Medical Physiology A Cellular and Molecular Approach, Updated 2nd Ed.

ELECTRICAL EXCITABILITY AND ACTION POTENTIALS

Edward G. Moczydlowski

Cellular communication in the nervous system is based on electrical and chemical signaling events that are mediated by ion channels. Certain types of cells, including neurons and myocytes, have a remarkable property called electrical excitability. In cells with this property, depolarization of the membrane above a certain threshold voltage triggers a spontaneous all-or-none response called an action potential. This action potential is a transient, regenerative electrical impulse in which the membrane potential (V_m) rapidly rises to a peak that is ~100 mV more positive than the normal, negative resting voltage (V_rest). Such signals, also called spikes, can propagate for long distances along nerve or muscle fibers. Conduction of action potentials allows information from sensory organs to be transmitted along afferent nerves leading to the brain. Conversely, the brain exerts voluntary and involuntary control over muscles and other effector organs by efferent nerves leading away from it.

In the first part of this chapter, we examine the biophysical and molecular basis of action potentials and the mechanisms that underlie their genesis and propagation. The second part deals with the structure and function of voltage-gated ion channel proteins. Finally, we examine the conduction properties of neurons—called cable properties—and how they determine the spread of action potentials along the axon.

MECHANISMS OF NERVE AND MUSCLE ACTION POTENTIALS

An action potential is a transient depolarization triggered by a depolarization beyond a threshold

The change in membrane potential that occurs during an action potential can be accurately measured by recording V_m with an intracellular microelectrode. Figure 7-1A is a diagram illustrating various features of a typical action potential recorded from an electrically stimulated nerve or muscle cell. If the depolarizing stimulus causes V_m to become more positive than a threshold voltage, the depolarization triggers an action potential. The initial depolarizing (positive-going) phase of an action potential consists of a rapid and smooth increase in V_m from the negative resting potential to a maximum positive value that typically lies between +10 and +40 mV. This sharp rise in V_m to the peak voltage of the action potential is then followed by a slower repolarizing (negative-going) phase. The part of the action potential that lies above 0 mV is called the overshoot. As we will see, the time course and shape of the repolarization phase vary considerably among different excitable tissues and cells. The repolarization phase may lead directly back to V_rest, or it may undershoot and give rise to a voltage minimum that is more negative than V_restbefore relaxing back to V_rest. Such an undershoot is an example of an afterhyperpolarization.

Figure 7-1 The action potential.

The threshold, amplitude, time course, and duration of the action potential depend on the following factors:

1. the gating (opening and closing) and permeability properties of specific types of ion channels—these properties depend on both V_m and time;

2. the intracellular and extracellular concentrations of the ions that pass through these channels, such as Na⁺, K⁺, Ca²⁺, and Cl⁻; and

3. membrane properties such as capacitance, resistance, and the geometry of the cell.

The shape of the action potential in a given cell reflects the specialized functions of that cell. For example, the brief action potentials of a nerve axon permit rapid signaling, whereas the prolonged, repetitive action potentials of cardiac and certain types of smooth muscle cells mediate the slow, rhythmic contractions of these tissues. Figure 7-1B compares action potentials recorded from an invertebrate nerve fiber (unmyelinated squid axon), a vertebrate nerve fiber (myelinated rabbit axon), a skeletal muscle fiber, and a cardiac atrial myocyte. This comparison illustrates the diversity in the duration and shape of the repolarizing phase of action potentials. The shape of the action potential is subject to hormonal modulation in certain cell types. As one example, the peptide hormone endothelin, produced by vascular endothelial cells, shortens the duration of the action potential when it is applied to a guinea pig atrial myocyte. Modulation of the shape and frequency of action potentials occurs by various biochemical regulatory mechanisms that affect the function of ion channels.

In contrast to an action potential, a graded response is proportional to stimulus intensity and decays with distance along the axon

Not all electrical activity in nerve or muscle cells is characterized by an all-or-none response. As shown earlier in Figure 6-12A, when we apply a small square pulse of hyperpolarizing current to a cell membrane, V_m gradually becomes more negative and then stabilizes (Fig. 7-2A). In such an experiment, the observed change in V_m approximates an exponential time course, with a time constant (see Chapter 6) that is determined by the product of membrane resistance and capacitance (τ = RC). Figure 7-2A also shows that progressively greater hyperpolarizing currents produce progressively larger V_m responses, but the time constant is always the same. The size of the graded voltage change (i.e., the steady-state ΔV_m) is proportional to the strength of the stimulus (i.e., the current), in accord with Ohm’s law.

Figure 7-2 Basic properties of action potentials. A, The upper panels show four graded hyperpolarizing stimuli and the V_m responses. The lower panels show four graded depolarizing stimuli and the V_m responses. Note that the two largest stimuli evoke identical action potentials. B, A stimulating electrode injects current at the extreme left of the cell. Four recording electrodes monitor V_m at equidistant sites to the right. If the stimulus is hyperpolarizing, the graded V_m responses decay with distance from the stimulus site. If the stimulus is depolarizing and large enough to evoke an action potential, a full action potential appears at each of the recording sites. However, the action potential arrives at the most distant sites with increasing delay.

If instead of imposing a hyperpolarizing stimulus we impose a small depolarizing stimulus, V_m changes to the same extent and with the same time course as we described for the hyperpolarizing stimulus, but in the opposite direction (Fig. 7-2A). The size of ΔV_m is also proportional to the size of the depolarizing stimulus—up to a point. If the membrane is excitable, a square wave depolarization above the threshold triggers an action potential, or voltage spike. Smaller or subthreshold depolarizations will not elicit an action potential. Hyperpolarizations are always ineffective. Thus, both hyperpolarizations and subthreshold depolarizations behave like graded voltage changes. That is, the magnitude of a cell’s voltage change increases proportionally with the size of the stimulus. Such graded responses can be seen in the response of certain cells to synaptic transmitters, to sensory stimuli (e.g., light), or, in the laboratory, to the injection of current into cells through a microelectrode.

Why do excitable cells exhibit threshold behavior? As V_m becomes progressively more and more positive, the gating process (i.e., transitions from closed to open states) of certain types of voltage-gated ion channels becomes activated. When V_m passes the threshold, opening of these voltage-gated channels initiates the runaway depolarization that characterizes the rising phase of the action potential. Thus, the firing of an action potential is a binary, all-or-none event; that is, the spike has a constant, nongraded voltage peak that occurs only if the depolarizing stimulus exceeds the threshold.

Thus far we have seen that graded responses and action potentials differ markedly from one another if we examine the cell at one particular site. However, graded responses and action potentials also behave very differently in the way that they spread along the membrane from the site of origin. Figure 7-2B illustrates how a graded hyperpolarizing response spreads along the axon of a neuron or along a skeletal muscle fiber. As the graded response spreads, its magnitude decays exponentially with the distance from the site of stimulation because of loss of energy to the medium. This decay is called electrotonic conduction. We see the same kind of electrotonic spread for a subthreshold, depolarizing stimulus. The electrotonic spread of graded responses is governed by the same physical principles that determine the spread of electrical current in an electrical cable. We briefly discuss cable theory at the end of this chapter.

Propagation of an action potential signal is very different from the spread of a graded signal. In a healthy axon or muscle fiber, action potentials propagate at a constant velocity (up to ~130 m/s), without change in amplitude or shape. The amplitude of a propagating action potential does not diminish with distance, as would a graded, subthreshold response, because excitation of voltage-gated channels in adjacent regions of the excitable membrane progressively regenerates the original response. Because the action potential in a given nerve fiber propagates at a constant velocity, the time delay between the stimulus and the peak of the action potential increases linearly with distance from the point of stimulus.

Excitation of a nerve or muscle depends on the product (strength × duration) of the stimulus and on the refractory period

In the preceding section, the importance of the magnitude (intensity) of the depolarizing stimulus emerged as a critical factor for firing of an action potential. However, the duration of the stimulus pulse is also important. A large stimulus is effective in triggering an action potential even at short duration, and a small stimulus may be effective at long duration (Fig. 7-3A). This strength-duration relationship arises because the same minimum electrical charge necessary to excite an action potential can come from a current that is either brief but large or prolonged but small. It is the product of strength and duration that determines excitability, and thus these two parameters are inversely related in their effectiveness. However, regardless of the stimulus strength, successful stimulation requires a minimum duration (vertical asymptote in Fig. 7-3A). Conversely, regardless of the stimulus duration, successful stimulation requires a minimum strength (horizontal asymptote in Fig. 7-3A). (See Note: Rheobase and Chronaxie)

Figure 7-3 Nerve and muscle excitability. The curve in A represents the combination of the minimum stimulus intensity and duration that is required to reach threshold and to evoke an action potential.

An important feature of excitable cells is their ability to fire repetitive action potentials. Once a cell fires an action potential, how quickly can it fire a second? If we were to impose a current step that produced a graded response, we could immediately add a second current step while the first persisted. As long as V_m did not exceed the threshold, the result would be a simple algebraic and instantaneous summation of the two graded responses. The situation for action potentials is quite different. First, action potentials never summate. Second, after one action potential fires, a finite time must elapse before it is possible to trigger a second. The time after initiation of an action potential when it is impossible or difficult to produce a second spike is the refractory period (Fig. 7-3B). The absolute refractory period lasts from initiation of the spike to a time after the peak when repolarization is almost complete. During this time, a second action potential cannot be elicited, regardless of the stimulus strength or duration. After this period, a second action potential can be evoked during the relative refractory period, but the minimal stimulus necessary for activation is stronger or longer than predicted by the strength-duration curve for the first action potential. The two phases of the refractory period arise from the gating properties of particular Na⁺ and K⁺ channels and the overlapping time course of their currents.

The action potential arises from changes in membrane conductance to Na⁺ and K⁺

Approximately 200 years after Luigi Galvani (1737–1798) discovered “animal electricity,” the electrochemical basis of the nerve action potential was finally elucidated by the combined application of modern electrical recording techniques and the theory of electrodiffusion (see Chapter 6). We now understand that the nerve action potential is a phenomenon involving voltage-dependent currents of Na⁺ and K⁺ that flow through distinct molecular pathways called Na⁺ channels and K⁺ channels. In 1963, the Nobel Prize in Physiology or Medicine was awarded to A. L. Hodgkin and A. F. Huxley for their quantitative description of these ionic currents in the squid giant axon in studies involving two-electrode voltage-clamp recordings. Invertebrate axons are unmyelinated, and axons in certain squid nerves have an unusually large diameter (500 to 1000 μm), which allows both external and internal ionic concentrations to be manipulated experimentally. The basic concepts underlying the Hodgkin-Huxley analysis have since been extended to a wide variety of voltage-dependent ionic currents. (See Note: Alan L. Hodgkin and Andrew F. Huxley; Two-Electrode Voltage Clamping)

The squid axon generates a very brief action potential signal without a significant plateau phase (Fig. 7-4). Ionic permeability changes underlying this impulse can be interpreted with a form of the constant-field equation (see Equation 6-9) that includes only Na⁺ and K⁺:

Figure 7-4 Changes in ionic conductance that underlie the action potential. (Data from Hodgkin AL, Huxley AF: A quantitative description of membrane current and its application to conduction and excitation in nerve. J Physiol [Lond] 1952; 117:500-544.)

According to Equation 7-1, the negative resting potential (about –60 mV) of the axon membrane corresponds to a K⁺/Na⁺ permeability ratio (P_K/P_Na) of ~14:1. The change in V_m to a value near +40 mV at the peak of the action potential must involve a transient and selective increase in the permeability to either Na⁺ or Ca²⁺ because the equilibrium potential of these cations lies in the positive voltage range (see Fig. 6-10). Experimentally, if [Na⁺]_o is reduced by replacing it with a nonelectrolyte such as sucrose, the nerve action potential decreases in amplitude. Complementary experiments measuring radioactive tracer fluxes of Na⁺ and K⁺ also demonstrate that action potentials are accompanied by a small influx of Na⁺ and an efflux of K⁺. These and related findings showed that the waveform of the squid action potential is produced by separate permeability pathways for Na⁺ and K⁺.

The time course of the action potential (Fig. 7-4) can be dissected into an initial, transient increase in Na⁺ conductance (and thus permeability), followed by a similar but delayed increase in K⁺ conductance. As one would predict from our discussion of Equation 6-12, a transient increase in Na⁺ conductance—relative to K⁺—would shift V_m toward the positive Na⁺ equilibrium potential (E_Na); the subsequent increase in K⁺ conductance would restore the original negative resting potential, which approaches the K⁺ equilibrium potential (E_K). We can thus attribute the depolarizing and repolarizing phases of the action potential to a transient reversal of the large resting conductance of K⁺ relative to Na⁺.

