**The cell membrane model includes various ionic conductances and electromotive forces in parallel with a capacitor**

The current carried by a particular ion varies with membrane voltage, as described by the *I-V* relationship for that ion (e.g., __Fig. 6-7__). This observation suggests that the contribution of each ion to the electrical properties of the cell membrane may be represented by elements of an electrical circuit. The various ionic gradients across the membrane provide a form of stored electrical energy, much like that of a battery. In physics, the voltage source of a battery is known as an **emf** (electromotive force). The equilibrium potential of a given ion can be considered an emf for that ion. Each of these batteries produces its own ionic **current** across the membrane, and the sum of these individual ionic currents is the total ionic current (see __Equation 6-8__). According to **Ohm's law,** the emf or voltage *(V)* and current *(I)* are related directly to each other by the **resistance** ** (R)**—or inversely to the reciprocal of resistance,

**(6-11)**

Thus, the slopes of the theoretical curves in __Figure 6-7__ represent conductances because *I* = *GV*. In a membrane, we can represent each ionic permeability pathway with an electrical conductance. Ions with high permeability or conductance move via a low-resistance pathway; ions with low permeability move via a high-resistance pathway. For cell membranes, *V*_{m} is measured in millivolts, membrane current (*I*_{m}) is given in amperes per square centimeter of membrane area, and membrane resistance (*R*_{m}) has the units of ohms × square centimeter. Membrane conductance (*G*_{m}), the reciprocal of membrane resistance, is thus measured in units of ohms^{−1} (or siemens) per square centimeter. **N6-11**

**N6-11**

**Electrical Units**

*Contributed by Ed Moczydlowski*

Unit of **resistance:** ohm. 1 ohm = 1 volt/ampere.

Unit of **conductance** (the reciprocal of resistance): siemens. 1 siemens = 1/ohm. In English, *siemens*—named after Ernst von Siemens—is used for both the singular and plural.

Unit of **charge:** coulomb. 1 coulomb = the electrical charge separated by the plates of a 1-farad capacitor charged to 1 volt.

Unit of **capacitance:** farad. 1 farad = 1 coulomb/volt. Thus, if we charge a 1-farad capacitor to 1 volt, the charge on each plate will be 1 coulomb.

Unit of **electrical work:** joule. 1 joule = 1 volt × 1 coulomb.

Currents of Na^{+}, K^{+}, Ca^{2+}, and Cl^{−} generally flow across the cell membrane via distinct pathways. At the molecular level, these pathways correspond to specific types of ion channel proteins (__Fig. 6-9__*A*). It is helpful to model the electrical behavior of cell membranes by a circuit diagram (see __Fig. 6-9__*B*). The electrical current carried by each ion flows via a separate parallel branch of the circuit that is under the control of a **variable resistor** and an emf. For instance, the variable resistor for K^{+} represents the conductance provided by K^{+} channels in the membrane (*G*_{K}). The emf for K^{+} corresponds to *E*_{K}. Similar parallel branches of the circuit in __Figure 6-9__*B* represent the other physiologically important ions. Each ion provides a component of the total conductance of the membrane, so *G*_{K} + *G*_{Na} + *G*_{Ca} + *G*_{Cl} sum to *G*_{m}.

**FIGURE 6-9** Electrical properties of model cell membranes. **A,** Four different ion channels are arranged in parallel in the cell membrane. **B,** The model represents each channel in **A** with a variable resistor. The model represents the Nernst potential for each ion as a battery in series with each variable resistor. The four parallel current paths correspond to the four parallel channels in **A.** Also shown is the membrane capacitance, which is parallel with each of the channels. **C,** On the left is an idealized capacitor, which is formed by two parallel conductive plates, each with an area *A* and separated by a distance *a.* On the right is a capacitor that is formed by a piece of lipid membrane. The two *plates* are, in fact, the electrolyte solutions on either side of the membrane.

The GHK voltage equation (see __Equation 6-9__) predicts steady-state *V*_{m}, provided the underlying assumptions are valid. We can also predict steady-state *V*_{m} (i.e., when the net current across the membrane is zero) with another, more general equation that assumes channels behave like separate ohmic conductances:

**(6-12)**

Thus, *V*_{m} is the sum of equilibrium potentials (*E*_{X}), each weighted by the ion's fractional conductance (e.g., *G*_{X}/*G*_{m}).

