Electrophysiology of the Cell Membrane

Principles of electrostatics explain why aqueous pores formed by channel proteins are needed for ion diffusion across cell membranes

The plasma membranes of most living cells are electrically polarized, as indicated by the presence of a transmembrane voltage—or a membrane potential —in the range of 0.1 V. In Chapter 5 , we discussed how the energy stored in this miniature battery can drive a variety of transmembrane transport processes. Electrically excitable cells such as brain neurons and heart myocytes also use this energy for signaling purposes. The brief electrical impulses produced by such cells are called action potentials. To explain these electrophysiological phenomena, we begin with a basic review of electrical energy.

Atoms consist of negatively (−) and positively (+) charged elementary particles, such as electrons (e ⁻ ) and protons (H ⁺ ), as well as electrically neutral particles (neutrons). Charges of the same sign repel each other, and those of opposite sign attract. Charge is measured in units of coulombs (C). The unitary charge of one electron or proton is denoted by e ₀ and is equal to 1.6022 × 10 ⁻¹⁹ C. Ions in solution have a charge valence (z) that is an integral number of elementary charges. For example, z = +2 for Ca ²⁺ , z = +1 for K ⁺ , and z = −1 for Cl ⁻ . The charge of a single ion ( q ₀ ), measured in coulombs, is the product of its valence and the elementary charge:

In an aqueous solution or a bulk volume of matter, the number of positive and negative charges is always equal. Charge is also conserved in any chemical reaction.

The attractive electrostatic force between two ions that have valences of z ₁ and z ₂ can be obtained from Coulomb's law. N6-1 This force ( ℱ ) is proportional to the product of these valences and inversely proportional to the square of the distance (a) between the two. The force is also inversely proportional to a dimensionless term called the dielectric constant ( ε _r ):

N6-1

Coulomb's Law

The attractive electrostatic force between two charged particles of opposite sign and the repulsive electrostatic force between two charged particles of the same sign are described by Coulomb's law. The coulombic force between two interacting particles with charges of q ₁ and q ₂ is

The above equation shows that the electrostatic force is directly proportional to the product of the charges and is inversely proportional to the square of the distance, a, between them. ε ₀ is a physical constant called the permittivity of free space (or the vacuum permittivity) and is equal to 8.854 × 10 ⁻¹² C ² N ⁻¹ m ⁻² , where C is coulomb, N is newton, and m is meter. The denominator of the equation also includes a dimensionless parameter called the dielectric constant ( ε _r ) , also known as the relative permittivity. The dielectric constant of a vacuum is defined as 1.0. The dielectric constant is a property that depends on the polarizability of the medium surrounding the two charges. Polariz­ability refers to the ability of molecules of the medium to orient themselves around ions to reduce electrostatic interactions. Polar water molecules are able to solvate ions effectively by orienting themselves around ions in solutions and thereby reducing coulombic forces between neighboring ions. The dielectric constant of water is therefore relatively high and has a value of ~80. For a nonpolar hydrocarbon, such as decane or the alkyl-chain interior of a phospholipid bilayer, ε is comparatively low and has a value of ~2.

Because the dielectric constant of water is ~40-fold greater than that of the hydrocarbon interior of the cell membrane, the electrostatic force between ions is reduced by a factor of ~40 in water compared with membrane lipid.

If we were to move an Na ⁺ ion from the extracellular to the intracellular fluid without the aid of any proteins, the Na ⁺ would have to cross the membrane by “dissolving” in the lipids of the bilayer. However, the energy required to transfer an Na ⁺ ion from water (high ε) to the interior of a phospholipid membrane (low ε) is ~36 kcal/mole. This value is 60-fold higher than molecular thermal energy at room temperature. Thus, the probability that an ion would dissolve in the bilayer (i.e., partition from an aqueous solution into the lipid interior of a cell membrane) is essentially zero. This analysis explains why inorganic ions cannot readily cross a phospholipid membrane without the aid of other molecules such as specialized transporters or channel proteins, which provide a favorable polar environment for the ion as it moves across the membrane ( Fig. 6-2 ).

Membrane potentials can be measured with microelectrodes as well as dyes or fluorescent proteins that are voltage sensitive

The voltage difference across the cell membrane, or the membrane potential (V _m ), is the difference between the electrical potential in the cytoplasm (ψ _i ) and the electrical potential in the extracellular space (ψ _o ). Figure 6-3 A shows how to measure V _m with an intracellular electrode. The sharp tip of a microelectrode is gently inserted into the cell and measures the transmembrane potential with respect to the electrical potential of the extracellular solution, defined as ground (i.e., ψ _o = 0). If the cell membrane is not damaged by electrode impalement and the impaled membrane seals tightly around the glass, this technique provides an accurate measurement of V _m . Such a voltage measurement is called an intracellular recording .