The Na⁺ and K⁺ currents that flow during the action potential are time and voltage dependent

The assumption of independent permeability pathways—or distinct channels for Na⁺ and K⁺—has been verified by ionic substitution and pharmacological experiments. Figure 7-5 illustrates the use of inhibitors to pharmacologically dissect Na⁺ and K⁺ currents (I_Na and I_K) from the total membrane current (I_m) in a typical excitable membrane preparation, such as a myelinated vertebrate nerve fiber bathed in a normal physiological solution. In a myelinated nerve, these currents flow through small segments of the axon that are not covered with myelin; these segments are called nodes of Ranvier. As we shall see, pharmacological dissection of Na⁺ and K⁺ currents allows us to determine how they vary with time and how they depend on V_m.

Figure 7-5 Dissection of Na⁺ and K⁺ currents by voltage-clamp analysis and pharmacology. A, In a typical voltage-clamp experiment, a sudden hyper polarization from –80 to –140 mV results in a transient capacitative current but no ionic currents. B, In a voltage-clamp experiment, a sudden de polarization from –80 to –20 mV results in a transient capacitative current followed first by an inward ionic current and then by an outward ionic current. C, Blockade of the outward current by TEA leaves only the inward current, which is carried by Na⁺. Conversely, a blockade of the inward current by TTX or STX leaves only the outward current, which is carried by K⁺.

Time Dependence of Na⁺ and K⁺ Currents Stepwise hyperpolarization of the nerve membrane (from a “holding potential” of –80 to –140 mV) by a voltage-clamp technique produces a transient capacitative current (see Chapter 6) but little or no ionic current (Fig. 7-5A). However, a step depolarization of equivalent magnitude produces a capacitative transient current that is followed by a large, time-dependent ionic current (Fig. 7-5B). This ionic current first flows inward, reaches a maximum in the inward direction, and then reverses to the outward direction. The initial inward current corresponds to a movement of cations into the axon. After the reversal of I_m, the outward current corresponds to an outward movement of cations. Ion substitution experiments—in which selected ions are removed from either the outside or the inside of the cell—have shown that the inward current corresponds to Na⁺ current and the outward current corresponds to K⁺ current. Applying a particular organic cation, tetraethylammonium (TEA), to an axon prevents the outward I_K and reveals the isolated inward I_Na (Fig. 7-5C, top I_m record). Conversely, adding either tetrodotoxin (TTX) or saxitoxin (STX)—which we discuss later—abolishes the inward I_Na and reveals the isolated outward I_K (Fig. 7-5C, bottom I_m record). TEA, TTX, and STX are cationic molecules that act as specific ion channel blockers. Millimolar concentrations of TEA block the outer entrance of certain neuronal K⁺ channels, and nanomolar concentrations of TTX (or STX) block the outer entrance of neuronal Na⁺ channels. Biophysical evidence suggests that these particular molecules act by binding in the outer vestibule of their respective channels, thus occluding the channel pore to permeant ions. Thus, the terms channel block and blocking agent are often used to describe their effect.

Voltage Dependence of Na⁺ and K⁺ Currents The ability to use specific inhibitors to resolve separate pathways for Na⁺ and K⁺ current in excitable membranes makes it possible to characterize how these ionic currents depend on V_m. Figure 7-6A illustrates an idealized family of records of total membrane current (I_m) recorded from a myelinated nerve axon. In each case, V_m was initially clamped to –60 mV and then rapidly shifted to a more positive value. The five traces in Figure 7-6A show the current evoked by depolarizations to –45, –30, 0, +30, and +60 mV. By repeating the same experiment in the presence of TEA or TTX, one can obtain the unique time course and voltage dependence of I_Na and I_K.

Figure 7-6 Voltage dependence of ionic currents. A, The top panels show the time course of the total ionic current. This is a voltage-clamp experiment on a frog node of Ranvier. Sudden shifting of V_m from a holding potential of –60 mV to –45, –30, 0, +30, and +60 mV elicits ionic currents that depend on V_m. B, These results are comparable to those in A, except that TEA abolished the outward K⁺ currents, leaving the Na⁺ current. Notice that the peak Na⁺ current varies with V_m. C, These results are comparable to those in A, except that TTX abolished the inward Na⁺ currents, leaving the K⁺ current. Notice that the peak K⁺ current varies with V_m. D, The blue curve is a plot of peak Na⁺ currents from experiments that are similar to those in B. The green curve is a plot of peak K⁺ currents from experiments that are similar to those in C. Notice that both the Na⁺ and K⁺ currents are linear or ohmic in the positive voltage range. In a more negative V_m range, the Na⁺ current exhibits negative resistance, that is, the magnitude of the current becomes more negative rather than more positive as V_m increases in the positive direction. (A-C, Data from Hille B: Common mode of action of three agents that decrease the transient change in sodium permeability in nerves. Nature 1966; 210:1220-1222; and Hille B: The selective inhibition of delayed potassium currents in nerve by tetraethylammonium ions. J Gen Physiol 1967; 50:1287-1302. D, Data from Cole KS, Moore JW: Ionic current measurements in the squid giant axon membrane. J Gen Physiol 1960; 44:123-167.)

The time course of I_Na obtained in the presence of TEA to block K⁺ channels is distinctly biphasic (Fig. 7-6B). Immediately after a depolarizing voltage step to a V_m of –30 mV, for example, the inward I_Na(downward-going) reaches a “peak” value and then returns to zero. The initial phase of this time course (before the peak) is called activation, and the later phase (after the peak) is called inactivation.

In contrast to I_Na, a depolarizing voltage step to a V_m of +60 mV, for example, causes the outward I_K to activate with a definite lag time that gives rise to a sigmoidal time course (Fig. 7-6C). Moreover, I_K takes longer to reach its maximal value (peak). Notice that the K⁺ current is sustained even at the end of the depolarizing pulse. Thus, I_K does not show significant inactivation during the same rapid time scale as does I_Na.

If we plot the peak Na⁺ and K⁺ currents obtained at each of the clamped voltages in Figure 7-6B and C versus the clamped voltages, we obtain the two I-V relationships shown in Figure 7-6D. Because the currents in Figure 7-6B and C represent the activity of many individual ion channels, the plots in Figure 7-6D are macroscopic current-voltage relationships. The I-V relationship for K⁺ is the more straightforward of the two. If we step V_m from –60 mV to increasingly more positive values, the peak I_K is outward and increases with voltage in a monotonic fashion, as expected from Ohm’s law (ΔI = ΔV/R). Because such nerve K⁺ channels pass current in the outward direction and activate with a time delay (Fig. 7-6C) under physiological conditions, the term delayed rectifier K⁺ current (or delayed, outwardly rectifying K⁺ channel) has been coined. We discuss this delayed outward rectifying K⁺ current and the K⁺ channel responsible for it in more detail later.

The voltage dependence of the peak Na⁺ current is biphasic. Stepping V_m from –60 mV to more positive values at first causes I_Na to become increasingly negative (i.e., inward) and then to reach a peak. This portion of the Na⁺ I-Vrelationship is sometimes referred to as the negative resistance region because the negative slope corresponds to an anomalous or negative resistance value according to Ohm’s law (ΔI = ΔV/R). At more positive values of V_m, the peak I_Na reverses direction and becomes more positive, with a nearly linear or ohmic dependence on voltage. (See Note: Ohmic I-V Curve)

Macroscopic Na⁺ and K⁺ currents result from the opening and closing of many channels

The complex macroscopic I-V relationships of the Na⁺ and K⁺ currents (Fig. 7-6D) reflect the single-channel conductance and gating of Na⁺ and K⁺ channels. The pore of an open channel is expected to have a linear or ohmic I-Vrelationship:

Here, i_x is the single-channel current and g_x is the single-channel conductance. We already introduced a similar relationship as Equation 6-15. Figure 7-7A illustrates the predicted linear behavior of single-channel currents as a function of V_m for hypothetical Na⁺ and K⁺ channels. Assuming a Na⁺ reversal potential (E_Na) of +50 mV, the Na⁺ current is zero at a V_m of +50 mV. Similarly, with an E_K of –80 mV, the K⁺ current is zero at a V_m of –80 mV. Assuming a unitary conductance of 20 pS for each channel, the two I-V relationships have the same slope. Note that these idealized single-channel I-V plots for Na⁺ and K⁺approximate the shape of the macroscopic peak I-V relationships of Figure 7-6D for the positive V_m range (i.e., in the right upper quadrant of Fig. 7-6D). In this V_m range, both the Na⁺ and K⁺ channels through which the currents flow are maximally activated at the peaks of their respective time courses. Thus, the macroscopic peak I-V relationships (Fig. 7-6D) are nearly linear in this range, just as they would be for idealized, fully open channels (Fig. 7-7A).

Figure 7-7 The microscopic basis of macroscopic I-V relationships. A, The blue line represents the I-V relationship of an idealized, open Na⁺ channel. The green line represents the I-V relationship of an idealized, open K⁺ channel. Because the channels are assumed to always be fully open (i.e., the conductance does not change with voltage), the current through them is linear or ohmic. B, The blue curve shows the open probability of Na⁺ channels. The equation in the inset will generate this curve if the values z_Na = 6.5 and V_0.5 = –50 mV are inserted. The green curve shows the open probability of K⁺channels. The equation in the inset will generate this curve if the values z_K = 5.3 and V_0.5 = –30 mV are inserted. C, We can obtain a reasonable estimate for the macroscopic Na⁺ current and the macroscopic K⁺ current by multiplying the single-channel current in A, the P_o in B, and the number of channels (N). We assume that there are 100 Na⁺ and 100 K⁺ channels.

However, in the negative voltage range, the macroscopic peak I-V relationships for Na⁺ and K⁺ in Figure 7-6D deviate from the linear (or ohmic) behavior in Figure 7-7A. Why, as the voltage is made more negative, does the inward Na⁺ current fail to increase further and even decrease (negative resistance)? Similarly, why, as the voltage becomes more negative, does the outward K⁺ current fall to zero long before the voltage reaches an E_K of –80 mV? The answer is that the probability that the Na⁺ and K⁺ channels are “open” (P_o)—and therefore able to conduct current—depends on voltage. We introduced the concept of open probability in Chapter 6. To see why V_m might affect P_o, we consider a simplified model.

Assume that a channel protein molecule may exist in either of two conformational states, closed (C) and open (O), and that these two conformational states are in equilibrium with one another:

C O

The equilibrium constant K_eq for this reaction is the ratio of the concentrations of open to closed channels, which can also be expressed as the ratio of the probability that the channel is open (P_o) to the probability that the channel is closed (P_c):

In the case of voltage-gated channel proteins, V_m changes affect K_eq and thus the distribution of channels between the open and closed states. The probability of a channel’s being open depends on V_m, according to a Boltzmann distribution (Fig. 7-7B). (See Note: Boltzmann Distribution of Voltage-Dependent Gating for Ion Channel Proteins)

If the valence (z) of the voltage-sensing part of the channel protein (i.e., the “gating charge”) is positive, the probability of channel opening should increase from 0 to 1 in a sigmoid fashion as V_m becomes more positive. Figure 7-7B shows the behavior of P_o for hypothetical Na⁺ and K⁺ channels that simulate Na⁺ and K⁺ channels in real cells.

To summarize, Figure 7-7A shows that once a single channel is open, the current flowing through the open channel is linearly related to V_m. Figure 7-7B shows that the likelihood that the channel is open depends on V_m in a sigmoid fashion. The actual macroscopic current (I_X) depends on the number of channels (N) in the area of membrane being sampled, the open probability, and the single-channel current, as we already pointed out in Equation 6-21:

Thus, we can use Equation 7-4 to compute the macroscopic currents (I) contributed by our hypothetical Na⁺ and K⁺ channels. We merely multiply the number of channels (which we assume to be 100 for both cations), the open probability for Na⁺ and K⁺ channels in Figure 7-7B, and the single-channel currents for Na⁺ and K⁺ in Figure 7-7A. If we compare the resulting hypothetical I_Na and I_K curves in Figure 7-7C,which are based on a simple theory, with actual data on macroscopic I-V relationships (Fig. 7-6D), we see that this model provides a reasonable description of voltage-sensitive ionic currents.

The Hodgkin-Huxley model predicts macroscopic currents and the shape of the action potential

Even before the concepts of single channels and channel proteins emerged, Hodgkin and Huxley in 1952 formulated voltage-dependent and time-dependent parameters to predict the ionic currents that underlie the action potential of the squid giant axon. Hodgkin and Huxley defined a series of three dimensionless parameters, n, m, and h, each of which can have a value between 0 and 1. The activation parameter ndescribes the probability that the K⁺ channels are open (Fig. 7-8A). The activation parameter m describes the probability that the Na⁺ channels are open (Fig. 7-8B, blue curve). Because Hodgkin and Huxley observed that the Na⁺ current inactivates, they introduced the inactivation parameter h to describe this process (Fig. 7-8B, violet curve). (See Note: Boltzmann Distribution of Voltage-Dependent Gating for Ion Channel Proteins)

Figure 7-8 Voltage-dependent parameters of the Hodgkin-Huxley model and their use in predicting the shape of the action potential. A, The n parameter describes the probability that each of four particles in the K⁺ channel is in the proper state for channel opening. It is believed that these four “particles” are the gates of the four K⁺-channel subunits. The parameter plotted here is the value of n at infinite time. B, The m parameter describes the probability that each of three particles in the Na⁺ channel is in the proper state for channel opening. The hparameter describes the probability that an inactivation particle is not in the proper state for inactivating the Na⁺ channel. Thus, a high h favors the open state of the channel. The parameters plotted here are the values of m and h at infinite time. C, Hodgkin and Huxley used data similar to those in A and B to compute the time course of an action potential in the squid giant axon. D, The actual data are very similar to the computed action potential in C. (Data from Hodgkin AL, Huxley AF: A quantitative description of membrane current and its application to conduction and excitation in nerve. J Physiol 1952; 117:500-544.)