One more parallel element, a **capacitor,** is needed to complete our model of the cell membrane as an electrical circuit. A capacitor is a device that is capable of storing separated charge. Because the lipid bilayer can maintain a separation of charge (i.e., a voltage) across its ~4-nm width, it effectively functions as a capacitor. In physics, a capacitor that is formed by two parallel plates separated by a distance *a* can be represented by the diagram in __Figure 6-9__*C**.* When the capacitor is charged, one of the plates bears a charge of +*Q* and the other plate has a charge of −*Q*. **N6-12** Such a capacitor maintains a potential difference *(V)* between the plates. **Capacitance (C)** is the magnitude of the charge stored per unit potential difference:

**N6-12**

**Charge Carried by a Mole of Monovalent Ions**

*Contributed by Ed Moczydlowski*

We can compute a macroscopic quantity of charge by using a conversion factor called the Faraday *(F)*. The Faraday is the charge of a mole of univalent ions, or *e*_{0} × Avogadro's number:

**(NE 6-15)**

**(6-13)**

Capacitance is measured in units of farads (F); **N6-11** 1 farad = 1 coulomb/volt. For the particular geometry of the parallel-plate capacitor in __Figure 6-9__*C**,* capacitance is directly proportional to the surface area *(A)* of one side of a plate, to the relative permittivity (dielectric constant) of the medium between the two plates (ε_{r}), and to the vacuum permittivity constant (ε_{0}), and it is inversely proportional to the distance *(a)* separating the plates.

**(6-14)**

Because of its similar geometry, the cell membrane has a capacitance that is analogous to that of the parallel-plate capacitor. The capacitance of 1 cm^{2} of most cell membranes is ~1 µF; that is, most membranes have a specific capacitance of 1 µF/cm^{2}. We can use __Equation 6-14__ to estimate the thickness of the membrane. If we assume that the average dielectric constant of a biological membrane is ε_{r} = 5 (slightly greater than the value of 2 for pure hydrocarbon), __Equation 6-14__ gives a value of 4.4 nm for *a*—that is, the thickness of the membrane. This value is quite close to estimates of membrane thickness that have been obtained by other physical techniques.

**The separation of relatively few charges across the bilayer capacitance maintains the membrane potential**

We can also use the capacitance of the cell membrane to estimate the amount of charge that the membrane actually separates in generating a typical membrane potential. For example, consider a spherical cell with a diameter of 10 µm and a [K^{+}]_{i} of 100 mM. This cell needs to lose only 0.004% of its K^{+} to charge the capacitance of the membrane to a voltage of −61.5 mV. **N6-13** This small loss of K^{+} is clearly insignificant in comparison with a cell's total ionic composition and does not significantly perturb concentration gradients. In general, cell membrane potentials are sustained by a very small separation of charge.

**N6-13**

**Charge Separation Required to Generate the Membrane Potential**

*Contributed by Ed Moczydlowski*

To generate a membrane potential, there must be a tiny separation of charge across the membrane. How large is that charge? Imagine that we have a spherical cell with a diameter of 10 µm. If [K^{+}]_{i} is 100 mM and [K^{+}]_{o} is 10 mM, the *V*_{m} according to the Nernst equation would be −61.5 mV (or −0.0615 V) for a perfectly K^{+}-selective membrane at 37°C. What is the charge *(Q)* on 1 cm^{2} of the “plates” of the membrane capacitor? We assume that the specific capacitance is 1 µF/cm^{2}. From __Equation 6-13__ on __page 150__ in the text, we know that

**(NE 6-16)**

where *Q* is measured in coulombs (C), *C* is in farads (F), and *V* is in volts (V). Thus,

**(NE 6-17)**

The Faraday is the charge of 1 mole of univalent ions—or 96,480 C. **N6-12** To determine how many moles of K^{+} we need to separate in order to achieve an electrical charge of 61.5 × 10^{−9} C cm^{−2} (i.e., the *Q* in the previous equation), we merely divide *Q* by the Faraday. Because *V*_{m} is negative, the cell needs to lose K^{+}:

**(NE 6-18)**

The surface area for a spherical cell with a diameter of 10 µm is 3.14 × 10^{−6}^{ }cm^{2}. Therefore,

**(NE 6-19)**

What fraction of the cell's total K^{+} content represents the charge separated by the membrane?

**(NE 6-20)**

Thus, in the process of generating a *V*_{m} of −61.5 mV, our hypothetical cell needs to lose only 0.004% of its total K^{+} content to charge the capacitance of the cell membrane.

Because of the existence of membrane capacitance, total membrane current has two components (see __Fig. 6-9__), one carried by ions through channels, and the other carried by ions as they charge the membrane capacitance.

