m, h, m3, and m3h to a sudden depolarization and repolarization. Note that according to Equation 4.20, GNa is proportional to m3h.
In these equations V' = Vm - Vr, where Vr is the resting voltage. All voltages are given in millivolts. Therefore, V' is the deviation of the membrane voltage from the resting voltage in millivolts, and it is positive if the potential inside the membrane changes in the positive direction (relative to the outside). The equations hold for the giant axon of the squid at a temperature of 6.3 °C.
Please note again that in the voltage clamp experiment the α and β are constants because the membrane voltage is kept constant during the entire procedure. During an unclamped activation, where the transmembrane voltage is continually changing, the transfer rate coefficients will undergo change according to the above equations.
|Vr - VNa||=||-115||mV|
|Vr - VK||=||+12||mV|
|Vr - VL||=||-10.613||mV|
Note that the value of VL is not measured experimentally, but is calculated so that the current is zero when the membrane voltage is equal to the resting voltage. The voltages in the axon are illustrated in Figure 4.17 in graphical form.
In Table 4.1 we summarize the entire set of Hodgkin-Huxley equations that describe the Hodgkin-Huxley model..
|GNa = GNa max m3h|
|GK = GK max n4 |
GL = constant
|TRANSFER RATE COEFFICIENTS|
Vr - VNa = -115
Vr - VK = +12
Vr - VL = -10.613 mV
Cm = 1 μF/cm²|
GNa max = 120 ms/cm²
GK max = 36 ms/cm²
GL = 0.3 ms/cm²
Fig. 4.18. Application of the Hodgkin-Huxley model to a propagating nerve impulse.
The figure illustrates the model for a unit length of axon. In the model the quantities ri and ro represent the resistances per unit length inside and outside the axon, respectively. Between the inside and outside of the membrane, describing the behavior of the membrane, is a Hodgkin-Huxley model. For the circuit in this figure, Equation 3.42 was derived in the previous chapter for the total membrane current, and it applies here as well:
In an axon with radius a, the membrane current per unit length is
|im = 2πaIm [µA/cm axon length]||(4.26)|
where Im = membrane current per unit area [µA/cm²].
The axoplasm resistance per unit length is:
where ρi = axoplasm resistivity [kΩcm]
In practice, when the extracellular space is extensive, the resistance of the external medium per unit length, ro, is so small that it may be omitted and thus from Equations 3.42, 4.26, and 4.27 we obtain:
Equation 4.10 evaluates the transmembrane current density based on the intrinsic properties of the membrane while Equation 4.28 evaluates the same current based on the behavior of the "load". Since these expressions must be equal, the Hodgkin-Huxley equation for the propagating nerve impulse may be written:
Under steady state conditions the impulse propagates with a constant velocity and it maintains constant form; hence it obeys the wave equation:
where Θ = the velocity of conduction [m/s].
Substituting Equation 4.30 into 4.29 permits the equation for the propagating nerve impulse to be written in the form:
This is an ordinary differential equation which can be solved numerically if the value of Θ is guessed correctly. Hodgkin and Huxley obtained numerical solutions that compared favorably with the measured values (18.8 m/s).
With modern computers it is now feasible to solve a parabolic partial differential equation, Equation 4.29, for Vm as a function of x and t (a more difficult solution than for Equation 4.31). This solution permits an examination of Vm during initiation of propagation and at its termination. One can observe changes in velocity and waveform under these conditions. The velocity in this case does not have to be guessed at initially, but can be deduced from the solution.
The propagation velocity of the nerve impulse may be written in the form:
|where||Θ||= propagation velocity [m/s]|
|K||= constant [1/s]|
|a||= axon radius [cm]|
|ρi||= axoplasm resistivity [Ωcm]|
This can be deduced from Equation 4.31 by noting that the equation is unchanged if the coefficient of the first term is held constant (= 1/K), it being assumed that the ionic conductances remain unaffected (Hodgkin, 1954). Equation 4.32 also shows that the propagation velocity of the nerve impulse is directly proportional to the square root of axon radius a in unmyelinated axons. This is supported by experiment; and, in fact, an empirical relation is:
|where||Θ||= propagation velocity [m/s]|
|d||= axon diameter [µm]|
This velocity contrasts with that observed in myelinated axons; there, the value is linearly proportional to the radius, as illustrated earlier in Figure 2.12. A discussion of the factors affecting the propagation velocity is given in Jack, Noble, and Tsien (1975).
Fig. 4.19. Sodium and potassium conductances (GNa and GK), their sum (Gm), and the membrane voltage (Vm) during a propagating nerve impulse. This is a numerical solution of Equation 4.32 (After Hodgkin and Huxley, 1952d.).