Hodgkin and Huxley developed an equation for total membrane current (I_m) and used it to predict the shape of the action potential in the squid giant axon. Figure 7-8C shows their predicted action potential, which is triggered by a brief depolarization. Figure 7-8D shows an actual recording. The close agreement between the Hodgkin-Huxley (HH) theory and experiment indicates that this model provides a reasonable description of nerve excitation. The fundamental observation of Hodgkin and Huxley was that a rapid increase in Na⁺ conductance causes the upswing or depolarizing phase of the action potential as V_m approaches E_Na, whereas inactivation of Na⁺ conductance and delayed activation of K⁺ conductance underlie the repolarization of V_m to its resting value near E_K. The importance of the HH model in electrophysiology is that it was the first analysis to accurately describe the time course and voltage dependence of ionic currents that occur during an action potential. (See Note: Classical Hodgkin–Huxley Model of the Action Potential)

In addition to delineating the basis of the action potential waveform, the HH model also explains threshold behavior and the refractory period. For an action potential to fire, an external stimulus must depolarize the membrane above threshold to activate a sufficient number of Na⁺ channels. The external stimulus can come from an electrode, a synaptic event, or propagation of a depolarizing wave along the cell membrane. What determines whether a stimulus will be sufficient to reach the threshold V_m for firing of an action potential? The number of Na⁺ channels activated by the stimulus is determined by the voltage dependence of the activation process (i.e., m parameter). Opposing the local depolarization that is produced by the current flowing through these Na⁺ channels are current losses that occur because of passive spread of the current through intracellular and extracellular fluid (see the later discussion of cable theory). Also opposing depolarization is the hyperpolarizing effect of currents through any open K⁺ or Cl⁻ channels in the membrane. Thus, the threshold is the level of depolarization at which the depolarizing effect of the open Na⁺ channels becomes sufficiently self-reinforcing to overcome these opposing influences. Once threshold is reached, further activation of Na⁺ channels rapidly drives V_m toward E_Na.

The basis of the absolute refractory period, the time during which a second action potential cannot occur under any circumstances, is Na⁺ channel inactivation. In other words, it is impossible to recruit a sufficient number of Na⁺channels to generate a second spike unless previously activated Na⁺ channels have recovered from inactivation (i.e., h parameter), a process that takes several milliseconds. The relative refractory period, during which a stronger than normal stimulus is required to elicit a second action potential, depends largely on delayed K⁺ channel opening (i.e., n parameter). In other words, for a certain period after the peak of the action potential, the increased K⁺conductance tends to hyperpolarize the membrane, so a stronger depolarizing stimulus is required to activate the population of Na⁺ channels that in the meantime have recovered from inactivation.

Another key feature of the HH model is that it implies that V_m activates a channel by inducing the movement of an electrically charged gating particle or voltage sensor across the membrane. Physically, this gating could occur by the movement of a charged portion of the channel protein through all or part of the transmembrane electrical field or by the reorientation of an electrical dipole (a neutral structure with positive and negative polarity) within the electrical field of the membrane. Thus, the HH model correctly predicted that activation of a voltage-gated Na⁺ channel or K⁺ channel should be accompanied by a small movement of gating charge, which should produce a gating current. This prediction was satisfied in 1973 when Armstrong and Bezanilla recorded a very small, rapid outward current that is activated by depolarization in a voltage-clamped squid axon in which the ionic current of the Na⁺channels is completely blocked by TTX (Fig. 7-5C, bottom I_m record). This tiny, transient gating current is almost finished by the time that the slower K⁺ current begins to flow. The properties of such gating currents account for the voltage dependence of channel activation kinetics. Although the key features of the HH theory are correct, modern patch-clamp studies of single Na⁺ and K⁺ channels have revealed that the kinetics of channel gating are much more complicated than originally assumed. Such complexity is to be expected inasmuch as the conformational dynamics of large protein molecules cannot generally be adequately described by simple models that incorporate only a few discrete states. (See Note: Evidence for Gating Currents)

PHYSIOLOGY OF VOLTAGE-GATED CHANNELS AND THEIR RELATIVES

A large superfamily of structurally related membrane proteins includes voltage-gated and related channels

In Chapter 6, we previewed families of channels that include voltage-gated Na⁺ channels, Ca²⁺ channels, and K⁺ channels. These voltage-gated channels are part of a larger superfamily of channel proteins called the voltage-gated–like (VGL) ion channel superfamily, which includes additional voltage-gated channels, as well as genetically related channels that are not strictly activated by voltage. Figure 7-9 shows a dendrogram with four branches corresponding to four distinct families belonging to the VGL superfamily. In this section, we discuss how structural relationships among these proteins determine their physiological functions.

Figure 7-9 Family tree of hypothetical evolutionary relationships among voltage-gated cation channels based on sequences of the S4 segments. This dendrogram of the superfamily of voltage-gated channels shows four distinct branches or families. Only a few examples of each are depicted. The family of nucleotide-gated channels is represented by mammalian channels that are gated by cAMP and cGMP. The family of K⁺ channels is presented by four types of Drosophila channels. The family of Na⁺ channels is represented by three types of Na⁺ channels from mammalian brain. Finally, the family of Ca²⁺channels is represented by mammalian channels from skeletal muscle, heart, and brain. (Data from Strong M, Chandy KG, Gutman GA: Molecular evolution of voltage-sensitive ion channel genes: On the origin of electrical excitability. Mol Biol Evol 1993; 10:221-242.)

Initial progress toward biochemical characterization of the voltage-gated ion channels responsible for the action potential began with the discovery of naturally occurring, specific, high-affinity neurotoxins such as TTX and STX and their use as biochemical probes. Tritium-labeled derivatives of TTX and STX were prepared chemically and used in radioligand-binding assays to directly measure the number of voltage-gated Na⁺ channels in excitable tissues.

The electroplax organ of the electric eel (Electrophorus electricus) proved to be a convenient source of tissue for the first successful biochemical purification of the Na⁺ channel protein by Agnew and coworkers in 1978. These Na⁺channels consist of a large glycosylated α subunit of ~200 kDa that contains the TTX binding site. Reconstitution experiments revealed that this subunit—by itself—mediates ionic selectivity for Na⁺, voltage-dependent gating and pharmacological sensitivity to various neurotoxins. Thus, the α subunit is the channel-forming protein. Similar biochemical purification procedures on rat skeletal muscle and brain led to the identification of analogous mammalian Na⁺ channel α subunits, which are protein products of related genes. (See Note: Electroplax Organ of the Electric Eel)

In addition to the α subunit, the functional complex of the rat skeletal muscle Na⁺ channel also contains a 38-kDa subunit, and the rat brain Na⁺ channel contains both a 33-and a 36-kDa subunit. These smaller subunits of mammalian Na⁺ channels are called β subunits and appear to play a role in modulating channel gating or channel expression. In mammals, four genes encode auxiliary β subunits—termed β₁–β₄—that preferentially associate with different α subunits in different tissues.

Molecular biological studies of voltage-gated channels began in 1984 with the cloning of the Electrophorus Na⁺ channel α subunit by the laboratory of Shosaku Numa. These investigators used antibodies raised against the purified α subunit to screen a cDNA library, and they isolated the cDNA encoding the electroplax Na⁺ channel. In addition, direct sequencing of channel peptides provided partial amino acid sequence information. Similar strategies led to the purification and cloning of voltage-gated Ca²⁺ channel proteins from skeletal muscle and brain tissue. The primary sequence of the α₁ subunit of the Ca²⁺channel is structurally homologous to the α subunit of the Na⁺channel.

In contrast to the biochemical approach used for Na⁺ and Ca²⁺ channels, the initial breakthrough in the molecular biology of K⁺ channels came with the study of Shaker mutants of the fruit fly Drosophila.These mutants are called Shaker because their bodies literally shake under the influence of ether anesthesia. This phenotype is due to defective voltage-gated K⁺ channels. The laboratory of L. Y. Jan and Y. N. Jan, and those of O. Pongs and M. Tanouye, used molecular genetic techniques to identify and clone the first K⁺ channel genes in 1987.

The hydropathy (see Chapter 2) plots for voltage-gated K⁺ channels (Fig. 7-10A) typically reveal six distinct peaks of hydrophobicity, corresponding to transmembrane segments S1 to S6—a conserved structural feature of all voltage-gated K⁺ channels. Transmembrane segments S1 to S6 have an α-helical secondary structure and are connected by cytoplasmic and extracellular linker regions (Fig. 7-10B).

Figure 7-10 Membrane topology of a single subunit of a voltage-gated K⁺ channel. A, This voltage-dependent K⁺ channel, a member of the Shaker family (Kv1.1), has six transmembrane segments (S1 to S6) with a high hydropathy index. Each of these six segments (highlighted in green or yellow) is presumed to traverse the membrane completely. In addition, the channel also has a smaller region (highlighted in red) with a somewhat lower hydropathy index, termed the P region. B, This model is based on the hydropathy data in A. The six membrane-spanning segments are assumed to be α helices. The S4 segment (highlighted in yellow) has a large number of positively charged lysine and arginine residues and is part of the voltage-sensing domain that comprises the entire S1–S4 region. S5 and S6—as well as the intervening P region—comprise the pore domain (see the box on page 191), which lines the pore of the channel. (Data from Shen NV, Pfaffinger PJ: Conservation of K+ channel properties in gene subfamilies. In Peracchia C [ed]: Handbook of Membrane Channels: Molecular and Cellular Physiology, pp 5-16. New York: Academic Press, 1994.)

Extensive mutagenesis studies on cloned channel genes have associated various channel functions and binding sites with particular domains. The S4 segment (Fig. 7-10) has four to seven arginine or lysine residues that occur at every third S4 residue in voltage-gated K⁺, Na⁺, and Ca²⁺ channels. Functional evidence indicates that these positively charged residues of the S4 segment have a major role in the voltage-sensing mechanism of channel activation.

The extracellular linker region between the S5 and S6 segments is termed the P region (for pore region) and contains residues that form the binding sites for toxins and external blocking molecules such as TEA. The P region also contains residues that are critical determinants of the ionic selectivity for permeant cations. Structural evidence indicates that the S6 transmembrane segment forms the internal aspect of the ion conduction pathway.

Since the discovery and recognition of diverse genes belonging to the voltage-gated channel superfamily, structural-biological studies have substantially advanced our understanding of the three-dimensional structure of certain channel proteins. In 1998, a major breakthrough in the structure of ion channel proteins occurred when MacKinnon and colleagues reported the crystal structure of a bacterial K⁺ channel protein called KcsA. This work revealed the three-dimensional structure of a protein that contained segments analogous to the S5-P-S6 part of voltage-gated channels, which forms the ion conduction pathway. For his work on the structural biology of ion channels, Roderick MacKinnon shared the 2003 Nobel Prize in Chemistry. (See Note: Roderick MacKinnon)

In 2005, another breakthrough by the MacKinnon laboratory revealed the entire structure of a mammalian voltage-gated K⁺ channel containing both the S1–S4 voltage-sensing domain and the S5-P-S6 pore domain (see Fig. 7-11and the box on page 191).

Figure 7-11 Crystal structure of the mammalian K⁺ channel, Kv1.2, at a resolution of 2.9 Å. A, Four α subunits of the channel, each in a unique color viewed from the extracellular side; a K⁺ ion is shown in the central open pore. B, Side view of the four α and four β subunits of the channel, each in a unique color with extracellular solution on the top and intracellular solution on the bottom. The transmembrane domain (TM) of each α subunit is preceded by an NH₂ terminus (T1 domain). The T1 domain is located over the intracellular entryway to the pore but allows access of K⁺ ions to the pore through “side portals.” The T1 domain is also a docking platform for the oxidoreductase β subunit. Each β subunit is colored according to the α subunit it contacts. C shows a side view of one α subunit and adjacent β subunit. Transmembrane segments are labeled S1 to S6. Tetramers of segments S5, pore helix, and S6 constitute the conduction pore in the shape of an inverted “teepee.” The selectivity filter lies in the wide portion (extracellular end) of the teepee. Helices S1 to S4 constitute the voltage sensors that are connected by a linker helix (S4-S5) to the pore. The PVP sequence (Pro-Val-Pro) on S6 is critical for gating. (From Long SB, Campbell EB, MacKinnon R: Crystal structure of a mammalian voltage-dependent Shaker family K⁺ channel. Science 2005; 309:897-903.)