**Ionic current is directly proportional to the electrochemical driving force (Ohm's law)**

__Figure 6-10__ compares the equilibrium potentials for Na^{+}, K^{+}, Ca^{2+}, and Cl^{−} with a resting *V*_{m} of −80 mV. In our discussion of __Figure 6-7__, we saw that *I*_{K} or *I*_{Na} becomes zero when *V*_{m} equals the reversal potential, which is the same as the *E*_{X} or emf for that ion. When *V*_{m} is more negative than *E*_{X}, the current is negative or inward, whereas when *V*_{m} is more positive than *E*_{X}, the current is positive or outward. Thus, the ionic current depends on the difference between the actual *V*_{m} and *E*_{X}. In fact, the ionic current through a given conductance pathway is proportional to the difference (*V*_{m} − *E*_{X}), and the proportionality constant is the ionic conductance (*G*_{X}):

**(6-15)**

**FIGURE 6-10** Electrochemical driving forces acting on various ions. For each ion, we represent the equilibrium potential (e.g., *E*_{Na} = +67 mV) as a horizontal bar and the net driving force for the ion (e.g., *V*_{m} − *E*_{Na} = −147 mV) as an arrow, assuming a resting potential (*V*_{m}) of −80 mV. The values for the equilibrium potentials are those for mammalian skeletal muscle given in __Table 6-1__, as well as a typical value for *E*_{Cl} in a nonmuscle cell. **N6-29**

**N6-29**

**Electrochemical Driving Forces and Predicted Direction of Net Fluxes**

*Contributed by Ed Moczydlowski*

In __Figure 6-10__, for Na^{+} and Ca^{2+}, the arrows—which indicate the driving force—point down, indicating that the driving force favors the passive *influx* of these ions. For K^{+}, the arrow points up, indicating that the driving force favors the passive *efflux* of K^{+}. For Cl^{−} in skeletal muscle cells, the arrow points up, indicating that the driving force favors a small passive *influx.* In other cells, the arrow for Cl^{−} points down, indicating that the driving force favors passive efflux.

This equation simply restates Ohm's law (see __Equation 6-11__). The term (*V*_{m} − *E*_{X}) is often referred to as the **driving force** in electrophysiology. In our electrical model of the cell membrane (see __Fig. 6-9__), this driving force is represented by the difference between *V*_{m} and the emf of the battery. The larger the driving force, the larger the observed current. Returning to the *I-V* relationship for K^{+} in __Figure 6-7__*A**,* when *V*_{m}is more positive than *E*_{K}, the driving force is positive, producing an outward (i.e., positive) current. Conversely, at *V*_{m} values more negative than *E*_{K}, the negative driving force produces an inward current. **N6-14**

**N6-14**

**Conductance Varies with Driving Force**

*Contributed by Ed Moczydlowski*

In **N6-8**, we pointed out that when [K^{+}]_{i} = [K^{+}]_{o}, the *I-V* relationship for K^{+} currents is linear and passes through the origin (see dashed line in __Fig 6-7__*A*). In this special case, the K^{+} conductance (*G*_{K}) is simply the slope of the line because, according to Ohm's law, *I*_{K} = *G*_{K} × *V*_{m}. In other words, *G*_{K} = Δ*I*_{K}/Δ*V*_{m}.

In webnote __Equation NE 6-8__, we also pointed out that when [K^{+}]_{i} *does not equal* [K^{+}]_{o}, the *I-V* relationship is curvilinear (see solid curve in __Fig. 6-7__*A* in the text) as described by the GHK current equation for K^{+}:

**(NE 6-21)**

The above equation is identical to __Equation 6-7__ in the text, but with K^{+} replacing the generic ion X. Note that for K^{+}, all of the *z* values are +1.

Because slope conductance for K^{+} (*G*_{K}) is the change in K^{+} current (*I*_{K}) divided by the change in membrane voltage (*V*_{m}), we could in principle derive an equation for *G*_{K} by taking the derivative of __Equation NE 6-21__ with respect to *V*_{m} (i.e., *G*_{K} = *dI*_{K}/*dV*_{m}). Because *V*_{m} appears three times in __Equation NE 6-21__ (and twice in an exponent), this derivative—that is, *G*_{K}—turns out to be extremely complicated (not shown). Nevertheless, it is possible to show that, in general, *G*_{K} increases with increasing values of *V*_{m}. For the special case in which *V*_{m} = *E*_{K}, the equation for *G*_{K} simplifies to

**(NE 6-22)**

It is clear from __Equation NE 6-22__ that *G*_{K} increases as *V*_{m} becomes more positive. However, this relationship is not linear because as *V*_{m} increases, *E*_{K} (the equilibrium potential for K^{+}) must also increase, and thus the [K^{+}]_{o} or the [K^{+}]_{i} terms in __Equation NE 6-22__ must also change.