The potential inside the membrane begins to increase before the sodium conductance starts to rise, owing to the local circuit current originating from the proximal area of activation. In this phase, the membrane current is mainly capacitive, because the sodium and potassium conductances are still low.
The local circuit current depolarizes the membrane to the extent that it reaches threshold and activation begins.
The activation starts with an increasing sodium conductance. As a result, sodium ions flow inward, causing the membrane voltage to become less negative and finally positive.
The potassium conductance begins to increase later on; its time course is much slower than that for the sodium conductance.
When the decrease in the sodium conductance and the increase in the potassium conductance are sufficient, the membrane voltage reaches its maximum and begins to decrease. At this instant (the peak of Vm), the capacitive current is zero (dV/dt = 0) and the membrane current is totally an ionic current.
The terminal phase of activation is governed by the potassium conductance which, through the outflowing potassium current, causes the membrane voltage to become more negative. Because the potassium conductance is elevated above its normal value, there will be a period during which the membrane voltage is more negative than the resting voltage - that is, the membrane is hyperpolarized.
Finally, when the conductances reach their resting value, the membrane voltage reaches its resting voltage..
Fig. 4.20. Sodium and potassium conductances GNa and GK, the ionic and capacitive components ImI and ImC of the membrane current Im, and the membrane voltage Vm during a propagating nerve impulse.
|3 (T - 6 . 3)/10||(4.33)|
where T is the temperature in °C.
Fig. 4.21. Membrane voltage during a nonpropagating nerve impulse of a squid axon
(A) calculated from Equation 4.10 with Im = 0 and
(B) measured (lower) at 6 °C temperature.
The numbers indicate the stimulus intensity in [nC/cm²]. Note the increasing latency as the stimulus is decreased until, finally, the stimulus falls below threshold.
Fig. 4.23. The membrane voltage of a propagating nerve impulse.
(A) Calculated from Equation 4.31. The temperature is 18.5 C and the constant K in Equation 4.32 has the value 10.47 [1/ms].
(B) Measured membrane voltage for an axon at the same temperature as (A).
Fig. 4.24. (A) The response during the refractory period calculated from Equation 4.10 at 6 C temperature. The axon is first stimulated with a stimulus intensity of 15 nC/cm², curve A. Curves B, C, and D represent the calculated response to a 90 nC/cm² stimulus at various instants of time after the curve A. Curve E represents the calculated response to a 90 nC/cm² stimulus for an axon in the resting state.
(B) The set of curves shows the corresponding experiments performed with a real axon at 9 C temperature. The time scale is corrected to reflect the temperature difference.
Fig. 4.25. (A) Calculated and (B) measured threshold. The calculated curves are numerical solutions of Equation 4.10. The stimulus intensity is expressed in [nC/cm²].
Fig. 4.26. Anode break phenomenon
(A) calculated from Equation 4.10 and
(B) measured from a squid axon at 6 C temperature.
The numbers attached to the curves give the initial depolarization in [mV]. The hyperpolarization is released at t = 0.
If a heat-polished glass microelectrode, called a micropipette, having an opening of about 0.5-1 µm, is brought into close contact with an enzymatically cleaned cell membrane, it forms a seal on the order of 50 MΩ . Even though this impedance is quite high, within the dimensions of the micropipette the seal is too loose, and the current flowing through the micropipette includes leakage currents which enter around the seal (i.e., which do not flow across the membrane) and which therefore mask the desired (and very small) ion-channel transmembrane currents.
If a slight suction is applied to the micropipette, the seal can be improved by a factor of 100-1000. The resistance across the seal is then 10-100 GΩ ("G" denotes "giga" = 109). This tight seal, called gigaseal, reduces the leakage currents to the point where it becomes possible to measure the desired signal - the ionic currents through the membrane within the area of the micropipette.
Formation of an outside-out or inside-out patch may involve major structural rearrangements of the membrane. The effects of isolation on channel properties have been determined in some cases. It is surprising how minor these artifacts of preparation are for most of the channel types of cell membranes.
Fig. 4.28. Registration of the flow of current through a single ion channel at the neuromuscular endplate of frog muscle fiber with patch clamp method. (From Sakmann and Neher, 1984.)
Fig. 4.29. Working hypothesis for a channel. The channel is drawn as a transmembrane macromolecule with a hole through the center. The functional regions - namely selectivity filter, gate, and sensor - are deducted from voltage clamp experiments and are only beginning to be charted by structural studies. (Redrawn from Hille, 1992.)