Figure 7-12 shows a comparison of the predicted membrane-folding diagrams of three families of voltage-gated channels: Na⁺, Ca²⁺, and K⁺ channels. The channel-forming subunit of each type of channel is called the α subunit for Na⁺ and K⁺ channels and the α₁ subunit for Ca²⁺ channels. Other identified accessory subunits are designated β₁ and β₂ for Na⁺ channels; α₂, β, γ, and δ for Ca²⁺ channels; and β for K⁺channels.

Figure 7-12 Subunit structure and membrane-folding models of voltage-gated channels. A, A voltage-gated Na⁺ channel is made up of a pseudo-oligomeric α subunit as well as membrane-spanning β₁ and β₂ subunits. Note that the domains I to IV of the α subunit are homologous to a single subunit of a voltage-gated K⁺ channel (see C). B, A voltage-gated Ca²⁺ channel is made up of a pseudo-oligomeric α₁ subunit as well as an extracellular α₂ subunit, a cytoplasmic β subunit, and membrane-spanning γ and δ subunits. Note that the domains I to IV of the α subunit are homologous to a single subunit of a voltage-gated K⁺ channel (see C). C, A voltage-gated K⁺ channel is made up of four α subunits as well as a cytoplasmic β subunit. (Data from Isom LL, De Jongh KS, Catterall WA: Auxiliary subunits of voltage-gated ion channels. Neuron 1994; 12:1183-1194.)

The α and α₁ subunits of this protein superfamily all contain the common S1-S6 structural motif composed of the S1-S4 voltage-sensing domain and the S5-P-S6 pore domain that we described earlier for K⁺channels. The α subunit of Na⁺ channels (Fig. 7-12A) and the α₁ subunit of Ca²⁺ channels (Fig. 7-12B) consist of four internally homologous repeats—domains I, II, III, and IV—each containing an S1-S6 motif. K⁺ channels (Fig. 7-12C) are likely to be an evolutionary precursor of the voltage-gated channel families inasmuch as their pore-forming α subunit contains only one S1-S6 motif. Voltage-gated K⁺channels are homo-oligomers of four α subunits, a tetramer (Fig. 7-11). Because Na⁺ and Ca²⁺ channels are composed of four internally homologous repeats of S1-S6, all α subunits of these families function as either tetrameric (K⁺ channels) or pseudotetrameric (Na⁺ and Ca²⁺ channels) units. Molecular evolution of the pseudotetrameric structure of Na⁺ and Ca²⁺ channels is believed to have occurred by consecutive gene duplication of a primordial gene containing the basic S1-S6 motif.

Crystal Structure of a Mammalian K⁺ Channel

In 2005, the MacKinnon laboratory solved the crystal structure of a rat voltage-gated K⁺ channel called Kv1.2, which is homologous to the Drosophila Shaker channel. This structure, which shows the channel in an open state, reveals that the S1-S4 domain containing the voltage-sensing S4 element is spatially separated from the K⁺ pore domain (S5-P-S6). The tetrameric Kv1.2 channel has a pinwheel shape when it is viewed from the extracellular surface (Fig. 7-11A). The central square portion of the Kv1.2 pinwheel is the pore—formed by the assembly of four S5-P-S6 domains, one from each monomer—and closely resembles the entire bacterial KcsA channel. The four wings of the pinwheel correspond to the four S1-S4 voltage sensor domains. The four Kv1.2 monomers (yellow, green, blue, and red in Figure 7-11A) form an interlinked assembly in which the S1-S4 voltage-sensing domain of any given monomer lies closest to the S5-P-S6 domain of an adjacent monomer. (See Note: Crystal Structure of the KcsA K⁺ Channel)

A lateral view of Kv1.2 shows an intracellular T1 domain formed by the four N-terminal segments of the channel (Fig. 7-11B). The T1 domain of Kv channels is also called the tetramerization domain because it helps assemble and maintain the tetrameric structure of the channel. This view also shows four separately attached intracellular β subunits. These β subunits of Kv channels are part of a separate gene family of soluble accessory proteins with structural homology to oxidoreductase enzymes. Certain variants of both the T1 domain and β subunits may contain an N-terminal inactivation peptide that produces the rapid N-type inactivation (ball-and-chain mechanism) of some Kv channels by plugging the intracellular entrance to the pore.

Figure 7-11C shows a lateral view of a single Kv1.2 monomer in an open configuration as well as a single β subunit. On depolarization, the S4 segment presumably moves within the membrane toward the extracellular side of the membrane. This mechanical movement of the S4 segment shifts an α-helical S4-S5 linker, causing a bending of the S6 transmembrane α helix from a linear configuration in the closed state to a curved configuration in the open state of the channel shown. Thus, voltage-dependent channel activation is an electromechanical coupling mechanism.

Na⁺ channels generate the rapid initial depolarization of the action potential

Because the equilibrium potential for Na⁺ and Ca²⁺ is in the positive voltage range for normal cellular ionic gradients, channels that are selectively permeable to these ions mediate electrical depolarization. However, prolonged cellular depolarization is an adverse condition inasmuch as it results in sustained contraction and rigor of muscle fibers, cardiac dysfunction, and abnormally elevated levels of intracellular Ca²⁺, which leads to cell death. Thus, it is critical that Na⁺ and Ca²⁺ channels normally reside in a closed conformation at the resting membrane potential. Their opening is an intrinsically transient process that is determined by the kinetics of channel activation and inactivation.

The primary role of voltage-gated Na⁺ channels is to produce the initial depolarizing phase of fast action potentials in neurons and skeletal and cardiac muscle. The selectivity of Na⁺ channels for Na⁺ is much higher than that for other alkali cations. The permeability ratio of Na⁺ relative to K⁺ (P_Na/P_K) lies in the range of 11 to 20 under physiological conditions. Voltage-gated Na⁺ channels are virtually impermeable to Ca²⁺ and other divalent cations under normal physiological conditions.

Although Na⁺ channels do not significantly conduct Ca²⁺ ions across the cell membrane, the voltage dependence of Na⁺ channel gating is nevertheless dependent on the extracellular Ca²⁺concentration ([Ca²⁺]_o). If [Ca²⁺]_o is progressively increased above the normal physiological level, the voltage activation range of Na⁺ channels progressively shifts to a more positive range. In Figure 7-13, this change is represented as a shift in the P_o versus V_mrelationship. Similarly, if [Ca²⁺]_o is decreased, the voltage activation range is shifted to more negative voltages. This phenomenon has important clinical implications because a negative shift corresponds to a reduced voltage threshold for action potential firing and results in hyperexcitability and spontaneous muscle twitching. Similarly, a positive voltage shift of Na⁺ channel gating corresponds to decreased electrical excitability (i.e., the threshold is now farther away from resting V_m), resulting in muscle weakness. Thus, metabolic disorders that result in abnormal plasma [Ca²⁺], such as hypoparathyroidism (low [Ca²⁺]) and hyperparathyroidism (high [Ca²⁺]), can cause marked neurologic and neuromuscular symptoms. The mechanism of this voltage shift in Na⁺ channel gating by extracellular divalent cations such as Ca²⁺ is thought to involve an alteration in the transmembrane electrical field that is sensed by the channel protein. Presumably, this effect is caused by Ca²⁺ binding or electrostatic screening of negative charges at the membrane surface.

Figure 7-13 Effect of extracellular Ca²⁺ concentration on Na⁺ channel activation. High [Ca²⁺]_o shifts the P_o versus V_m to more positive voltages (e.g., less excitable). Thus, hypo calcemia leads to hyper excitability.

Humans have at least ten homologous genes that encode the pore-forming α subunit of Na⁺ channels (Table 7-1). The isoforms encoded by these genes are expressed in different excitable tissues and can be partially discriminated on the basis of their sensitivity to TTX. Four of the isoforms (Nav1.1, 1.2, 1.3, and 1.6) are differentially expressed in various regions of the brain. One isoform (Nav1.4) is the major isoform in skeletal muscle. This muscle Na⁺channel is also uniquely sensitive to blockade by a peptide toxin called μ-conotoxin from a venomous marine snail. Natural mutations in the human gene for this Na⁺channel result in a variety of human genetic diseases, such as hyperkalemic periodic paralysis, and in several types of myotonia (see the box titled Na⁺ Channel Genetic Defects). Heart ventricular muscle expresses a TTX-insensitive isoform (Nav1.5) that also appears in skeletal muscle after denervation. Various natural mutations in the heart Na⁺ channel cause irregularities in heartbeat characterized by a particular type of long QT syndrome. Neurons from the dorsal root ganglia express Nav1.6, 1.7, 1.8, and 1.9 Na⁺channel isoforms, the last two of which are TTX insensitive. Various natural mutations in human Nav1.7 underlie genetic diseases characterized either by enhanced sensitivity to pain or deficiency in the perception of pain, indicating a role for Nav1.7 in nociception. (See Note: Effects of μ-Conotoxin)

Table 7-1 Na⁺ Channel α Subunits

Na⁺ channels are blocked by neurotoxins and local anesthetics

Studies of the mechanism of action of neurotoxins have provided important insight into channel function and structure. The guanidinium toxins TTX and STX (Fig. 7-5C) are specific blocking agents of Na⁺channels that act on the extracellular side of the cell membrane.

TTX is produced by certain marine bacteria and is apparently accumulated in some tissues of various invertebrates, amphibians, and fish. The internal organs of certain fish, such as the puffer fish that is consumed in Japan, often contain lethal amounts of TTX. The flesh of such fish must be carefully prepared to prevent food poisoning.

STX is produced by specific species of marine dinoflagellates that are responsible for “red tide” in the ocean as well as by freshwater cyanobacteria, which can poison ponds and rivers. It is the agent responsible for paralytic shellfish poisoning, which is caused by human ingestion of toxic shellfish that have accumulated STX-producing plankton. Death from TTX and STX intoxication, which ultimately results from respiratory paralysis, can be prevented by the timely administration of mechanical respiration.

As mentioned earlier, the snail peptide μ-conotoxin similarly blocks muscle Na⁺ channels by binding near the external binding site for TTX and STX.

TTX, STX, and μ-conotoxin are important pharmacological probes because they can be used to functionally discriminate among several distinct isoforms of Na⁺ channels (Table 7-1). Other important neurotoxins that act on Na⁺channels include batrachotoxin (a steroidal alkaloid from certain tropical frogs and birds), various plant alkaloids (veratridine, grayanotoxin, aconitine), natural plant insecticides (pyrethrins), brevetoxins (cyclic polyethers from dinoflagellates), and two distinct classes (α and β) of peptide scorpion toxins. Members of this diverse group of neurotoxins act primarily by altering the gating kinetics of Na⁺ channels by promoting both a longer duration of channel opening and channel opening under voltage conditions in which Na⁺ channels are normally closed or inactivated.

Local anesthetics are a large group of synthetic drugs that are generally characterized by an aromatic moiety linked to a tertiary amine substituent through an ester or amide linkage (Fig. 7-14A). Drug development of local anesthetics began with the recognition by Carl Koller in 1884 that the plant alkaloid cocaine numbs sensation in the tongue, in addition to producing psychoactive effects by its actions on the central nervous system (CNS). Attempts to synthesize safer alternatives to cocaine led to procaine, which mimics the local anesthetic effect of cocaine without the CNS effects.

Figure 7-14 Effect of local anesthetics. A, The three clinically useful local anesthetics shown here are synthetic analogues of the plant alkaloid cocaine. B, In the presence of lidocaine, the relative Na⁺ current decays with time during repetitive stimulation. However, the inhibition becomes more pronounced as the rate of stimulation increases from 1/s to 8/s. (Data from Hille B: Local anesthetics: Hydrophilic and hydrophobic pathways for the drug-receptor reaction. J Gen Physiol 1977; 69:497-515.)

Local anesthetics that are used clinically, such as procaine, lidocaine, and tetracaine, reversibly block nerve impulse generation and propagation by inhibiting voltage-gated Na⁺ channels. The action of these drugs is “use dependent,” which means that inhibition of Na⁺ current progresses in a time-dependent manner with increasing repetitive stimulation or firing of action potentials (Fig. 7-14B). Use dependenceoccurs because the drug binds most effectively only after the Na⁺ channel has already opened. This use-dependent action of the drug further enhances inhibition of nerve impulses at sites where repetitive firing of action potentials takes place. Local anesthetics are used to control pain during dental procedures, many types of minor surgery, and labor in childbirth.