__Equation NE 6-22__ describes *G*_{K} at exactly one point—when *V*_{m} = *E*_{K} at −95 mV. At other values of *V*_{m}, the appropriate expression for *G*_{K} is far more complicated than __Equation NE 6-22__. Nevertheless, it is clear from the graph in __Figure 6-7__*A* that the slope of the *I-V* relationship (i.e., *G*_{K}) increases with *V*_{m}. Thus, the slope of the curve in __Figure 6-7__*A* is relatively shallow (i.e., low *G*_{K}) for the inward currents at relatively negative *V*_{m} values (lower portion of the plot) and steeper (i.e., high *G*_{K}) for outward currents at more positive *V*_{m} values (upper portion of the plot).

In __Figure 6-10__, the arrows represent the magnitudes and directions of the driving forces for the various ions. For a typical value of the resting potential (−80 mV), the driving force on Ca^{2+} is the largest of the four ions, followed by the driving force on Na^{+}. In both cases, *V*_{m} is more negative than the equilibrium potential and thus draws the positive ion *into* the cell. The driving force on K^{+} is small. *V*_{m} is more positive than *E*_{K} and thus pushes K^{+} out of the cell. In muscle, *V*_{m} is slightly more positive than *E*_{Cl} and thus draws the anion inward. In most other cells, however, *V*_{m} is more negative than *E*_{Cl} and pushes the Cl^{−} out.

**Capacitative current is proportional to the rate of voltage change**

The idea that ionic channels can be thought of as conductance elements (*G*_{X}) and that ionic current (*I*_{X}) is proportional to driving force (*V*_{m} − *E*_{X}) provides a framework for understanding the electrical behavior of cell membranes. Current carried by inorganic ions flows through open channels according to the principles of electrodiffusion and Ohm's law, as explained above. However, when *V*_{m} is changing—as it does during an action potential—another current due to the membrane capacitance also shapes the electrical responses of cells. This current, which flows only while *V*_{m} is changing, is called the capacitative current. How does a capacitor produce a current? When voltage across a capacitor changes, the capacitor either gains or loses charge. This movement of charge onto or off the capacitor is an electrical (i.e., the capacitative) current.

The simple membrane circuit of __Figure 6-11__*A**,* which is composed of a capacitor (*C*_{m}) in parallel with a resistor (*R*_{m}) and a switch, can help illustrate how capacitative currents arise. Assume that the switch is open and that the capacitor is initially charged to a voltage of *V*_{0}, which causes a separation of charge *(Q)* across the capacitor. According to the definition of capacitance (see __Equation 6-13__), the charge stored by the capacitor is a product of capacitance and voltage.

**(6-16)**

**FIGURE 6-11** Capacitative current through a resistance-capacitance (RC) circuit.

As long as the switch in the circuit remains open, the capacitor maintains this charge. However, when the switch is closed, the charge on the capacitor discharges through the resistor, and the voltage difference between the circuit points labeled “In” and “Out” in __Figure 6-11__*A* decays from *V*_{0} to a final value of zero (see __Fig. 6-11__*B*). This voltage decay follows an exponential time course. The time required for the voltage to fall to 37% of its initial value is a characteristic parameter called the **time constant (τ),** which has units of time: **N6-15**

**N6-15**

**Units of the “Time Constant”**

*Contributed by Emile Boulpaep, Walter Boron*

As described in __Equation 6-17__ in the text (shown here as __Equation NE 6-23__), the time constant (τ) is

**(NE 6-23)**

where *R* is resistance (in ohms) and *C* is capacitance (in farads). The units of τ are thus

**(NE 6-24)**

Because an ohm is a volt per ampere, and a farad is a coulomb per volt,

**(NE 6-25)**

Because electrical current (in amperes) is the number of charges (in coulombs) moving per unit time (in seconds), an ampere is a coulomb per second:

**(NE 6-26)**

Thus, the unit of the “time constant” is seconds.

**(6-17)**

Thus, the time course of the decay in voltage is

**(6-18)**

__Figure 6-11__*C* shows that the **capacitative current ( I**

**N6-16**

**Time Constant of Capacitative Current**

*Contributed by Ed Moczydlowski*

In __Figure 6-11__ in the text, we saw that closing a switch (panel **A**) causes the voltage to decline exponentially with a time constant τ (panel **B**), and it causes a current to flow maximally at time zero and then to decay with the same time constant as voltage. In other words, the capacitative current flows only while voltage is changing. Why? Current is charge flowing per unit time. Thus, we can obtain the **capacitative current ( I**