For an amphibian or mammalian skeletal muscle cell, resting V _m is typically about −90 mV, which means that the interior of the resting cell is ~90 mV more negative than the exterior. There is a simple relationship between the electrical potential difference across a membrane and another parameter, the electric field (E) : N6-2

N6-2

Electric Fields and Potentials

A useful way to represent the electrical force (ℱ) acting on a charged particle is by the concept of an electric field. The electric field (E) is defined as the force that a particle with positive charge q ₀ would sense in the vicinity of a charge source. Forces are vector parameters that are described by a magnitude and a direction. The direction of an electrostatic force is defined by the direction in which a positive charge would move: namely, away from a positively charged source or toward a negatively charged source. Similarly, the direction of an electric field is the direction in which a positive test charge would move within the field. The definition of an electric field is

Although the net charge of any bulk system must be equal to zero, other forms of energy, such as chemical energy, can be used to separate positive and negative charges. The electrical potential (ψ) describes the potential energy that arises from such a separation of charge. The electrical potential difference (Δψ) is a measure of the work ( W ₁₂ ) needed to move a test charge q ₀ between two points (1 and 2) in an electric field:

The electrical potential difference (V) is measured in volts (i.e., joules per coulomb). Because work is also equal to force times distance, the electrical potential difference may also be expressed in terms of the magnitude of the force required to move a test charge ( q ₀ ) over a distance ( a , in centimeters), along the same direction as the force. With the help of the preceding two equations, we can therefore define the electrical potential difference in terms of the electric field (volts per centimeter):

Thus, the voltage difference between two points is the product of electric field and the distance between those points. Conversely, the electric field is the voltage difference divided by the distance:

Accordingly, for a V _m of −0.1 V and a membrane thickness of a = 4 nm (i.e., 40 × 10 ⁻⁸ cm), the magnitude of the elec­tric field is ~250,000 V/cm. Thus, despite the small transmembrane voltage, cell membranes actually sustain a very large electric field. Below, we discuss how this electric field influences the activity of a particular class of membrane signaling proteins called voltage-gated ion channels (see pp. 182–199 ).

Skeletal muscle cells, cardiac cells, and neurons typically have resting membrane potentials of approximately −60 to −90 mV; smooth-muscle cells have membrane potentials in the range of −55 mV; and the V _m of the human erythrocyte is only about −9 mV. However, certain bacteria and plant cells have transmembrane voltages as large as −200 mV. For very small cells such as erythrocytes, small intracellular organelles such as mitochondria, and fine processes such as the synaptic endings of neurons, V _m cannot be directly measured with a microelectrode. Instead, spectroscopic techniques allow the membrane potentials of such inaccessible membranes to be measured indirectly (see Fig. 6-3 B ). Such techniques involve labeling of the cell or membrane with an appropriate organic dye molecule and monitoring of the absorption or fluorescence of the dye. The optical signal of the dye molecule can be independently calibrated as a function of V _m .

Another approach for monitoring changes in V _m optically is to use cells that express genetically engineered voltage-sensing proteins that have been coupled to a modified version of the jellyfish green fluorescent protein (GFP). N6-3 For their work on GFP, Osamu Shimomura, Martin Chalfie, and Roger Tsien shared the 2008 Nobel Prize in Chemistry. N6-4 Whether V _m is measured directly by a microelectrode or indirectly by a spectroscopic technique, virtually all biological membranes are found to have a nonzero membrane potential. This transmembrane voltage is an important determinant of any physiological transport process that involves the movement of charge.

Measurements of V _m have shown that many types of cells are electrically excitable. Examples of excitable cells are neurons, muscle fibers, heart cells, and secretory cells of the pancreas. In such cells, V _m exhibits characteristic time-dependent changes in response to electrical or chemical stimulation. When the cell body, or soma, of a neuron is electrically stimulated, electrical and optical methods for measuring V _m detect an almost identical response at the cell body (see Fig. 6-3 C ). The optical method provides the additional insight that V _m changes are similar but delayed in the more distant neuronal processes inaccessible to a microelectrode (see Fig. 6-3 D ). When the cell is not undergoing such active responses, V _m usually remains at a steady value that is called the resting potential. In the next section, we discuss the origin of the membrane potential and lay the groundwork for understanding its active responses.

Membrane potential is generated by ion gradients

Chapter 5 introduced the concept that some integral membrane proteins are electrogenic transporters in that they generate an electrical current that sets up an electrical potential across the membrane. One class of electrogenic transporters includes the ATP-dependent ion pumps. These proteins use the energy of ATP hydrolysis to produce and to maintain concentration gradients of ions across cell membranes. In animal cells, the Na-K pump and Ca pump are responsible for maintaining normal gradients of Na ⁺ , K ⁺ , and Ca ²⁺ . The reactions catalyzed by these ion transport enzymes are electrogenic because they lead to separation of charge across the membrane. For example, enzymatic turnover of the Na-K pump (see pp. 115–117 ) results in the translocation of three Na ⁺ ions out of the cell and two K ⁺ ions into the cell, with a net movement of one positive charge out of the cell. In addition to electrogenic pumps, cells may express secondary active transporters that are electrogenic, such as the Na/glucose cotransporter (see pp. 121–122 ).

It may seem that the inside negative V _m originates from the continuous pumping of positive charges out of the cell by the electrogenic Na-K pump. The resting potential of large cells—whose surface-to-volume ratio is so large that ion gradients run down slowly—is maintained for a long time even when metabolic poisons block ATP-dependent energy metabolism. This finding implies that an ATP-dependent pump is not the immediate energy source underlying the membrane potential. Indeed, the squid giant axon normally has a resting potential of −60 mV. When the Na-K pump in the giant axon membrane is specifically inhibited with a cardiac glycoside (see p. 117 ), the immediate positive shift in V _m is only 1.4 mV. Thus, in most cases, the direct contribution of the Na-K pump to the resting V _m is very small.