Before proceeding it is useful to introduce a general description of a channel protein (illustrated in Figure 4.29). Although based on recognized channel features, the figure is nevertheless only a "working hypothesis." It contains in cartoon form the important electrophysiological properties associated with "selectivity" and "gating", which will be discussed shortly. The overall size of the protein is about 8 nm in diameter and 12 nm in length (representing 1800-4000 amino acids arranged in one or several polypeptide chains); its length substantially exceeds the lipid bilayer thickness so that only a small part of the molecule lies within the membrane. Of particular importance to researchers is the capacity to distinguish protein structures that lie within the membrane (i.e., hydrophobic elements) from those lying outside (i.e., hydrophilic extracellular and cytoplasmic elements). We have seen that membrane voltages are on the order of 0.1 V; these give rise to transmembrane electric fields on the order of 106 V/m. Fields of this intensity can exert large forces on charged residues within the membrane protein, as Figure 4.29 suggests, and also cause the conformational changes associated with transmembrane depolarization (the alteration in shape changes the conductance of the aqueous pore). In addition, ionic flow through aqueous channels, may be affected by fixed charges along the pore surface.
|Resting Open Inactivated|
An example is the sodium channel, mentioned earlier in this chapter. At the single-channel level, a transthreshold change in transmembrane potential increases the probability that a resting (closed) channel will open. After a time following the opening of a channel, it can again close as a result of a new channel process - that of inactivation. Although inactivation of the squid axon potassium channel was not observed on the time scale investigated, new information on single channels is being obtained from the shaker potassium channel from Drosophila melanogaster which obeys the more general scheme described above (and to which we return below). In fact, this preparation has been used to investigate the mechanism of inactivation. Thus a relative good picture has emerged.
The single-channel behavior illustrated in Figure 4.31 demonstrates the stochastic nature of single-channel openings and closings. Consistent with the Hodgkin-Huxley model is the view that this potassium channel has the probability n4 of being open. As a result, if GK max is the conductance when all of the channels are open, then the conductance under other conditions GK = GK max·n4 And, of course, this is precisely what the Hodgkin-Huxley equation (4.13) states.
One can interpret n as reflecting two probabilities: (1) that a subunit of the potassium channel is open, and (2) that there are four such subunits, each of which must be in the open condition for the channel itself to be open. Hodgkin and Huxley gave these probabilities specific form by suggesting the existence of gating particles as one possible physical model. Such particles have never been identified as such; however, the channel proteins are known to contain charged "elements" (see Figure 4.29), although in view of their overall electroneutrality, may be more appropriately characterized as dipole elements. The application of a depolarizing field on this dipole distribution causes movement (i.e., conformational changes) capable of opening or closing channel gates. In addition, such dipole movement, in fact, constitutes a capacitive gating current which adds to that associated with the displacement of charges held at the inside/outside of the membrane. If the applied field is increased gradually, a point is finally reached where all dipoles are brought into alignment with the field and the gating current reaches a maximum (saturation) value. In contrast, the current associated with the charge stored at the internal/external membrane surface is not limited and simply increases linearly with the applied transmembrane potential. Because of these different characteristics, measurements at two widely different voltage clamps can be used to separate the two components and reveal the gating currents themselves (Bezanilla, 1986).
from which one obtains the differential equation
Since the total number of subunits, N, must satisfy N = Nc(t) + No(t), where N is a fixed quantity, then the above equation becomes
Dividing Equation 4.36 through by N and recognizing that n = No/N as the statistical probability that any single subunit is open, we arrive at
which corresponds exactly to Equation 4.12. This serves to link the Hodgkin-Huxley description of a macroscopic membranes with the behavior of a single component channel. Specifically the transfer rate coefficients α and β describe the transition rates from closed to open (and open to closed) states. One can consider the movement of n-particles, introduced by Hodgkin and Huxley, as another way of describing in physical terms the aforementioned rates. (Note that n is a continuous variable and hence "threshold" is not seen in a single channel. Threshold is a feature of macroscopic membranes with, say, potassium, sodium, and leakage channels and describes the condition where the collective behavior of all channel types allows a regenerative process to be initiated which constitutes the upstroke of an action pulse.) In the above the potassium channel probability of being open is, of couse, n4.
While the description above involved the simultaneous behavior of a large number of equivalent channels, it also describes the statistics associated with the sequential behavior of a single channel (i.e., assuming ergodicity). If a membrane voltage step is applied to the aforementioned ensemble of channels, then the solution to Equation 4.36 is:
It describes an exponential change in the number of open subunits and also describes the exponential rise in probability n for a single subunit. But if there is no change in applied voltage, one would observe only random opening and closings of a single channel. However, according to the fluctuation-dissipation theorem (Kubo, 1966), the same time constants affect these fluctuations as affect the macroscopic changes described in Equation 4.38. Much work has accordingly been directed to the study of membrane noise as a means of experimentally accessing single-channel statistics (DeFelice, 1981)..
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