**(NE 6-27)**

By definition, the derivative of charge with respect to time is current (i.e., *I*_{C} = *dQ/dt*). Thus, if voltage is constant (i.e., *dV/dt* = 0), no capacitative current can flow. In __Figure 6-11__*C, I*_{C} is zero before the switch is closed (i.e., before the downward deflection of I_{C}) and again is zero at “infinite” time, when the voltage is stable at 0. On the other hand, when the voltage is changing, the above equation indicates that *I*_{C} is nonzero and is directly proportional to *C* and to the rate at which the voltage is changing. Note, however, that *V* and *I*_{C} relax with the same time constant. To understand the exponential time course, note that Ohm's law can be used to express the current through the resistor in __Figure 6-11__*A* as *V/R.* If *V/R* is substituted for *I*_{C} in __Equation NE 6-27__, we have

**(NE 6-28)**

We can rearrange the above differential equation to solve for *V:*

**(NE 6-29)**

We can now solve this differential equation to obtain the time course of the decay in voltage:

**(NE 6-30)**

__Equation NE 6-30__ is the same as __Equation 6-18__ in the text. Thus, the voltage falls exponentially with time. We now go back to the first equation and plug in our newly derived expression for *V:*

**(NE 6-31)**

Thus, the capacitative current decays with the same time constant as does voltage. At time zero, the current is −*V*_{0}/*R*, and at infinite time the current is zero.

In __Figure 6-11__, current and voltage change freely. __Figure 6-12__ shows two related examples in which either current or voltage is abruptly changed to a fixed value, held constant for a certain time, and returned to the original value. This pattern is called a square pulse. In __Figure 6-12__*A**,* we control, or “clamp,” the current and allow the voltage to follow. When we inject a square pulse of current across the membrane, the voltage changes to a new value with a rounded time course determined by the *RC* value of the membrane. In __Figure 6-12__*B**,* we clamp voltage and allow the current to follow. When we suddenly change voltage to a new value, a transient capacitative current flows as charge flows onto the capacitor. The capacitative current is maximal at the beginning of the square pulse, when charge flows most rapidly onto the capacitor, and then falls off exponentially with a time constant of *RC.* When we suddenly decrease the voltage to its original value, *I*_{C} flows in the direction opposite that observed at the beginning of the pulse. Thus, *I*_{C} appears as brief spikes at the beginning and end of the voltage pulse.

**FIGURE 6-12** Voltage and current responses caused by the presence of a membrane capacitance. **N6-30**

**N6-30**

**Voltage and Current Transients due to Membrane Capacitance**

*Contributed by Ed Moczydlowski*

In panel **A** of __Figure 6-12__ in the text (current clamp), we instruct the electronics to suddenly increase the current that we are injecting into the cell and to hold this new current at a constant value. The sudden increase in the current flowing through the membrane causes *V*_{m} to rise exponentially until we fully charge the membrane capacitance (*C*_{m}). Thus, *V*_{m} rises with a time constant **N6-15** of *R*_{m} × *C*_{m} (*R*_{m} is membrane resistance). At infinite time, the charge on the capacitor is at its maximal value, and all the current flowing through the membrane flows through *R*_{m}, the “ohmic” membrane resistance.

In panel **B** of __Figure 6-12__ in the text (voltage clamp), we instruct the electronics to inject enough current into the cell to suddenly increase in the membrane potential (*V*_{m}) of the cell. The current required to charge the membrane capacitance (*C*_{m}) is at first extremely large. However, as we charge the membrane capacitance, that current decays exponentially with a time constant **N6-15** of *R*_{m} × *C*_{m}. At infinite time, the membrane capacitance is fully charged, and no current is required to hold the command voltage. However, this current decays exponentially, with a time course also determined by the *R* × *C* of the membrane.

**A voltage clamp measures currents across cell membranes**

Electrophysiologists use a technique called **voltage clamping** to deduce the properties of ion channels. In this method, specialized electronics are used to inject current into the cell to set the membrane voltage to a value that is different from the resting potential. The device then measures the total current required to clamp *V*_{m} to this value. A typical method of voltage clamping involves impaling a cell with two sharp electrodes, one for monitoring *V*_{m} and one for injecting the current. __Figure 6-13__*A* illustrates how the technique can be used with a *Xenopus* (i.e., frog) oocyte. **N6-17** When the voltage-sensing electrode detects a difference from the intended voltage, called the command voltage, a feedback amplifier rapidly injects opposing current to maintain a constant *V*_{m}. The magnitude of the injected current needed to keep *V*_{m}constant is equal, but opposite in sign, to the membrane current and is thus an accurate measurement of the **total membrane current ( I**

**FIGURE 6-13** Two-electrode voltage clamp. **A,** Two microelectrodes impale a *Xenopus* oocyte. One electrode monitors membrane potential (*V*_{m}) and the other passes enough current (*I*_{m}) through the membrane to clamp *V*_{m} to a predetermined command voltage (*V*_{command}). **B,** In the left panel, the membrane is clamped for 10 ms to a hyperpolarized potential (40 mV more negative). Because a hyperpolarization does not activate channels, no ionic currents flow. Only transient capacitative currents flow after the beginning and end of the pulse. In the right panel, the membrane is clamped for 10 ms to a depolarized potential (40 mV more positive). Because the depolarization opens voltage-gated Na^{+} channels, a large inward Na^{+} current flows, in addition to the transient capacitative current. Adding the transient capacitative currents in the left panel to the total current in the right panel cancels the transient capacitative currents (*I*_{c}) and yields the pure Na^{+} current shown at the bottom in the right panel.