In contrast, many experiments have shown that cell membrane potentials depend on ionic concentration gradients. In a classic experiment, Paul Horowicz and Alan Hodgkin measured the V _m of a frog muscle fiber with an intracellular microelectrode. The muscle fiber was bathed in a modified physiological solution in which replaced Cl ⁻ , a manipulation that eliminates the contribution of anions to V _m . In the presence of normal extracellular concentrations of K ⁺ and Na ⁺ for amphibians ([K ⁺ ] _o = 2.5 mM and [Na ⁺ ] _o = 120 mM), the frog muscle fiber has a resting V _m of approximately −94 mV. As [K ⁺ ] _o is increased above 2.5 mM by substitution of K ⁺ for Na ⁺ , V _m shifts in the positive direction. As [K ⁺ ] _o is decreased below 2.5 mM, V _m becomes more negative ( Fig. 6-4 ). For [K ⁺ ] _o values >10 mM, the V _m measured in Figure 6-4 is approximately a linear function of the logarithm of [K ⁺ ] _o . Numerous experiments of this kind have demonstrated that the immediate energy source of the membrane potential is not the active pumping of ions but rather the potential energy stored in the ion concentration gradients themselves. Of course, it is the ion pumps—and the secondary active transporters that derive their energy from these pumps—that are responsible for generating and maintaining these ion gradients.

One way to investigate the role of ion gradients in determining V _m is to study this phenomenon in an in vitro (cell-free) system. Many investigators have used an artificial model of a cell membrane called a planar lipid bilayer. This system consists of a partition with a hole ~200 µm in diameter that separates two chambers filled with aqueous solutions ( Fig. 6-5 ). It is possible to paint a planar lipid bilayer having a thickness of only ~4 nm across the hole, thereby sealing the partition. By incorporating membrane proteins and other molecules into planar bilayers, one can study the essential characteristics of their function in isolation from the complex metabolism of living cells. Transmembrane voltage can be measured across a planar bilayer with a voltmeter connected to a pair of Ag/AgCl electrodes that are in electrical contact with the solution on each side of the membrane via salt bridges. This experimental arrangement is much like an intracellular voltage recording, except that both sides of the membrane are completely accessible to manipulation.

The ionic composition of the two chambers on opposite sides of the bilayer can be adjusted to simulate cellular concentration gradients. Suppose that we put 4 mM KCl on the left side of the bilayer and 155 mM KCl on the right side to mimic, respectively, the external and internal concentrations of K ⁺ for a mammalian muscle cell. To eliminate the osmotic flow of water between the two compartments (see p. 128 ), we also add a sufficient amount of a nonelectrolyte (e.g., mannitol) to the side with 4 mM KCl. N6-5 We can make the membrane selectively permeable to K ⁺ by introducing purified K ⁺ channels or K ⁺ ionophores into the membrane. N6-6 Assuming that the K ⁺ channels are in an open state and are impermeable to Cl ⁻ , the right (“internal”) compartment quickly becomes electrically negative with respect to the left (“external”) compartment because positive charge (i.e., K ⁺ ) diffuses from high to low concentration. However, as the negative voltage develops in the right compartment, the negativity opposes further K ⁺ efflux from the right compartment. Eventually, the voltage difference across the membrane becomes so negative that further net K ⁺ movement halts. At this point, the system is in equi­librium, and the transmembrane voltage reaches a value of 92.4 mV, right-side negative. In the process of generating the transmembrane voltage, a separation of charge has occurred in such a way that the excess positive charge on the left side (low [K ⁺ ]) balances the same excess negative charge on the right side (high [K ⁺ ]). Thus, the stable voltage difference (−92.4 mV) arises from the separation of K ⁺ ions from their counterions (in this case Cl ⁻ ) across the bilayer membrane.

For mammalian cells, Nernst potentials for ions typically range from −100 mV for K ⁺ to +100 mV for Ca ²⁺

The model system of a planar bilayer (impermeable membrane), unequal salt solutions (ionic gradient), and an ion-selective channel (conductance pathway) contains the minimal components essential for generating a membrane potential. The hydrophobic membrane bilayer is a formidable barrier to inorganic ions and is also a poor conductor of electricity. Poor conductors are said to have a high resistance to electrical current—in this case, ionic current. On the other hand, ion channels act as molecular conductors of ions. They introduce a conductance pathway into the membrane and lower its resistance.

In the planar-bilayer experiment of Figure 6-5 , V _m originates from the diffusion of K ⁺ down its concentration gradient. Membrane potentials that arise by this mechanism are called diffusion potentials. At equilibrium, the diffusion potential of an ion is the same as the equilibrium potential ( E _X ) given by the Nernst equation previously introduced as Equation 5-8 .

The Nernst equation predicts the equilibrium membrane potential for any concentration gradient of a particular ion across a membrane. E _X is often simply referred to as the Nernst potential. The Nernst potentials for K ⁺ , Na ⁺ , Ca ²⁺ , and Cl ⁻ are written as E _K , E _Na , E _Ca , and E _Cl , respectively.

The linear portion of the plot of V _m versus the logarithm of [K ⁺ ] _o for a frog muscle cell (see Fig. 6-4 ) has a slope that is ~58.1 mV for a 10-fold change in [K ⁺ ] _o , as predicted by the Nernst equation. Indeed, if we insert the appropriate values for R and F into Equation 6-4 , select a temperature of 20°C, and convert the logarithm base e (ln) to the logarithm base 10 (log ₁₀ ), we obtain a coefficient of −58.1 mV, and the Nernst equation becomes

For a negative ion such as Cl ⁻ , where z = −1, the sign of the slope is positive:

For Ca ²⁺ ( z = +2), the slope is half of −58.1 mV, or approximately −30 mV. Note that a Nernst slope of 58.1 mV is the value for a univalent ion at 20°C. For mammalian cells at 37°C, this value is 61.5 mV.