Ca²⁺ channels contribute to action potentials in some cells and also function in electrical and chemical coupling mechanisms

Ca²⁺ channels play important roles in the depolarization phase of certain action potentials, in coupling electrical excitation to secretion or muscle contraction, and in other signal transduction processes. Because [Ca²⁺]_o is ~1.2 mM, whereas [Ca²⁺]_i is only ~10⁻⁷ M, a huge gradient favors the passive influx of Ca²⁺ into cells. At the relatively high [Ca²⁺]_o that prevails under physiological conditions, voltage-gated Ca²⁺ channels are highly selective for Ca²⁺, with permeability to Ca²⁺ being ~1000-fold greater than permeability to Na⁺. Other alkaline earth divalent cations such as Sr²⁺ and Ba²⁺ also readily permeate through Ca²⁺ channels and are often used as substitute ions for recording the activity of Ca²⁺ channels in electrophysiological studies. However, if [Ca²⁺]_o is reduced to a nonphysiological level of less than 10⁻⁶M with the use of chelating agents, Ca²⁺ channels can also conduct large currents of monovalent alkali cations, such as Na⁺ and K⁺. Thus, in terms of its intrinsic ionic selectivity, the Ca²⁺ channel is functionally similar to the Na⁺ channel, except that high-affinity binding of Ca²⁺ in the pore effectively prevents permeation of all other physiological ions except Ca²⁺.

The mechanism of this extraordinary selectivity behavior is based on ion-ion interactions within the pore. For the Ca²⁺ channel to conduct current, at least two Ca²⁺ ions must bind simultaneously close to sites within the channel. Interactions between individual ions within the narrow region of the channel pore appear to control ion selectivity and the ionic flux. Variations on this general mechanism, referred to as multi-ion conduction, have also been described for many other classes of ion channels, notably K⁺ channels. Multi-ion conduction generally appears to play an important role in determining the permeation properties of channels that have a high degree of ionic selectivity, such as Ca²⁺ channels and K⁺ channels.

Na⁺ Channel Genetic Defects

Several human genetic diseases have been traced to inheritable defects in the genes for skeletal and cardiac muscle Na⁺ channels. The skeletal muscle gene SCN4A is located on human chromosome 17, and the cardiac muscle gene SCN5A is located on chromosome 3. One of the muscle disorders is called hyperkalemic periodic paralysis because muscle weakness is triggered by an elevation in serum [K⁺] that may occur after vigorous exercise or ingestion of foods rich in K⁺. A second muscle disorder is called paramyotonia congenita. This form of periodic paralysis may be induced in afflicted individuals by exposure to cold temperature and results in symptoms of myotonia (muscle stiffness) associated with abnormal repetitive firing of muscle action potentials. Long QT syndrome is an inherited defect in heart rhythm that can lead to sudden death from cardiac arrhythmia. A deletion of three amino acids, ΔKPQ, in the linker region between repetitive domains III and IV of the heart Na⁺ channel is one type of mutation that causes this disease. We will see later in the box titled Human Heart Defects Linked to Mutations of K⁺ Channels that defects in cardiac K⁺ channels can also cause a long QT syndrome. As shown in Figure 7-15, a number of mutations responsible for skeletal muscle diseases have also been identified and mapped within the folding diagram of the muscle Na⁺ channel α subunit. These mutations generally occur in one of the putative membrane-spanning segments (S3, S4, S5, and S6). Two paramyotonia congenita mutations have also been located in the intracellular linker segment between repeats III and IV; this linker plays an important role in Na⁺ channel inactivation. Electrophysiological analysis of some of these mutations suggests that abnormal kinetics of Na⁺ channel gating is the underlying cause of the profound symptoms associated with these diseases. For example, the occasional failure of the mutant heart Na⁺ channel to inactivate results in long bursts of openings and abnormal prolongation of the action potential. (See Note: Erythromelalgia)

Figure 7-15 Some naturally occurring mutations of human Na⁺ channels. Mutations in the Na⁺ channel of human skeletal muscle can cause at least two genetic diseases. Hyperkalemic periodic paralysis can be caused by mutations in membrane-spanning segment S5 of domain II and S6 of domain IV. Paramyotonia congenita can be caused by mutations in membrane-spanning segment S3 of domain IV and S4 of domain IV and also by mutations in the intracellular segment that links domains III and IV. (Data from Catterall WA: Cellular and molecular biology of voltage-gated sodium channels. Physiol Rev 1992; 72:S15-S48.)

One of the major functions of voltage-gated Ca²⁺ channels is to contribute to the depolarizing phase of action potentials in certain cell types. The gating of voltage-gated Ca²⁺ channels is slower than that of Na⁺channels. Whereas Na⁺ channels are most important in initiating action potentials and generating rapidly propagating spikes in axons, Ca²⁺ channels often give rise to a more sustained depolarizing current, which is the basis for the long-lived action potentials observed in cardiac cells, smooth muscle cells, secretory cells, and many types of neurons.

The exquisite selectivity of Ca²⁺ channels under physiological conditions endows them with special roles in cellular regulation. If a depolarizing electrical stimulus or a signal transduction cascade activates these Ca²⁺ channels, the subsequent influx of Ca²⁺ raises [Ca²⁺]_i, and Ca²⁺ can thereby serve as an important second messenger in regulating the activity of a multitude of intracellular proteins and enzymes. Thus, in serving as a major gateway for Ca²⁺influx across the plasma membrane, Ca²⁺ channels have not only an electrical function in membrane depolarization but also an important biochemical function in signal transduction.

Ca²⁺ channels also play a pivotal role in a special subset of signal transduction processes known as excitation-contraction coupling and excitation-secretion coupling. Excitation-contraction (EC) coupling refers to the process by which an electrical depolarization at the cell membrane leads to cell contraction, such as the contraction of a skeletal muscle fiber. In EC coupling of skeletal muscle, one class of plasma membrane Ca²⁺ channel that is located in the transverse tubule membrane of skeletal muscle serves as the voltage sensor and forms a direct structural linkage to intracellular Ca²⁺ release channels that are located in the sarcoplasmic reticulum membrane. In contrast, Ca²⁺channels play a different role in EC coupling in cardiac muscle, where Ca²⁺ channels in the plasma membrane mediate an initial influx of Ca²⁺. The resultant increase in [Ca²⁺]_i triggers an additional release of Ca²⁺ stored in the sarcoplasmic reticulum by a process known as Ca²⁺-induced Ca²⁺ release (see Chapter 9).

Excitation-secretion coupling is the process by which depolarization of the plasma membrane causes release of neurotransmitters in the nervous system and the secretion of hormones in the endocrine system. Such processes require an increase in [Ca²⁺]_i through the plasma membrane to trigger exocytosis of synaptic and secretory vesicles. Thus, in providing a primary signal for the initiation of cellular contraction and neurotransmitter/hormone release, Ca²⁺ channels are a fundamental locus of control.

Because Ca²⁺ channels must fulfill diverse roles, higher vertebrates use a family of genes that encode structurally homologous but functionally diverse Ca²⁺ channels. Mammals have at least 10 distinct genes for the channel-forming α₁ subunit of Ca²⁺ channels (Table 7-2). Biochemical and cloning work has also identified four accessory subunits of Ca²⁺ channels: α₂, δ, β, and γ (Fig. 7-12B). The α₂ and δ subunits are the products of a single gene; after translation, proteolytic cleavage of the polypeptide yields α₂ and δ. Coexpression studies have shown that these accessory subunits can greatly influence the kinetics, voltage sensitivity, and peak currents that are exhibited by various α₁ channel subunits. This structural complexity and diversity at the genetic level are mirrored by a diversity of Ca²⁺ currents that have been differentiated in various cell types on the basis of their functional characteristics.

Table 7-2 Properties and Classification of Ca²⁺ Channel α Subunits

Ca²⁺ channels are characterized as L-, T-, P/Q-, N-, and R-type channels on the basis of kinetic properties and inhibitor sensitivity

An example of the functional diversity of Ca²⁺ channels is illustrated in Figure 7-16, which shows two different types of voltage-gated Ca²⁺ channels that have been identified in cardiac ventricular cells by the patch-clamp technique. If the cell-attached patch, initially clamped at –50 mV, is suddenly depolarized to +10 mV, currents appear from a large-conductance (18 to 25 pS), slowly inactivating Ca²⁺ channel (Fig. 7-16A). However, if the same patch is initially clamped at –70 mV and depolarized to only –20 mV, currents appear instead from a small-conductance (8 pS), rapidly inactivating Ca²⁺ channel (Fig. 7-16B). These two types of Ca²⁺ channels are respectively named L-type (for long-lived) and T-type (for transient) channels. T-type channels are activated at a lower voltage threshold (more negative than –30 mV) than are other types of Ca²⁺ channels and are also inactivated over a more negative voltage range. These characteristics of T-type channels permit them to function briefly in the initiation of action potentials and to play a role in the repetitive firing of cardiac cells and neurons. Other types of Ca²⁺ channels, including L-, N-, P/Q-, and R-type channels, which are activated at a higher voltage threshold (more positive than –30 mV), mediate the long-lived plateau phase of slow action potentials and provide a more substantial influx of Ca²⁺ for contractile and secretory responses. N-, P/Q-, and R-type Ca²⁺channels appear to mediate the entry of Ca²⁺ into certain types of presynaptic nerve terminals and thus play an important role in facilitating the release of neurotransmitters.

Figure 7-16 Current records from two types of Ca²⁺ channel. A, This is an experiment on guinea pig ventricular myocytes in which cell-attached patches were used. The authors studied the currents that are carried by Ba²⁺ through these L-type Ca²⁺ channels because they conduct Ba²⁺ even better than Ca²⁺. Shown in the middle panel are seven single-channel current records that were obtained during and after a shift of the cytosolic voltage from –50 to +10 mV. Note that the channel activity (i.e., downward deflections) begins only after depolarization and continues more or less at the same level throughout the depolarization. The lower panel shows the average of many records that are similar to those shown in the middle panel. B, The experiments summarized for these T-type Ca²⁺ channels were identical in design to those shown in A, except that the depolarizing step shifted cytosolic voltage from –70 to –20 mV. Note that once again, the channel activity begins only after depolarization (middle panel). However, the channel activity is transient; it wanes during a sustained depolarization, as confirmed by the average current shown in the lower panel. (Data from Nilius B, Hess P, Lansman JB, Tsien RW: A novel type of cardiac calcium channel in ventricular cells. Nature 1985; 316:443-446.)

In addition to discrimination on the basis of gating behavior, Ca²⁺ channel isoforms can also be distinguished by their sensitivity to different drugs and toxins (Table 7-2). Ca²⁺ channel blockers are an important group of therapeutic agents. Figure 7-17 shows the structures of representatives of three different classes of Ca²⁺ channel blockers: 1,4-dihydropyridines (DHPs), phenylalkylamines, and benzothiazepines. These synthetic compounds are used in the treatment of cardiovascular disorders such as angina pectoris (see Chapter 24) and hypertension and also are being evaluated for their potential in treatment of various diseases of the CNS.

Figure 7-17 Antagonists and agonists of L-type Ca²⁺ channels. A, 1,4-Dihydropyridines. One, nitrendipine, is an antagonist; another, Bay K8644, is an agonist. B, Phenylalkylamines. Verapamil is an antagonist. C, Benzothiazepines. Diltiazem is an antagonist.

DHPs such as nitrendipine selectively block L-type Ca²⁺ channels. Phenylalkylamines (e.g., verapamil) and benzothiazepines (e.g., diltiazem) also inhibit L-type Ca²⁺ channels; however, these other two classes of drugs act at sites that are distinct from the site that binds DHPs. Particular DHP derivatives, such as Bay K8644, actually enhance rather than inhibit Ca²⁺ channel currents. DHPs can have the contrasting effects of either inhibitors (antagonists) or activators (agonists) because they act not by plugging the channel pore directly but by binding to a site composed of transmembrane helices S5 and S6 in domain III and S6 in domain IV. Drug binding in this region probably induces various conformational changes in channel structure and thereby perturbs Ca²⁺ permeation and gating behavior.

Other molecules that are useful in discriminating Ca²⁺ channel isoforms are present in the venom of the marine snail Conus geographus and the funnel web spider Agelenopsis aperta. The snail produces a peptide called ω-conotoxin GVIA, which selectively blocks N-type Ca²⁺ channels; the spider produces the peptide ω-agatoxin IVA, which selectively blocks P/Q-type Ca²⁺ channels. In contrast, an R-type neuronal Ca²⁺ channel is resistant to these two peptide toxins.

The summary of the basic properties of L-, T-, N-, P/Q-, and R-type Ca²⁺ channels contained in Table 7-2 indicates their presumed correspondence to 10 known genes that encode α₁ subunits.

K⁺ channels determine resting potential and regulate the frequency and termination of action potentials

K⁺ channels are the largest and most diverse family of voltage-gated ion channels. Humans have at least 78 distinct genes encoding K⁺ channels with the complete S1 to S6 motif. Ion conduction through most types of K⁺ channels is very selective for K⁺ according to the permeability sequence K⁺ > Rb⁺ > NH₄⁺ >> Cs⁺ > Li⁺, Na⁺, Ca²⁺. Under normal physiological conditions, the permeability ratio P_K/P_Na is greater than 100 and Na⁺ can block some K⁺ channels. Some K⁺ channels can pass Na⁺ current in the complete absence of K⁺. This finding is analogous to the behavior of Ca²⁺ channels, which can pass Na⁺ and K⁺currents in the absence of Ca²⁺.