**N6-17**

**Two-Electrode Voltage Clamping**

*Contributed by Ed Moczydlowski*

Historically, the technique of two-electrode voltage clamping was first used to analyze the ionic currents in a preparation known as the perfused squid giant axon. Certain nerve fibers of the squid are so large that their intracellular contents can be extruded and the hollow fiber can be perfused with physiological solutions of various ionic composition. Electrodes in the form of thin wires can be inserted into the axon to clamp the axon membrane potential along its length and measure the current. This technique was used by Alan L. Hodgkin and Andrew F. Huxley in 1952 to deduce the nature of ionic conductance changes that underlie the nerve action potential. For this work, Hodgkin and Huxley shared (with J.C. Eccles) the Nobel Prize in Physiology or Medicine in 1963. **N6-32** The Hodgkin-Huxley analysis is discussed further beginning on __page 176__.

Another, more recent, application of the two-electrode voltage-clamp technique is called **oocyte recording** (see __Fig. 6-13__*A*). A large oocyte from the African clawed frog, *Xenopus laevis,* is simultaneously impaled with two micropipette electrodes that serve to clamp the voltage and record current. Native *Xenopus* oocytes have only small endogenous currents, but they can be induced to express new currents by preinjecting the cell with messenger RNA (mRNA) transcribed from an isolated gene that codes for an ion channel protein. The oocyte system can therefore be used to characterize the conductance behavior of a relatively pure population of ion channels that are expressed in the plasma membrane after protein translation of the injected mRNA by the oocyte. This approach has proven to be an invaluable assay system for isolating complementary DNA molecules coding for many different types of channels and electrogenic transporters (see __Chapter 5__). This approach also has become a standard technique used to study the molecular physiology and pharmacology of ion channels.

**N6-32**

**Alan L. Hodgkin and Andrew F. Huxley**

For more information about Alan Hodgkin and Andrew Huxley and the work that led to their Nobel Prize, visit __http://www.nobel.se/medicine/laureates/1963/index.html__ (accessed October 2014).

*I*_{m} is the sum of the individual currents through each of the parallel branches of the circuit in __Figure 6-9__*B**.* For a simple case in which only one type of ionic current (*I*_{X}) flows through the membrane, *I*_{m} is simply the sum of the capacitative current and the ionic current:

**(6-19)**

__Equation 6-19__ suggests a powerful way to analyze how ionic conductance (*G*_{X}) changes with time. For instance, if we abruptly change *V*_{m} to another value and then hold *V*_{m} constant (i.e., we *clamp* the voltage), the capacitative current flows for only a brief time at the voltage transition and disappears by the time that *V*_{m} reaches its new steady value (see __Fig. 6-12__*B*). Therefore, after *I*_{C} has decayed to zero, any additional changes in *I*_{m} must be due to changes in *I*_{X}. Because *V*_{m} is clamped and the ion concentrations do not change (i.e., *E*_{X} is constant), only one parameter on the right side of __Equation 6-19__ is left free to vary, *G*_{X}. In other words, we can directly monitor changes in *G*_{X} because this conductance parameter is directly proportional to *I*_{m} when *V*_{m} is constant (i.e., clamped).

__Figure 6-13__*B* shows examples of records from a typical voltage-clamp experiment on an oocyte expressing voltage-gated Na^{+} channels. In this experiment, a cell membrane is initially clamped at a resting potential of −80 mV. *V*_{m} is then stepped to −120 mV for 10 ms (a step of −40 mV) and finally returned to −80 mV. Such a negative-going *V*_{m} change is called a **hyperpolarization.** With this protocol, only brief spikes of current are observed at the beginning and end of the voltage step and are due to the charging of membrane capacitance. No current flows in between these two spikes.

What happens if we rapidly change *V*_{m} in the opposite direction by shifting the voltage from −80 to −40 mV (a step of +40 mV)? Such a positive-going change in *V*_{m} from a reference voltage is called a **depolarization.** In addition to the expected transient *capacitative* current, a large, inward, time-dependent current flows. This current is an *ionic* current and is due to the opening and closing kinetics of a particular class of channels called voltage-gated Na^{+} channels, which open only when *V*_{m} is made sufficiently positive. We can remove the contribution of the capacitative current to the total current by subtracting the inverse of the rapid transient current recorded during the hyperpolarizing pulse of the same magnitude. The remaining slower current is inward (i.e., downward or negative going) and represents *I*_{Na}, which is directly proportional to *G*_{Na} (see __Equation 6-19__).