At [K ⁺ ] _o values above ~10 mM, the magnitude of V _m and the slope of the plot in Figure 6-4 are virtually the same as those predicted by the Nernst equation (see Equation 6-5 ), which suggests that the resting V _m of the muscle cell is almost equal to the K ⁺ diffusion potential. When V _m follows the Nernst equation for K ⁺ , the membrane is said to behave like a potassium electrode because ion-specific electrodes monitor ion concentrations according to the Nernst equation.

Table 6-1 lists the expected Nernst potentials for K ⁺ , Na ⁺ , Ca ²⁺ , Cl ⁻ , and as calculated from the known concentration gradients of these physiologically important inorganic ions for mammalian skeletal muscle and typical nonmuscle cells. For a mammalian muscle cell with a V _m of −80 mV, E _K is ~15 mV more negative than V _m , whereas E _Na and E _Ca are about +67 and +123 mV, respectively, far more positive than V _m . E _Cl is ~9 mV more negative than V _m in muscle cells but slightly more positive than the typical V _m of −60 mV in most other cells.

TABLE 6-1

Ion Concentration Gradients in Mammalian Cells

ION	OUT (mM)	IN (mM)	OUT/IN	E X * (mV)
Skeletal Muscle
K +	4.5	155	0.026	−95
Na +	145	12	12	+67
Ca 2+	1.0	10 −4	10,000	+123
Cl −	116	4.2	28	−89
	24	12	2	−19
Most Other Cells
K +	4.5	120	0.038	−88
Na +	145.4	15	9.67	+61
Ca 2+	1.0	10 −4	10,000	+123
Cl −	116	20	5.8	−47
	24	15	1.6	−13

* Nernst equilibrium potential of X at 37°C.

What determines whether the cell membrane potential follows the Nernst equation for K ⁺ or Cl ⁻ rather than that for Na ⁺ or Ca ²⁺ ? As we shall see in the next two sections, V _m depends on the relative permeabilities of the cell membrane to the various ions and the concentrations of ions on both sides of the membrane. Thus, the stability of V _m depends on the constancy of plasma ion concentrations—an important aspect of the homeostasis of the milieu intérieur (see pp. 3–4 ). Changes in the ionic composition of blood plasma can therefore profoundly affect physiological function. For example, conditions of low or high plasma [K ⁺ ]—termed hypokalemia and hyperkalemia (see pp. 792–793 ), respectively—result in neurological and cardiac impairment (see Box 37-1 ) due to changes in V _m of excitable cells.

Currents carried by ions across membranes depend on the concentration of ions on both sides of the membrane, the membrane potential, and the permeability of the membrane to each ion

Years before ion channel proteins were discovered, physiologists devised a simple but powerful way to predict the membrane potential, even if several different kinds of permeable ions are present at the same time. The first step, which we discuss in this section, is to compute an ionic current, that is, the movement of a single ion species through the membrane. The second step, which we describe in the following section, is to obtain V _m by summing the currents carried by each species of ion present, assuming that each species moves independently of the others.

The process of ion permeation through the membrane is called electrodiffusion because both electrical and concentration gradients are responsible for the ionic current. To a first approximation, the permeation of ions through most channel proteins behaves as though the flow of these ions follows a model based on the Nernst-Planck electrodiffusion theory, which was first applied to the diffusion of ions in simple solutions. This theory leads to an important equation in medical physiology called the constant-field equation, which predicts how V _m will respond to changes in ion concentration gradients or membrane permeability. Before introducing this equation, we first consider some important underlying concepts and assumptions.

Without knowing the molecular basis for ion movement through the membrane, we can treat the membrane as a “black box” characterized by a few fundamental parameters ( Fig. 6-6 ). We must assume that the rate of ion movement through the membrane depends on (1) the external and internal concentrations of the ion X ([X] _o and [X] _i , respectively), (2) the transmembrane voltage ( V _m ), and (3) a permeability coefficient for the ion X ( P _X ). In addition, we make four major assumptions about how the ion X behaves in the membrane:

• 1

  The membrane is a homogeneous medium with a thickness a.

• 2

  The voltage difference varies linearly with distance across the membrane (see Fig. 6-6 ). This assumption is equivalent to stating that the electric field —that is, the change in voltage with distance—is constant N6-2 throughout the thickness of the membrane. This requirement is therefore called the constant-field assumption.

• 3

  The movement of an ion through the membrane is independent of the movement of any other ions. This assumption is called the independence principle.

• 4

  The permeability coefficient P _X is a constant (i.e., it does not vary with the chemical or electrical driving forces). P _X ( units: centimeters per second) is defined as P _X = D _X β/ a. D _X is the diffusion coefficient for the ion in the membrane, β is the membrane/water partition coefficient for the ion, and a is the thickness of the membrane. Thus, P _X describes the ability of an ion to dissolve in the membrane (as described by β) and diffuse from one side to the other (as described by D _X ) over the distance a.

With these assumptions, we can calculate the current carried by a single ion X ( I _X ) through the membrane by using the basic physical laws that govern (1) the movement of molecules in solution (Fick's law of diffusion; see Equation 5-13 ), (2) the movement of charged particles in an electric field (electrophoresis), and (3) the direct proportionality of current to voltage (Ohm's law). The result is the Goldman-Hodgkin-Katz (GHK) current equation, named after the pioneering electrophysiologists who applied the constant-field assumption to Nernst-Planck electrodiffusion:

I _X , or the rate of ions moving through the membrane, has the same units as electrical current: amperes (coulombs per second).
N6-7 Thus, the GHK current equation relates the current of ion X through the membrane to the internal and external concentrations of X, the transmembrane voltage, and the permeability of the membrane to X. The GHK equation thus allows us to predict how the current carried by X depends on V _m . This current-voltage (I-V) relationship is important for understanding how ionic currents flow into and out of cells.