Given such strong K⁺ selectivity and an equilibrium potential near –80 mV, the primary role of K⁺ channels in excitable cells is inhibitory. K⁺ channels oppose the action of excitatory Na⁺ and Ca²⁺ channels and stabilize the resting, nonexcited state. Whereas some K⁺ channels are major determinants of the resting potential, the voltage dependence and kinetics of other K⁺ channels in excitable cells have specialized functions, such as mediating the repolarization and shaping of action potentials, controlling firing frequency, and defining the bursting behavior of rhythmic firing. Such functions are broadly important in regulating the strength and frequency of all types of muscle contraction, in terminating transmitter release at nerve terminals, and in attenuating the strength of synaptic connections. Finally, in epithelia, K⁺ channels also function in K⁺ absorption and secretion.

Ca²⁺ Channel and Autoimmune Genetic Defects

Ca²⁺ channels have been linked to a large variety of genetic diseases. In mice, an interesting mutation results in muscular dysgenesis, or failure of normal skeletal muscle to develop. These mice lack a functional Ca²⁺ channel α₁ subunit in their skeletal muscle. They die shortly after birth, but their cultured muscle cells provide an assay system to investigate the mechanism of EC coupling. Contraction of such defective muscle cells can be rescued by expression of cloned genes for either the skeletal Cav1.1 (CACNA1S gene) or the cardiac Cav1.2 (CACNA1C gene) L-type Ca²⁺ channels. As discussed in Chapter 9, a physiologically distinguishing feature of EC coupling in normal skeletal versus cardiac muscle is that skeletal muscle does not require extracellular Ca²⁺, whereas cardiac muscle does. Indeed, when the rescue is accomplished with skeletal Cav1.1, contraction does not require extracellular Ca²⁺; when the rescue is accomplished with cardiac α_1C, contraction does require extracellular Ca²⁺. Such studies have provided strong support for the concept that EC coupling in skeletal muscle takes place by direct coupling of Cav1.1 to the Ca²⁺ release channels of the sarcoplasmic reticulum; in cardiac muscle, EC coupling occurs as Ca²⁺ entering through α_1C-containing channels induces the release of Ca²⁺ from internal stores. Mutagenesis experiments with chimeric α₁subunits containing artificially spliced segments of the cardiac and skeletal channel isoforms have shown that the intracellular linker region between repeats II and III is the domain of the α₁ subunit that determines the skeletal versus the cardiac type of EC coupling.

A human pathologic condition called Lambert-Eaton syndrome has been characterized as an impairment of presynaptic Ca²⁺ channels at motor nerve terminals. Lambert-Eaton syndrome is an autoimmune disorder that is most often seen in patients with certain types of cancer, such as small cell lung carcinoma. Patients afflicted with this condition produce antibodies against presynaptic Ca²⁺ channels that somehow reduce the number of such channels able to function in the depolarization-induced influx of Ca²⁺ for neurotransmitter release.

Hypokalemic periodic paralysis (not to be confused with hyperkalemic periodic paralysis, discussed earlier in the box titled Na⁺ Channel Genetic Defects) is an autosomal dominant muscle disease of humans. Affected family members have a point mutation in the CACNA1S gene encoding the skeletal Cav1.1, located in transmembrane segment S4 of domain II. This finding explains the basis for a human disorder involving defective EC coupling of skeletal muscle. Certain other rare human genetic diseases result in neurologic symptoms of migraine (severe headache) and ataxia (a movement disorder). One of these diseases, familial hemiplegic migraine, is caused by point mutations at various locations in the human CACNA1A gene encoding Cav2.1. These locations include the S4 region of domain I, the P region of domain II, and the S6 helices of domains I and IV. Another such genetic disease caused by mutations in the human CACNA1A gene encoding Cav2.1 is called episodic ataxia type 2, a condition associated with the occurrence of ataxia originating from the cerebellum. Discovery of the genetic origin of such diseases has led to the realization that delicate perturbations of Ca²⁺ channel activity can have profound consequences on proper function of the human nervous system.

Before molecular cloning revealed the structural relationships among the various kinds of K⁺ channels, electrophysiologists classified K⁺ currents according to their functional properties and gating behavior. They grouped the macroscopic K⁺ currents into four major types:

1. delayed outward rectifiers;

2. transient outward rectifiers (A-type currents);

3. Ca²⁺-activated K⁺ currents; and

4. inward rectifiers.

These four fundamental K⁺ currents are the macroscopic manifestation of five distinct families of genes (Table 6.2):

1. Kv channels (voltage-gated K⁺ channels related to the Shaker family);

2. Small conductance K_Ca channels (Ca²⁺-activated K⁺ channels), including, SK_Ca and IK_Ca channels;

3. Large-conductance K_Ca channels (Ca²⁺-activated K⁺ channels, including BK_Ca and Na⁺-activated K⁺ channels);

4. Kir channels (inward rectifier K⁺ channels); and

5. K2P channels (two-pore K⁺ channels).

In the next three sections, we discuss the various families of K⁺ channels and their associated macroscopic currents.

The Kv (or shaker-related) family of K⁺ channels mediates both the delayed outward rectifier current and the transient A-type current

The K⁺ current in the HH voltage-clamp analysis of the squid giant axon is an example of a delayed outward rectifier. Figure 7-18A shows that this current activates with a sigmoidal lag phase (i.e., it is delayedin time, as in Fig. 7-6C). Figure 7-18B is an I-V plot of peak currents obtained in experiments such as that in Figure 7-18A; it shows that the outward current rises steeply at positive voltages (i.e., it is an outward rectifier).

Figure 7-18 Outwardly rectifying K⁺ channels. A, Note that in a voltage-clamp experiment, a depolarizing step in V_m activates the current, but with a delay. B, The current-voltage relationship is shown for a delayed outward rectifying K⁺ channel, as in A. C, This A-type K⁺current is active at relatively negative values of V_m and tends to hyperpolarize the cell. In a spontaneously spiking neuron, a low level of the A-type current allows V_m to rise relatively quickly toward the threshold, which produces a relatively short interspike interval and thus a high firing rate. D, In a spontaneously spiking neuron, a high level of the A-type current causes V_m to rise relatively slowly toward the threshold, which produces a relatively long interspike interval and thus a low firing rate. E, These experiments were performed on four different types of K⁺ channels (Kv1.1, 1.2, 1.3, and 1.4) from mammalian brain and expressed in Xenopus oocytes. Shown are the results of voltage-clamp experiments in which V_m was stepped from –80 mV to 0 mV. The left panel, at high time resolution, shows that some of these channels activate more slowly than others. The right panel, at a longer time scale, shows that inactivation gradually speeds up from Kv1.1 to Kv1.4. F, The left panel shows N-type inactivation, so called because the N or amino terminus of the protein is essential for inactivation. Each of the four subunits is thought to have an N-terminal “ball” tethered by a “chain” that can swing into place to block the pore. The right panel shows a variant in which certain β subunits can provide the ball-and-chain for Kv channel α subunits that themselves lack this capability at their N termini. (Data from Stühmer W, Ruppersberg JP, Schroter KH, et al: Molecular basis of functional diversity of voltage-gated potassium channels in mammalian brain. EMBO J 1989; 8:3235-3244.)

A second variety of K⁺ current that is also outwardly rectifying is the transient A-type K⁺ current. This current was first characterized in mollusk neurons, but similar currents are common in the vertebrate nervous system. A-type currents are activated and inactivated over a relatively rapid time scale. Because their voltage activation range is typically more negative than that of other K⁺ currents, they are activated in the negative V_m range that prevails during the after-hyperpolarizing phase of action potentials. In neurons that spike repetitively, this A-type current can be very important in determining the interval between successive spikes and thus the timing of repetitive action potentials. For example, if the A-type current is small, V_m rises relatively quickly toward the threshold, and consequently the interspike interval is short and the firing frequency is high (Fig. 7-18C). However, if the A-type current is large, V_mrises slowly toward the threshold, and therefore the interspike interval is long and the firing frequency is low (Fig. 7-18D). Because the nervous system often encodes information as a frequency-modulated signal, these A-type currents play a critical role.

The channels responsible for both the delayed outward rectifier and the transient A-type currents belong to the Kv channel family (where v stands for voltage-gated). The prototypic protein subunit of these channels is the Shaker channel of Drosophila. All channels belonging to this family contain the conserved S1-S6 core that is characteristic of the Shaker channel (Fig. 7-10) but may differ extensively in the length and sequence of their intracellular N-terminal and C-terminal domains. The voltage-sensing element in the S4 segment underlies activation by depolarization; the S4 segment actually moves outward across the membrane with depolarizing voltage, thus increasing the probability of the channel’s being open (see the box titled Crystal Structure of a Mammalian K⁺ Channel).

The Kv channel family has multiple subclasses (see Table 6-2). Individual members of this Kv channel family, whether in Drosophila or humans, exhibit profound differences in gating kinetics that are analogous to delayed rectifier (slow activation) or A-type (rapid inactivation) currents. For example, Figure 7-18E shows the macroscopic currents of four subtypes of rat brain Kv1 (or Shaker) channels heterologously expressed in frog oocytes. All of these Kv1 channel subtypes (Kv1.1 to Kv1.4) exhibit sigmoidal activation kinetics when they are examined on a brief time scale—in the millisecond range (left side of Fig. 7-18E). That is, these channels display some degree of “delayed” activation. Different Kv channels exhibit different rates of activation. Thus, these currents can modulate action potential duration by either keeping it short (e.g., in nerve and skeletal muscle) when the delayed rectifier turns on quickly or keeping it long (e.g., in heart) when the delayed rectifier turns on slowly.

Kv1 channels also differ markedly in their inactivation kinetics when they are observed over a long time scale—in the range of seconds (right side of Fig. 7-18E). Kv1.1 exhibits little time-dependent inactivation (i.e., the current is sustained throughout the stimulus). On the other hand, the Kv1.4 channel completely inactivates in less than 1 second. Kv1.2 and Kv1.3 show intermediate behavior.

How are Kv channels inactivated? The structural basis for one particular type of K⁺ channel inactivation, known as N-type inactivation, is a stretch of ~20 amino acid residues at the N terminus of some fast-inactivating Kv channels. This domain acts like a ball to block or to plug the internal mouth of the channel after it opens, thereby resulting in inactivation (Fig. 7-18F). Thus, this process is also known as the ball-and-chain mechanism of K⁺channel inactivation. Particular kinds of β subunits that are physically associated with some isoforms of Kv channels have structural elements that mimic this N-terminal ball domain and rapidly inactivate K⁺ channel α subunits that lack their own inactivation ball domain (Fig. 7-11).

Various delayed rectifier K⁺ channels are blocked by either internal or external application of quaternary ammonium ions such as TEA. We already have described an example of how TEA can inhibit the outward rectifier K⁺current (Fig. 7-5C) in pharmacological dissection of the currents underlying the action potential. Many transient A-type K⁺ currents are inhibited by another organic cation, 4-aminopyridine. Two distinct families of peptide toxins—charybdotoxins of scorpion venom and dendrotoxins of mamba snake venom—can discriminate particular subtypes of Kv and K_Ca channels, depending on the particular amino acids present in the P region.

Two families of K_Ca K⁺ channels mediate Ca²⁺-activated K⁺ currents

Ca²⁺-activated K⁺ channels—K_Ca channels—appear to be present in the plasma membrane of cells in many different tissues. In patch-clamp experiments, they are easily recognized because the opening probability of individual channels increases at positive values of V_m (Fig. 7-19A). P_o also increases with increasing [Ca²⁺] on the intracellular surface of the membrane patch (Fig. 7-19B). Figure 7-19C shows how increasing [Ca²⁺]_i causes a negative shift in the P_o versus V_m plot for these channels. A particular type of K_Ca channel called the maxi-K_Ca or BK (for “big” K⁺) channel is noted for its large unitary conductance (~300 pS) and distinctive gating activity.

Figure 7-19 Ca²⁺-activated K⁺ channels (K_Ca). A, Shown is an experiment on K_Ca channels that are expressed in Xenopus oocytes and studied by use of a patch pipette in an inside-out configuration. When V_m is held at –60 mV, there is very little channel activity. On the other hand, when V_m is +80 mV, both channels in the patch are open most of the time. B, The experiment is the same as in A except that V_m is always held at +40 mV and the [Ca²⁺] on the cytosolic side of the patch varies from 1 to 10 to 100 μM. Note that channel activity increases with increasing [Ca²⁺]_i. C, Combined effects of changing V_m and [Ca²⁺]_i. Shown is a plot of relative open probability (P_o) of the K_Ca channels versus V_m at three different levels of Ca²⁺. The data come from experiments such as those shown in B. (Data from Butler A, Tsunoda S, McCobb DP, et al: mSlo, a complex mouse gene encoding “maxi” calcium-activated potassium channels. Science 1993; 261:221-224.)