The ionic current in __Figure 6-13__*B* (lower right panel) is called a **macroscopic current** because it is due to the activity of a large population of channels sampled from a whole cell. Why did we observe Na^{+}current only when we shifted *V*_{m} in a positive direction from the resting potential? As described below, such Na^{+} channels are actually members of a large family of voltage-sensitive ion channels that are activated by *depolarization.* A current activated by depolarization is commonly observed when an electrically excitable cell, such as a neuron, is voltage clamped under conditions in which Na^{+} is the sole extracellular cation.

A modern electrophysiological method called **whole-cell voltage clamping** involves the use of a single microelectrode both to monitor *V*_{m} and to pass current into the cell. In this method, one presses onto the cell surface a glass micropipette electrode with a smooth, fire-polished tip that is ~1 µm in diameter (__Fig. 6-14__*A*). Applying slight suction to the inside of the pipette causes a high-resistance seal to form between the circular rim of the pipette tip and the cell membrane. The piece of sealed membrane is called a **patch,** and the pipette is called a **patch pipette.** Subsequent application of stronger suction causes the patch to rupture, creating a continuous, low-resistance pathway between the inside of the cell and the pipette. In this configuration, **whole-cell currents** can be recorded (see __Fig. 6-14__*B*). Because single cells can be dissociated from many different tissues and studied in culture, this method has proved very powerful for analyzing the physiological roles of various types of ion channels and their regulation at the cellular level. The approach for recording whole-cell currents with a patch pipette was introduced by Erwin Neher and Bert Sakmann, who received the Nobel Prize in Physiology or Medicine in 1991. **N6-18**

**FIGURE 6-14** Patch-clamp methods. (Data from Hamill OP, Marty A, Neher E, et al: Improved patch-clamp techniques for high-resolution current recording from cells and cell-free membrane patches. Pflugers Arch 391:85–100, 1981.)

**N6-18**

**Erwin Neher and Bert Sakmann**

For more information about Erwin Neher and Bert Sakmann and the work that led to their Nobel Prize, visit __http://www.nobel.se/medicine/laureates/1991/index.html__ (accessed October 2014).

**The patch-clamp technique resolves unitary currents through single channel molecules**

Voltage-clamp studies of ionic currents at the whole-cell (i.e., macroscopic) level led to the question of how many channels are involved in the production of a macroscopic current. Electrophysiologists realized that if the area of a voltage-clamped membrane was reduced to a very small fraction of the cell surface area, it might be possible to observe the activity of a single channel.

This goal was realized when Neher and Sakmann developed the **patch-clamp technique.** Applying suction to a patch pipette creates a high-resistance seal between the glass and the cell membrane, as described in the preceding section for whole-cell voltage clamping. However, rather than rupturing the enclosed membrane patch as in the whole-cell approach, one keeps the tiny membrane area within the patch intact and records current from channels within the patch. A current recording made with the patch pipette attached to a cell is called a **cell-attached recording** (see __Fig. 6-14__*A*). After a cell-attached patch is established, it is also possible to withdraw the pipette from the cell membrane to produce an **inside-out patch configuration** by either of two methods (see __Fig. 6-14__*E* __and Fig. 6-14__*F–H*). In this configuration, the *intracellular* surface of the patch membrane faces the bath solution. One can also arrive at the opposite orientation of the patch of membrane by starting in the cell-attached configuration (see __Fig. 6-14__*A*), rupturing the cell-attached patch to produce a whole-cell configuration (see __Fig. 6-14__*B*), and then pulling the pipette away from the cell (see __Fig. 6-14__*C*). When the membranes reseal, the result is an **outside-out patch configuration** in which the *extracellular* patch surface faces the bath solution (see __Fig. 6-14__*D*).

The different patch configurations summarized in __Figure 6-14__ are useful for studying drug channel interactions, receptor-mediated processes, and biochemical regulatory mechanisms that take place at either the inner or external surface of cell membranes.

**Single channel currents sum to produce macroscopic membrane currents**

__Figure 6-15__ illustrates the results of a patch-clamp experiment that are analogous to the macroscopic experiment results on the right-hand side of __Figure 6-13__*B**.* Under the diagram of the voltage step in __Figure 6-15__*A* are eight current records, each of which is the response to an identical step of depolarization lasting 45 ms. The smallest, nearly rectangular transitions of current correspond to the opening and closing of a single Na^{+} channel. When two or three channels in the patch are open simultaneously, the measured current level is an integral multiple of the single channel or “unitary” transition.