N6-7

Calculating an Ionic Current from an Ionic Flow

On page 147 of the text, we pointed out that the current carried by ion X through the membrane ( I _x ) has the units of amperes, which is the same as coulombs per second (the coulomb is the fundamental unit of charge). In order to compute how many moles per second of X are passing through the membrane, we need to convert from coulombs to moles. We can compute a macroscopic quantity of charge by using a conversion factor called the Faraday (F) . The Faraday is the charge (in coulombs) of a mole of univalent ions. Put another way, F is the product of the elementary charge ( e ₀ ; see p. 141 ) and Avogadro's number:

Thus, given an ionic current, we can easily compute the flow of the ion:

Figure 6-7 A shows how the K ⁺ current ( I _K ) depends on V _m , as predicted by Equation 6-7 for the normal internal (155-mM) and external (4.5-mM) concentrations of K ⁺ . By convention, a current of ions flowing into the cell (inward current) is defined in electrophysiology as a negative-going current, and a current flowing out of the cell (outward current) is defined as a positive current. (As in physics, the direction of current is always the direction of movement of positive charge. This convention means that an inward flow of Cl ⁻ is an outward current.) For the case of 155 mM K ⁺ inside the cell and 4.5 mM K ⁺ outside the cell, an inward current is predicted at voltages that are more negative than −95 mV and an outward current is predicted at voltages that are more positive than −95 mV (see Fig. 6-7 A ). The value of −95 mV is called the reversal potential ( V _rev ), because it is precisely at this voltage that the direction of current reverses (i.e., the net current equals zero). If we set I _K equal to zero in Equation 6-7 and solve for V _rev , we find that the GHK current equation reduces to the Nernst equation for K ⁺ (see Equation 6-5 ). Thus, the GHK current equation for an ion X predicts a reversal potential ( V _rev ) equal to the Nernst potential ( E _X ) for that ion; that is, the current is zero when the ion is in electrochemical equilibrium. At V _m values more negative than V _rev , the net driving force on a cation is inward; at voltages that are more positive than V _rev , the net driving force is outward. N6-8

N6-8

Shape of the I-V Relationship

In the text, we introduced the GHK current equation as Equation 6-7 (shown here as Equation NE 6-8 ):

In the nonphysiological case in which [K ⁺ ] _i and [K ⁺ ] _o are equal to [K ⁺ ], the above equation reduces to

In this case, the relationship between the K ⁺ current ( I _K ) and V _m should be a straight line that passes through the origin, as shown by the dashed line in Figure 6-7 A .

Similarly, in the nonphysiological case in which [Na ⁺ ] _i and [Na ⁺ ] _o are equal to [Na ⁺ ], the GHK current equation reduces to

Again, the preceding equation predicts that the relationship between the Na ⁺ current ( I _Na ) and V _m also should be a straight line, as shown by the dashed line in Figure 6-7 B in the text. These relationships are “ohmic” because they follow Ohm's law: Δ I = Δ V/R, N6-31 where R in this equation represents resistance. Thus, the slope of the line is 1/ R or the conductance:

Comparing the above equation with the two that precede it, we see that—for the special case in which the ion concentrations ([X]) are identical on both sides of the membrane—the conductance is

Thus, according to the GHK current equation, the membrane's conductance to an ion is proportional to the membrane's permeability and also depends on ion concentration.

What does the GHK current equation predict for more realistic examples in which [K ⁺ ] _i greatly exceeds [K ⁺ ] _o , or [Na ⁺ ] _i is much lower than [Na ⁺ ] _o ? The solid curve in Figure 6-7 A in the text is the prediction of the GHK current equation for the normal internal (155 mM) and external (4.5 mM) concentrations of K ⁺ . By convention, a current of ions flowing into the cell (inward current) is defined in electrophysiology as a negative-going current, and a current flowing out of the cell (outward current) is defined as a positive current. (As in physics, the direction of current is always the direction of movement of positive charge. This means that an inward flow of Cl ⁻ is an outward current.) The nonlinear behavior of the I-V relationship in Figure 6-7 A in the text is solely due to the asymmetric internal and external concentrations of K ⁺ . Because K ⁺ is more concentrated inside than outside, the outward K ⁺ currents will tend to be larger than the inward K ⁺ currents. That is, the K ⁺ current will tend to exhibit outward rectification, as shown by the solid I-V curve in Figure 6-7 A . Such I-V rectification is known as Goldman rectification. It is due solely to asymmetric ion concentrations and does not reflect an asymmetric behavior of the channels through which the ion moves.

For the case of 155 mM K ⁺ inside the cell and 4.5 mM K ⁺ outside the cell, the GHK current equation predicts an inward current at voltages more negative than −95 mV and an outward current for voltages more positive than −95 mV. The value of −95 mV is called the reversal potential ( V _rev ) because it is precisely at this voltage that the direction of current reverses (i.e., the net current equals zero). If we set I _K equal to zero in the GHK current equation and solve for V _rev , we find that this rather complicated equation reduces to the Nernst equation for K ⁺ (see Equation 6-5 in the text, shown here as Equation NE 6-13 ):

Thus, the GHK current equation for an ion X predicts a reversal potential ( V _rev ) equal to the Nernst potential ( E _X ) for that ion; that is, the current is zero when the ion is in electrochemical equilibrium. At voltages more negative than V _rev , the net driving force on a cation is inward; at voltages more positive than V _rev , the net driving force is outward.