Human Heart Defects Linked to Mutations of K⁺ Channels

A congenital cardiac abnormality in some people results in lengthening of the QT interval of the electrocardiographic signal—long QT syndrome—which corresponds to a prolonged cardiac action potential. Affected children and young adults can exhibit an arrhythmic disturbance of the ventricular heartbeat that results in sudden death. As we have already seen in the box titled Na⁺ Channel Genetic Defects, one form of a long QT syndrome involves defects in cardiac Na⁺ channels. However, several forms of this syndrome are caused by mutations in cardiac K⁺ channel proteins. Some families have mutations in the KCNQ1 gene encoding KvLQT1, a 581-residue protein belonging to the Kv family of voltage-gated K⁺ channels. Another form of this disease involves mutations in the KCNH2 gene encoding HERG, which is related to the ether-a-go-go Drosophila mutant, a more distant relative of the Kv channels. Both KvLQT1 and HERG K⁺ channels participate in repolarization of the cardiac action potential. Such defective repolarization can lead to premature heartbeats or asynchronous ventricular contraction, with subsequent death. The KvLQT1 K⁺ channel also physically associates with another small membrane protein called minK. Mutations in minK also cause a form of long QT syndrome. K⁺ channels are also crucial for proper function of the auditory system. Thus, congenital deafness is commonly associated with mutations in some of these K⁺ channels.

In principle, K_Ca channels provide a stabilizing mechanism to counteract repetitive excitation and intracellular Ca²⁺ loading. K_Ca channels mediate the afterhyperpolarizing phase of action potentials (Fig. 7-1A) in cell bodies of various neurons. They have also been implicated in terminating bursts of action potentials in bursting neuronal pacemaker cells. Thus, the gradual increase in [Ca²⁺]_i that occurs during repetitive firing triggers the opening of K_Cachannels, which results in hyperpolarization and a quiescent interburst period that lasts until intracellular Ca²⁺ accumulation is reversed by the action of Ca²⁺ pumps. K_Ca channels are also present at high density in many types of smooth muscle cells, where they appear to contribute to the relaxation of tension by providing a hyperpolarizing counterbalance to Ca²⁺-dependent contraction. In a number of nonexcitable cells, K_Ca channels are activated during cell swelling and contribute to regulatory volume decrease (see Chapter 5).

Drosophila genetics also led the way to identification of the first of several genes that encode members of the K_Ca channel family. Electrophysiological studies of the Slowpoke mutation in flies showed that this mutation eliminated a fast, Ca²⁺-activated K⁺ current that is present in larval muscle and neurons. Subsequent cloning and sequencing of the Slowpoke gene product revealed a channel-forming subunit that has an S1-S6 core domain similar to that of the Kv family, but it also contains a unique C-terminal domain of ~850 residues (Fig. 7-19). Because BK_Ca channels—like Kv channels—have a voltage-sensing domain that is analogous to S4, they are also activated by positive voltage. Structure-function studies on this class of K⁺ channel indicate that the unique C-terminal domain contains the Ca²⁺-binding sites that function in channel activation.

In addition to the BK_Ca family, another K⁺ channel gene family includes intermediate- and small-conductance Ca²⁺-activated K⁺ channels, respectively termed IK_Ca and SK_Ca. Unlike BK_Ca channels, the closely related IK_Ca and SK_Cachannels are voltage insensitive and are activated by the Ca²⁺- binding protein calmodulin (see Chapter 3). In some cells, IK_Ca and SK_Ca channels participate in action potential repolarization and afterhyperpolarization, thus regulating action potential firing frequency. Certain types of these channels function in the activation of lymphocytes.

The Kir K⁺ channels mediate inward rectifier K⁺ currents, and K2P channels may sense stress

In contrast to delayed rectifiers and A-type currents—which are outwardly rectifying K⁺ currents—the inward rectifier K⁺ current (also known as the anomalous rectifier) actually conducts more K⁺ current in the inward direction than in the outward direction. Such inwardly rectifying, steady-state K⁺ currents have been recorded in many types of cells, including heart, skeletal muscle, and epithelia. Physiologically, these channels help clamp the resting membrane potential close to the K⁺ equilibrium potential and prevent excessive loss of intracellular K⁺ during repetitive activity and long-duration action potentials. In epithelial cells, these inwardly rectifying K⁺ currents are important because they stabilize V_m in the face of electrogenic ion transporters that tend to depolarize the cell (see Chapter 3).

In contrast to the Kv and K_Ca channel families, the channel-forming subunits of the inward rectifier (Kir) K⁺ channel family are smaller proteins (~400 to 500 residues) that do not contain a complete S1-S6 core domain. However, they do have a conserved region that is similar to the S5-P-S6 segment of Kv channels (Fig. 7-20A; see the box titled Crystal Structure of a Mammalian K⁺ Channel). The conserved P region is the most basic structural element that is common to all K⁺ channels. The lack of an S1-S4 voltage-sensing domain in inward rectifier channels accounts for the observation that unlike Kv channels, Kir K⁺ channels are not steeply activated by voltage.

Figure 7-20 Inwardly rectifying K⁺ channels. A, This family of channels has only two membrane-spanning segments that correspond to the S5-P-S6 domain of the voltage-gated K⁺ channels. B, The GIRK1 channels were expressed in Xenopus oocytes and studied by use of a patch pipette in the inside-out configuration. V_m was clamped to values between –100 mV and +60 mV, and [Mg²⁺] was 2.5 mM on the cytosolic side. Note that channel activity increases at more negative voltages but is virtually inactive at positive voltages. C, The I-V plot shows that there is inward rectification only in the presence of Mg²⁺ on the cytosolic side. In the absence of Mg²⁺, the I-V relationship is nearly linear or ohmic. D, As shown in the left panel, cytosolic Mg²⁺ occludes the channel pore and prevents the exit of K⁺. However, even in the presence of Mg²⁺, K⁺ can move into the cell by displacing the Mg²⁺. (Data from Kubo Y, Reuveny E, Slesinger PA, et al: Primary structure and functional expression of a rat G protein–coupled muscarinic potassium channel. Nature 1993; 364:802-806.)

Figure 7-20B shows a series of single-channel currents that were obtained from a Kir channel, with equal concentrations of K⁺ on both sides of the membrane as well as Mg²⁺ on the cytosolic side. Under these conditions, the channel conducts K⁺ current only in the inward direction. An I-V plot (Fig. 7-20C) derived from data such as these shows typical inward rectification of the unitary current. At negative values of V_m, the inward current decreases linearly as voltage becomes more positive, and no outward current is present at positive values of V_m. However, when Mg²⁺ is omitted from the cytosolic side of the membrane, the channel now exhibits a linear or ohmic I-V curveeven over the positive range of V_m values. Thus, the inward rectification is due to intracellular block of the channel by Mg²⁺. Inhibition of outward K⁺ current in the presence of intracellular Mg²⁺ results from voltage-dependent binding of this divalent metal ion. Positive internal voltage favors the binding of Mg²⁺ to the inner mouth of this channel (Fig. 7-20D), as would be expected if the Mg²⁺ binding site is located within the transmembrane electrical field. Because Mg²⁺ is impermeant, it essentially blocks outward K⁺ current. However, negative values of V_m pull the Mg²⁺ out of the channel. Moreover, incoming K⁺ tends to displace any remaining Mg²⁺. Thus, the Kir channel favors K⁺influx over efflux. Intracellular polyamines such as spermine and spermidine—which, like Mg²⁺, carry a positive charge—also produce inward rectification of inward rectifier channels. These organic cations are important channel-modulating factors that also determine the current-voltage behavior of this particular class of ion channels.

The Kir family of K⁺ channels exhibits various modes of regulation. One Kir subfamily (the G protein–activated, inwardly rectifying K⁺ channels or GIRKs) is regulated by the βγ subunits of heterotrimeric G proteins (see Chapter 3). For example, stimulation of the vagus nerve slows the heartbeat because the vagal neurotransmitter acetylcholine binds to postsynaptic muscarinic receptors in the heart that are coupled to G proteins. The binding of acetylcholine to its receptor causes the release of G protein βγ subunits, which diffuse to a site on neighboring GIRK channels to activate their opening. The resulting increase in outward K⁺ current hyperpolarizes the cardiac cell, thereby slowing the rate at which V_m approaches the threshold for firing action potentials and lowering the heart rate. GIRK channels are also activated by the membrane phospholipid PIP₂. Thus, G protein–coupled receptors that activate phospholipase C lead to the release of PIP₂, thereby activating GIRK channels. (See Note: Hyperpolarization by Activation of GIRKs)

The members of another subfamily of Kir K⁺ channels, the K_ATP channels, are directly regulated by adenine nucleotides. K_ATP channels are present in the plasma membrane of many cell types, including skeletal muscle, heart, neurons, insulin-secreting β cell of the pancreas, and renal tubule. These channels are inhibited by intracellular adenosine triphosphate (ATP) and activated by adenosine diphosphate (ADP) in a complex fashion. They are believed to provide a direct link between cellular metabolism on the one hand and membrane excitability and K⁺ transport on the other. For example, if cellular ATP levels fall because of oxygen deprivation, such channels could theoretically open and hyperpolarize the cell to suppress firing of action potentials and further reduce energy expenditure. In the pancreatic β cell, an increase in glucose metabolism increases the ATP/ADP ratio. This increased ratio inhibits enough K_ATP channels to cause a small depolarization, which in turn activates voltage-gated Ca²⁺ channels and results in insulin secretion (see Chapter 51).

K_ATP channels are the target of a group of synthetic drugs called sulfonylureas that include tolbutamide and glibenclamide. Sulfonylureas are used in the treatment of type 2 (or non–insulin-dependent) diabetes mellitus because they inhibit pancreatic K_ATP channels and stimulate insulin release. Newer and chemically diverse synthetic drugs called K⁺ channel openers (e.g., pinacidil and cromakalim) activate K_ATPchannels. The therapeutic potential of K⁺channel openers is being explored in light of their ability to relax various types of smooth muscle. The ability of sulfonylurea drugs to inhibit K_ATP channels depends on an accessory subunit called SUR (for sulfonylurea receptor). This protein is a member of the ATP-binding cassette family of proteins (see Chapter 5), which includes two nucleotide-binding domains.

The newest family of K⁺ channels is that of the two-pore or K2P channels, which consist of a tandem repeat of the basic Kir topology (see Fig. 6-21F). Because the monomeric subunit of K2P channels contains two linked S5-P-S6 pore domains of the basic Shaker Kv channel, the functional K2P channel is likely to be a dimer of the monomer subunit, which is itself a pseudodimer. K2P channels have been implicated in genesis of the resting membrane potential. K⁺ channels encoded by the 15 human genes for K2P channels may be activated by various chemical and physical signals including PIP₂, membrane stretch, heat, intracellular pH, and general anesthetics. These channels are thought to be involved in a wide range of sensory and neuronal functions.

PROPAGATION OF ACTION POTENTIALS

The propagation of electrical signals in the nervous system involves local current loops

The extraordinary functional diversity of ion channel proteins provides a large array of mechanisms by which the membrane potential of a cell can be changed to evoke an electrical signal or biochemical response. However, channels alone do not control the spread of electrical current. Like electricity in a copper wire, the passive spread of current in biological tissue depends on the nature of the conducting and insulating medium. Important factors include geometry (i.e., cell shape and tissue anatomy), electrical resistance of the aqueous solutions and cell membrane, and membrane capacitance. Furthermore, the electrotonic spread of electrical signals is not limited to excitable cells.

Efficient propagation of a change in V_m is essential for the local integration of electrical signals at the level of a single cell and for the global transmission of signals across large distances in the body. As we discussed earlier in this chapter (Fig. 7-2), action potentials propagate in a regenerative manner without loss of amplitude as long as the depolarization spreads to an adjacent region of excitable membrane and does so with sufficient strength to depolarize the membrane above its threshold. However, many types of nonregenerative, subthreshold potentials also occur and spread for short distances along cell membranes. These graded responses, which we also discussed earlier, contrast with the all-or-nothing nature of action potentials. Such nonregenerative signals include receptor potentials generated during the transduction of sensory stimuli and synaptic potentials generated by the opening of agonist-activated channels.

With a graded response, the greater the stimulus, the greater the voltage response. For example, the greater the intensity of light that is shined on a mammalian photoreceptor cell in the retina, the greater the hyperpolarization produced by the cell. Similarly, the greater the concentration of acetylcholine that is applied at a postsynaptic neuromuscular junction, the greater the resulting depolarization (i.e., synaptic potential). Of course, if this depolarization exceeds the threshold in an excitable cell, an all-or-nothing action potential is initiated. The generation of a physiological response from a graded potential change critically depends on its electrotonic spread to other regions of the cell. Like the subthreshold voltage responses produced by injection of a current into a cell through a microelectrode, the electrotonic spread of graded responses declines with distance from the site of initiation. Graded signals dissipate over distances of a few millimeters and thus have only local effects; propagated action potentials can travel long distances through nerve axons.