**FIGURE 6-15** Outside-out patch recordings of Na^{+} channels. **A,** Eight single-current responses—in the same patch on a myotube (a cultured skeletal muscle cell)—to a depolarizing step in voltage (cytosolic side of patch negative). **B,** Average current. The record in black shows the average of many single traces, such as those in **A.** The blue record shows the average current when tetrodotoxin blocks the Na^{+} channels. (Data from Weiss RE, Horn R: Single-channel studies of TTX-sensitive and TTX-resistant sodium channels in developing rat muscle reveal different open channel properties. Ann N Y Acad Sci 479:152–161, 1986.)

The opening and closing process of ion channels is called **gating.** Patch-clamp experiments have demonstrated that macroscopic ionic currents represent the gating of single channels that have discrete unitary currents. Averaging consecutive microscopic Na^{+} current records produces a time-dependent current (see __Fig. 6-15__*B*) that has the same shape as the macroscopic *I*_{m} shown in __Figure 6-13__*B**.* If one does the experiment in the same way but blocks Na^{+} channels with tetrodotoxin, the averaged current is equivalent to the zero current level, which indicates that Na^{+} channels are the only channels present within the membrane patch.

Measuring the current from a single channel in a patch at different clamp voltages reveals that the size of the discrete current steps depends on voltage (__Fig. 6-16__*A*). Plotting the **unitary current (i)** of single channels versus the voltage at which they were measured yields a single channel

**FIGURE 6-16** Voltage dependence of currents through single Cl^{−} channels in outside-out patches. **A,** The channel is a GABA_{A} receptor channel, which is a Cl^{−} channel activated by GABA. Identical solutions, containing 145 mM Cl^{−}, were present on both sides of the patch. **B,** The magnitudes of the single channel current transitions (y-axis) vary linearly with voltage (x-axis). (Data from Bormann J, Hamill OP, Sakmann B: Mechanism of anion permeation through channels gated by glycine and γ-aminobutyric acid in mouse cultured spinal neurones. J Physiol 385:243–286, 1987.)

The slope of a single channel *I-V* relationship is a measure of the conductance of a single channel, the **unitary conductance (g).** Every type of ion channel has a characteristic value of

How do we know that the unitary current in fact corresponds to just a single channel? One good indication is that such conductance measurements are close to the theoretical value expected for ion diffusion through a cylindrical, water-filled pore that is long enough to span a phospholipid membrane and that has a diameter large enough to accept an ion. The unitary conductance of typical channels corresponds to rates of ion movement in the range of 10^{6} to 10^{8} ions per second per channel at 100 mV of driving force. These rates of ion transport through single channels are many orders of magnitude greater than typical rates of ion transport by ion pumps (~500 ions/s) or by the fastest ion cotransporters and exchangers (~50,000 ions/s). The high ionic flux through channels places them in a unique class of transport proteins whose unitary activity can be resolved by patch-clamp current recordings.

**Single channels can fluctuate between open and closed states**

When a channel has opened from the **closed state** (zero current) to its full unitary conductance value, the channel is said to be in the **open state.** Channel gating thus represents the transition between closed and open states. A single channel record is actually a record of the conformational changes of a single protein molecule as monitored by the duration of opening and closing events.

Examination of the consecutive records of a patch recording, such as that in __Figure 6-15__*A**,* shows that the gating of a single channel is a probabilistic process. On average, there is a certain probability that a channel will open at any given time, but such openings occur randomly. For example, the average record in __Figure 6-15__*B* indicates that the probability that the channels will open is highest ~4 ms after the start of the depolarization.

The process of channel gating can be represented by kinetic models that are similar to the following hypothetical two-state scheme.

**(6-20)**

This scheme indicates that a channel can reversibly change its conformation between closed (C) and open (O) states according to first-order reactions that are determined by an opening rate constant (*k*_{o}) and a closing rate constant (*k*_{c}). The **probability of channel opening ( P**

We already have seen in __Figure 6-15__ that the *average* of many single channel records from a given patch produces a time course that is similar to a macroscopic current recorded from the same cell. The same is true for the *sum* of the individual single channel current records. This conclusion leads to an important relationship: macroscopic ionic current is equal to the product of the **number of channels (N)** within the membrane area, the unitary current of single channels, and the probability of channel opening:

**(6-21)**

Comparison of the magnitude of macroscopic currents recorded from large areas of voltage-clamped membrane with the magnitude of unitary current measured by patch techniques indicates that the surface density of ion channels typically falls into the range of 1 to 1000 channels per square micrometer of cell membrane, depending on the channel and cell type.