Figure 6-7 B in the text shows a similar treatment for Na ⁺ . Again, the dashed line that passes through the origin refers to the artificial situation in which [Na ⁺ ] _i and [Na ⁺ ] _o are each equal to 145 mM. This line describes an ohmic relationship. The solid curve in Figure 6-7 B shows the I-V relationship for a physiological set of Na ⁺ concentrations: [Na ⁺ ] _o = 145 mM, [Na ⁺ ] _i = 12 mM. The relationship is nonlinear solely because of the asymmetric internal and external concentrations of Na ⁺ . Because Na ⁺ is more concentrated outside than inside, the inward Na ⁺ currents will tend to be larger than the outward Na ⁺ currents. That is, the Na ⁺ current will tend to exhibit inward rectification. Again, such I-V rectification is known as Goldman rectification.

Figure 6-7 B shows the analogous I-V relationship predicted by Equation 6-7 for physiological concentrations of Na ⁺ . In this case, the Na ⁺ current ( I _Na ) is inward at V _m values more negative than V _rev (+67 mV) and outward at voltages that are more positive than this reversal potential. Here again, V _rev is the same as the Nernst potential, in this case, E _Na .

Membrane potential depends on ionic concentration gradients and permeabilities

In the preceding section, we discussed how to use the GHK current equation to predict the current carried by any single ion, such as K ⁺ or Na ⁺ . If the membrane is permeable to the monovalent ions K ⁺ , Na ⁺ , and Cl ⁻ —and only to these ions—the total ionic current carried by these ions across the membrane is the sum of the individual ionic currents:

The individual ionic currents given by Equation 6-7 can be substituted into the right-hand side of Equation 6-8 . Note that for the sake of simplicity, we have not considered currents carried by electrogenic pumps or other ion transporters; we could have added extra “current” terms for such electrogenic transporters. At the resting membrane potential (i.e., V _m is equal to V _rev ), the sum of all ion currents is zero (i.e., I _total = 0). When we set I _total to zero in the expanded Equation 6-8 and solve for V _rev , we get an expression known as the GHK voltage equation or the constant-field equation:

Because we derived Equation 6-9 for the case of I _total = 0, it is valid only when zero net current is flowing across the membrane. This zero net current flow is the steady-state condition that exists for the cellular resting potential, that is, when V _m equals V _rev . The logarithmic term of Equation 6-9 indicates that resting V _m depends on the concentration gradients and the permeabilities of the various ions. However, resting V _m depends primarily on the concentrations of the most permeant ion. N6-9

N6-9

Contribution of Ions to Membrane Potential

In the text, we introduced Equation 6-9 (shown here as Equation NE 6-14 )

and pointed out that the resting V _m depends mostly on the concentrations of the most permeant ion. This last statement is only true on the condition that the most permeant ion is also present at a reasonable concentration. It would be more precise to state that V _m depends on a series of permeability-concentration products. Thus, an ion contributes to V _m to the extent that its permeability-concentration product dominates the above equation. An interesting example is the H ⁺ ion, which we omit from Equation NE 6-14 . Although its permeability P _H may be quite high in some cells, H ⁺ concentrations on both sides of the membrane are usually extremely low (at a pH of 7, [H ⁺ ] is 10 ⁻⁷ M). Thus, even though P _H may be large, the product P _H × [H ⁺ ] is usually negligibly small, so that H ⁺ usually does not contribute noticeably to V _m via a P _H × [H ⁺ ] term, which is why we omitted it from the above equation.

The principles underlying Equation 6-9 show why the plot of V _m versus [K ⁺ ] _o in Figure 6-4 —which summarizes data obtained from a frog muscle cell—bends away from the idealized Nernst slope at very low values of [K ⁺ ] _o . Imagine that we expose a mammalian muscle cell to a range of [K ⁺ ] _o values, always substituting extracellular K ⁺ for Na ⁺ , or vice versa, so that the sum of [K ⁺ ] _o and [Na ⁺ ] _o is kept fixed at its physiological value of 4.5 + 145 = 149.5 mM. To simplify matters, we assume that the membrane permeability to Cl ⁻ is very small (i.e., P _Cl ≅ 0). We can also rearrange Equation 6-9 by dividing the numerator and denominator by P _K and representing the ratio P _Na / P _K as α. At 37°C, this simplified equation becomes

Figure 6-8 shows that when α is zero—that is, when the membrane is impermeable to Na ⁺ — Equation 6-10 reduces to the Nernst equation for K ⁺ (see Equation 6-4 ), and the plot of V _m versus the logarithm of [K ⁺ ] _o is linear. If we choose an α of 0.01, however, the plot bends away from the ideal at low [K ⁺ ] _o values. This bend reflects the introduction of a slight permeability to Na ⁺ . As we increase this P _Na further by increasing α to 0.03 and 0.1, the curvature becomes even more pronounced. Thus, as predicted by Equation 6-10 , increasing the permeability of Na ⁺ relative to K ⁺ tends to shift V _m in a positive direction, toward E _Na . In some skeletal muscle cells, an α of 0.01 best explains the experimental data.