Electrotonic spread of voltage changes along the cell occurs by the flow of electrical current that is carried by ions in the intracellular and extracellular medium along pathways of the least electrical resistance. Both depolarizations and hyperpolarizations of a small area of membrane produce local circuit currents. Figure 7-21A illustrates how the transient voltage change that occurs during an action potential at a particular active site results in local current flow. The cytosol of the active region, where the membrane is depolarized, has a slight excess of positive charge compared with the adjacent inactive regions of the cytosol, which have a slight excess of negative charge. This charge imbalance within the cytosol causes currents of ions to flow from the electrically excited region to adjacent regions of the cytoplasm. Because current always flows in a complete circuit along pathways of least resistance, the current spreads longitudinally from positive to negative regions along the cytoplasm, moves outward across membrane conductance pathways (“leak channels”), and flows along the extracellular medium back to the site of origin, thereby closing the current loop. Because of this flow of current (i.e., positive charge), the region of membrane immediately adjacent to the active region becomes more depolarized, and V_m eventually reaches threshold. Thus, an action potential is generated in this adjacent region as well. Nerve and muscle fibers conduct impulses in both directions if an inactive fiber is excited at a central location, as in this example. However, if an action potential is initiated at one end of a nerve fiber, it will travel only to the opposite end and stop because the refractory period prevents backward movement of the impulse. Likewise, currents generated by subthreshold responses migrate equally in both directions. (See Note: Charge Separation Required to Generate the Membrane Potential)

Figure 7-21 Local current loops during action-potential propagation. A, In an unmyelinated axon, the ionic currents flow at one instant in time as a result of the action potential (“active” zone). In the “inactive” zones that are adjacent to the active zone, the outward currents lead to a depolarization. If the membrane is not in an absolute refractory period and if the depolarization is large enough to reach threshold, the immediately adjacent inactive zones will become active and fire their own action potential. In the more distant inactive zones, the outward current is not intense enough to cause V_m to reach threshold. Thus, the magnitudes of the outward currents decrease smoothly with increasing distance from the active zone. B, In this example, the “active” zone consists of a single node of Ranvier. In a myelinated axon, the ionic current flows only through the nodes, where there is no myelin and the density of Na⁺ channels is very high. Ionic current does not flow through the internodal membrane because of the high resistance of myelin. As a result, the current flowing down the axon is conserved, and the current density at the nodes is very high. This high current density results in the generation of an action potential at the node. Thus, the regenerative action potential propagates in a “saltatory” manner by jumping from node to node. Note that the action potential is actually conducted through the internodal region by capacitative current due to charge displacement across the membrane arising from the resistance-capacitance properties of the membrane (see Fig. 6-11).

Myelin improves the efficiency with which axons conduct action potentials

The flow of electrical current along a cylindrical nerve axon has often been compared with electrical flow through an undersea cable. Similar principles apply to both types of conducting fiber. An underwater cable is designed to carry an electrical current for long distances with little current loss; therefore, it is constructed of a highly conductive (low resistance) metal in its core and a thick plastic insulation wrapped around the core to prevent loss of current to the surrounding seawater. In contrast, the axoplasm of a nerve fiber has much higher resistance than a copper wire, and the nerve membrane is inherently electrically leaky because of background channel conductance. Therefore, in a biological fiber such as a nerve or muscle cell, some current is passively lost into the surrounding medium, and the amplitude of the signal rapidly dissipates over a short distance.

Animal nervous systems use two basic strategies to improve the conduction properties of nerve fibers: (1) increasing the diameter of the axon, thus decreasing the internal resistance of the cable; and (2) myelination, which increases the electrical insulation around the cable. As axon diameter increases, the conduction velocity of action potentials increases because the internal resistance of the axoplasm is inversely related to the internal cross-sectional area of the axon. Unmyelinated nerve fibers of the invertebrate squid giant axon (as large as ~1000 μm in diameter) are a good example of this type of size adaptation. These nerve axons mediate the escape response of the squid from its predators and can propagate action potentials at a velocity of ~25 m/s.

In vertebrates, myelination of smaller diameter (~1 to 5 μm) nerve axons serves to improve the efficiency of impulse propagation, especially over the long distances that nerves traverse between the brain and the extremities. Axons are literally embedded in myelin, which consists of concentrically wound wrappings of the membranes of glial cells (see Chapter 11). The thickness of the myelin sheath may amount to 20% to 40% of the diameter of a nerve fiber, and the sheath may consist of as many as 300 membrane layers. The glial cells that produce myelin are called Schwann cells in the periphery and oligodendrocytes in the brain. Because resistors in series add directly and capacitors in series add as the sum of the reciprocal, the insulating resistance of a myelinated fiber with 300 membrane layers is increased by a factor of 300 and the capacitance is decreased to 1/300 that of a single membrane. This large increase in membrane resistance minimizes loss of current across the leaky axonal membrane and forces the current to flow longitudinally along the inside of the fiber.

In myelinated peripheral nerves, the myelin sheath is interrupted at regular intervals, forming short (~1 μm) uncovered regions called nodes of Ranvier. The length of the myelinated axon segments between adjacent unmyelinated nodes ranges from 0.2 to 2 mm. In mammalian axons, the density of voltage-gated Na⁺ channels is very high in the nodal membrane. The unique anatomy of myelinated axons results in a mode of impulse propagation known as saltatory conduction. Current flow that is initiated at an excited node flows directly to adjacent nodes with little loss of transmembrane current through the internode region (Fig. 7-21B). In other words, the high membrane resistance in the internode region effectively forces the current to travel from node to node.

The high efficiency of impulse conduction in such axons allows several adjacent nodes in the same fiber to fire an action potential virtually simultaneously as it is being propagated. Thus, saltatory conduction in a myelinated nerve can reach a very high velocity, up to 130 m/s. The action potential velocity in a myelinated nerve fiber can thus be severalfold greater than that in a giant unmyelinated axon, even though the axon diameter in the myelinated fiber may be more than two orders of magnitude smaller. During conduction of an action potential in a myelinated axon, the intracellular regions between nodes also depolarize. However, no transmembrane current flows in these internodal regions, and therefore no dissipation of ion gradients occurs. The nodal localization of Na⁺ channels conserves ionic concentration gradients that must be maintained at the expense of ATP hydrolysis by the Na-K pump.

The cable properties of the membrane and cytoplasm determine the velocity of signal propagation

Following the analogy of a nerve fiber as an underwater cable, cable theory allows one to model the pathways of electrical current flow along biomembranes. The approach is to use circuit diagrams that were first employed to describe the properties of electrical cables. Figure 7-22A illustrates the equivalent circuit diagram of a cylindrical electrical cable or membrane that is filled and bathed in a conductive electrolyte solution. The membrane itself is represented by discrete elements, each with a transverse membrane resistance (r_m) and capacitance (c_m) connected in parallel (a representation we used earlier, in Fig. 6-11A). Consecutive membrane elements are connected in series by discrete resistors, each of which represents the electrical resistance of a finite length of the external medium (r_o) or internal medium (r_i). The parameters r_m, c_m, r_o, and r_i refer to a unit length of axon (Table 7-3).

Figure 7-22 Passive cable properties of an axon. A, The axon is represented as a hollow, cylindrical “cable” that is filled with an electrolyte solution. All of the electrical properties of the axon are represented by discrete elements that are expressed in terms of the length of the axon. r_i is the resistance of the internal medium. Similarly, r_o is the resistance of the external medium. r_m and c_m are the membrane resistance and capacitance per discrete element of axon length. B, When current is injected into the axon, the current flows away from the injection site in both directions. The current density smoothly decays with increasing distance from the site of injection. C, Because the current density decreases with distance from the site of current injection in B, the electrotonic potential (V) also decays exponentially with distance in both directions. V_o is the maximum change in V_m that is at the site of current injection.

Table 7-3 Cable Parameters

How do the various electrical components of the cable model influence the electrotonic spread of current along an axon? To answer this question, we inject a steady electrical current into an axon with a microelectrode to produce a constant voltage (V₀) at a particular point (x = 0) along the length of the axon (Fig. 7-22B). This injection of current results in the longitudinal spread of current in both directions from point x = 0. The voltage (V) at various points along the axon decays exponentially with distance (x) from the point of current injection (Fig. 7-22C), according to the following equation:

The parameter λ has units of distance and is referred to as the length constant or the space constant. One length constant away from the point of current injection, V is 1/e, or ~37% of the maximum value of V₀. The decaying currents that spread away from the location of a current-passing electrode are called electrotonic currents. Similarly, the spread of subthreshold voltage changes away from a site of origin is referred to as electrotonic spread, unlike the regenerative propagation of action potentials.

The length constant depends on the three resistance elements in Figure 7-22A: (See Note: Resistance and Capacitance Units for Cable Properties)

We can simplify this expression by noting that internal resistance is much larger than external resistance, so the contribution of r_o to the denominator can be ignored. Thus,

The significance of the length constant is that it determines how far the electrotonic spread of a local change in membrane potential is able to influence neighboring regions of membrane. The longer the length constant, the farther down the axon a voltage change spreads.

How does the diameter of an axon affect the length constant? To answer this question, we must replace r_m and r_i (expressed in terms of axon length) in Equation 7-7 with the specific resistances R_m and R_i(expressed in terms of the area of axon membrane or cross-sectional area of axoplasm). Making the substitutions according to the definitions in Table 7-3, we have (See Note: Units of “Length Constant”)

Thus, the length constant (λ) is directly proportional to the square root of the axon radius (a). Equation 7-8 confirms basic intuitive notions about what makes an efficiently conducting electrical cable:

1. The greater the specific membrane resistance (R_m) and cable radius, the greater the length constant and the less the loss of signal.

2. The greater the resistance of the internal conductor (R_i), the smaller the length constant and the greater the loss of signal.

These relationships also confirm measurements of length constants in different biological preparations. For example, the length constant of a squid axon with a diameter of ~1 mm is ~13 mm, whereas that of a mammalian nerve fiber with a diameter of ~1 μm is ~0.2 mm.

So far, we have been discussing the spatial spread of voltage changes that are stable in time. In other words, we assumed that the amount of injected current was steady. What happens if the current is not steady? For example, what happens at the beginning of a stimulus when we (or a physiological receptor) first turn the current “on”? To answer these questions, we need to know how rapidly V_m changes in time at a particular site, which is described by a second cable parameter called the membrane time constant (τ_m). Rather than determining the spread of voltage changes in space, as the length constant does, the time constant influences the spread of voltage changes in time and thus the velocity of signal propagation. We previously discussed the time constant with respect to the time course of the change in V_m caused by a stepwise pulse of current (see Fig. 6-12A). Because the membrane behaves like an RC circuit, the voltage response to a square current pulse across a small piece of membrane follows an exponential time course with a time constant that is equal to the product of membrane resistance and capacitance:

We introduced this expression earlier as Equation 6-17. The shorter the time constant, the more quickly a neighboring region of membrane will be brought to threshold and the sooner the region will fire an action potential. Thus, the shorter the time constant, the faster the speed of impulse propagation, and vice versa. In contrast, conduction velocity is directly proportional to the length constant. The greater the length constant, the farther a signal can spread before decaying below threshold and the greater the area of membrane that the stimulus can excite. These relationships explain why, in terms of relative conduction velocity, a high-resistance, low-capacitance myelinated axon has a distinct advantage over an unmyelinated axon of the same diameter for all but the smallest axons (<1 μm in diameter; see Chapter 12).

In summary, the cable parameters of length constant and time constant determine the way in which graded potentials and action potentials propagate over space and time in biological tissue. These parameters are in turn a function of material properties that include resistance, capacitance, and geometric considerations. The dependence of impulse conduction velocity on fiber diameter has been studied experimentally and analyzed theoretically for unmyelinated and myelinated nerve axons. For unmyelinated axons, conduction velocity increases roughly with the square root of the axon’s diameter, just as the length constant increases with the square root of the axon’s diameter or radius (Equation 7-8). In contrast, the conduction velocity of myelinated fibers is a linear function of diameter and increases ~6 m/s per 1-μm increase in outer diameter. Thus, a mammalian myelinated axon with an outer diameter of ~4 μm has roughly the same impulse velocity as a squid giant axon with a diameter of 500 μm! However, for myelinated fibers with a very small diameter (<1 μm), the adverse effect of high internal resistance of the axoplasm predominates, and conduction is slower than in unmyelinated axons of the same outer diameter. For outer diameters that are greater than ~1 μm, the increased membrane resistance and reduced capacitance caused by myelination result in much faster conduction velocities.

The physiological importance of myelin in action potential propagation is most dramatically illustrated in the pathology that underlies human demyelinating diseases such as multiple sclerosis. As discussed more fully in Chapter 11, multiple sclerosis is an autoimmune disorder in which the myelin sheath surrounding CNS axons is progressively lost (see Chapter 12 for the box on Demyelinating Diseases). Gradual demyelination is responsible for an array of neurological symptoms that involve various degrees of paralysis and altered or lost sensation. As myelin is eliminated, the loss of membrane resistance and increased capacitance mean that propagated action potentials may ultimately fail to reach the next node of Ranvier and thus result in nerve blockage.

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