The constant-field equation (see Equation 6-9 ) and simplified relationships derived from it (e.g., Equation 6-10 ) show that steady-state V _m depends on the concentrations of all permeant ions, weighted according to their relative permeabilities. Another very useful application of the constant-field equation is determination of the ionic selectivity of channels. For example, when [K ⁺ ] _o is in the normal range, a particular K ⁺ channel in human cardiac myocytes (i.e., the TWIK-1 K2P channel introduced below in Table 6-2 and Fig. 6-20 F ) has an extremely low α—as we could calculate from V _rev and the ion concentrations using Equation 6-10 . However, under conditions of hypokalemia (plasma [K ⁺ ] < 3.5 mM), α becomes substantially larger, which causes a depolarization that can trigger cardiac arrhythmias that may lead to cardiac arrest and sudden death. N6-10

TABLE 6-2

Major Families of Human Ion-Channel Proteins

Data from references listed in
N6-26 .

CHANNEL FAMILY	DESCRIPTION AND SUBFAMILIES	HUMAN GENE SYMBOLS (NUMBER OF GENES): PROTEIN NAMES	NOTED PHYSIOLOGICAL FUNCTIONS	KNOWN HUMAN GENETIC AND AUTOIMMUNE DISEASES	NOTES FOR TOPOLOGY FIGURE
1 Connexin channels	Hexameric gap junction channels	GJA (7)	Cell-cell communication, electrical coupling and cytoplasmic diffusion of molecules between interconnected cells; mediate Ca 2+ waves of coupled cells	GJA1: oculodentodigital dysplasia GJA3, 8: congenital cataract GJA5: familial atrial standstill and fibrillation	Hexamer of 4-TM subunits ( Fig. 6-20 A )
		GJB (7)		GJB1: Charcot-Marie-Tooth disease GJB2, 3, 6: keratitis-ichthyosis-deafness syndrome GJB3, 4: erythrokeratodermia variabilis
		GJC (3) GJD (3) GJE (1)		GJC2: spastic paraplegia, lymphedema
2 Potassium channels (canonical members of VGL channel superfamily)	Homo- or heterotetrameric voltage-gated channels (Kv channels)	KCNA (8): Shaker-related or Kv1 KCNB (2): Shab-related or Kv2	Electrical signaling; repolarization of action potentials; frequency encoding of action potentials	KCNA1: episodic ataxia 1 and myokymia 1 KCNA5: atrial fibrillation 7	Tetramer of 6-TM subunits ( Fig. 6-20 B )
		KCNC (4): Shaw-related or Kv3		KCNC3: spinocerebellar ataxia 3
		KCND (3): Shal-related or Kv4 KCNF (1): modulatory KCNG (4): modulatory
		KCNH (8): eag-related	KCNH2 (cardiac HERG): promiscuously drug-sensitive K + channel responsible for acquired long QT syndrome	KCNH2: long QT syndrome 2, short QT syndrome 1
		KCNQ (5): KvLQT-related		KCNQ1 (cardiac KvLQT1): long QT syndrome 1, Romano-Ward syndrome, Jervell and Lange-Nielsen syndrome 1 and congenital deafness, atrial fibrillation 3, short QT syndrome 2 KCNQ2, 3: benign familial neonatal seizures, early infantile epileptic encephalopathy 7 KCNQ4: deafness 2A
		KCNS (3): modulatory
		KCNV (2)		KCNV2: cone dystrophy with night blindness
	Tetrameric small- and intermediate-conductance Ca 2+ -activated K + channels	KCNN (4) KCNN1, 2, 3: SK Ca 1, 2, 3 = K Ca 2.1, 2.2, 2.3 KCNN4: IK Ca = SK Ca 4 = K Ca 3.1	Repolarization of APs; slow phase of AP afterhyperpolarization; regulation of AP interspike interval and firing frequency; activated by Ca 2+ -calmodulin; voltage-insensitive		Tetramer of 6-TM subunits ( Fig. 6-20 C )
	Tetrameric large-conductance Ca 2+ -, Na + -, or H + - activated K + channels	KCNMA (1): K Ca 1.1 = Slo1= BK Ca	Slo1 (BK Ca ): voltage- and Ca 2+ -activated K + channel; mediation of fast component of AP afterhyperpolarization; feedback regulation of contractile tone of smooth muscle; feedback regulation release of neurotransmitters at nerve terminals and auditory hair cells	KCNMA1: generalized epilepsy and paroxysmal dyskinesia	Tetramer of 7-TM subunits ( Fig. 6-20 D )
		KCNT1 (1): K Ca 4.1 = Slo2.1 = Slick KCNT2 (1): K Ca 4.2 = Slo2.2 = Slack	Slo2.1 (Slick) and Slo2.2 (Slack): low intrinsic voltage dependence and synergistically activated by internal Na + and Cl −
		KCNU (1): KCa5.1 = Slo3	Slo3: activated by voltage and internal pH; involved in sperm capacitation and acrosome; exclusively expressed in spermatocytes and mature spermatozoa
	Homo- or heterotetrameric inward-rectifier channels	KCNJ (16): Kir	Genesis and regulation of resting membrane potential, regulation of electrical excitability		Tetramer of 2-TM subunits ( Fig. 6-20 E )
		KCNJ1, 10, 13: Kir1.1 = ROMK1, Kir1.2, 1.4	Renal outer medullary K + channel	KCNJ1: Bartter syndrome 2 KCNJ10: SESAME complex disorder KCNJ13: snowflake vitreoretinal degeneration
		KCNJ2, 12, 4, 14: Kir2.1, 2, 3, 4 = IRK1, 2, 3, 4	IRK: strong inward rectifiers; blocked by intracellular Mg 2+ and polyamines, activated by PIP 2	KCNJ2: long QT syndrome 7 (Anderson syndrome), short QT syndrome 3, atrial fibrillation 9
		KCNJ3, 6, 9, 5: Kir3.1, 2, 3, 4 = GIRK1, 2, 3, 4	GIRK: G protein–coupled K + channels	KCNJ5: long QT syndrome 13, familial hyperaldosteronism 3
		KCNJ8,11: Kir6.1, 2 = K ATP	K ATP : coupling of metabolism to excitability, release of insulin in pancreas	KCNJ11: familial persistent hyperinsulinemic hypoglycemia 2, neonatal diabetes mellitus
		KCNJ18: Kir2.6		KCNJ18: thyrotoxic hypokalemic periodic paralysis
	Dimeric 2-TM tandem two-pore channels	KCNK (15): K2P KCNK1: TWIK-1	Genesis and regulation of resting membrane potential; regulation of AP firing frequency; sensory perception of touch, stretch, and temperature; involved in mechanism of general anesthesia; activated by chloroform, halothane, heat, internal pH, PIP 2 , fatty acids, G proteins	KCNK9: Birk-Barel mental retardation syndrome KCNK18: Migraine with or without aura 13	Dimer of 4-TM subunits ( Fig. 6-20 F )
3 Hyperpolarization-activated cyclic nucleotide–gated cation channels (VGL superfamily member)	Tetrameric cation-selective HCN channels	HCN (4)	Na + /K + selective, cAMP- and cGMP-activated, I f current in heart; hyperpolarization-activated I h current in heart and neurons; generation of AP automaticity in heart and CNS neurons; mediate depolarizing current that triggers the next AP in rhythmically firing cells	HCN4: sick sinus syndrome 2, Brugada syndrome 8 (tachyarrhythmia)	Tetramer of 6-TM subunits ( Fig. 6-20 G )
4 Cyclic nucleotide–gated cation channels (VGL superfamily member)	Tetrameric CNG channels	CNGA (4)	Cation nonselective channels permeable to Na + , K + , and Ca 2+ ; sensory transduction mechanism in vision, and olfaction, cGMP- and cAMP-gated cation-selective channels in rods, cones, and olfactory receptor neurons	CNGA1 , CNGB1: retinitis pigmentosa CNGA3: achromatopsia 2 (total colorblindness)	Tetramer of 6-TM subunits ( Fig. 6-20 H )
		CNGB (2)		CNGB3: Stargardt disease 1 (macular degeneration), achromatopsia 3
5 Transient receptor potential cation channels (VGL superfamily member)	Tetrameric TRP channels	TRPA (1)	Cation nonselective channels permeable to Na + , K + , and Ca 2+ ; involved in polymodal sensory transduction of pain, itch, thermosensation, various chemicals, osmotic and mechanical stress, taste (TRPM5); TRPV family is also called the vanilloid receptor family, which includes the capsaicin receptor (TRPV1); TRPM8 is the menthol receptor	TRPA1: familial episodic pain syndrome	Tetramer of 6-TM subunits ( Fig. 6-20 I )
		TRPC (6)		TRPC6: focal segmental glomerulosclerosis (proteinuric kidney disease)
		TRPV (6)		TRPV4: hereditary motor and sensory neuropathy
		TRPM (8)		TRPM1: congenital stationary night blindness TRPM4: progressive familial heart block TRPM6, 7: hypomagnesia with secondary hypocalcemia
		PKD (3)		PKD1, 2, 3: polycystic kidney disease
		MCOLN (3)		MOCLN1: mucolipidosis IV
6 NAADP receptor Ca 2+ -release channels (VGL superfamily member)	Dimeric 6-TM tandem two-pore Ca 2+ channels (related to TRP, CatSper, and Cav channels)	TPCN (2)	Ca 2+ -selective channels activated by NAADP that mediate release of Ca 2+ from acidic stores and lysosomes	TPCN2: Genetic differences are linked to variations in human skin, hair, and eye pigmentation	Dimer of 12-TM subunits ( Fig. 6-20 J )
7 Voltage-gated sodium channels (VGL superfamily member) (see Table 7-1 )	Pseudo-tetrameric voltage-gated channels (Nav channels)	SCN (10): Nav	Na + selective, voltage-activated channels that mediate the depolarizing upstroke of propagating APs in neurons and muscle; blocked by local anesthetics SCN7A (Nax, Nav2.1) senses plasma [Na + ] in brain circumventricular organs	SCN1A: generalized epilepsy with febrile seizures SCN2A: infantile epileptic encephalopathy SCN4A: hyperkalemic periodic paralysis, paramyotonia congenita, potassium-aggravated myotonia SCN5A: cardiac long QT syndrome 3 SCN9A: primary erythermalgia, paroxysmal extreme pain disorder, congenital indifference to pain	Monomer of 4 × 6 TMs ( Fig. 6-20 K )
8 Voltage-gated calcium channels (VGL superfamily member) (see Table 7-2 )	Pseudo-tetrameric voltage-gated channels (Cav channels)	CACNA (10): Cav	CACNA genes encode Ca 2+ -selective, voltage-activated channels that mediate prolonged depolarizing phase of APs in muscle and neurons; entry of Ca 2+ via Cav triggers release of transmitter and hormone secretion; molecular target of Ca-blocker drugs	CACNA1A: episodic ataxia 2, familial hemiplegic migraine, spinocerebellar ataxia 6 CACNA1A, 1B: antibodies to channel proteins cause Lambert-Eaton syndrome CACNA1C: Timothy syndrome arrhythmia, Brugada syndrome 3 CACNA1F: congenital stationary night blindness, X-linked cone-rod dystrophy 3 CACNA1H: idiopathic generalized epilepsy 6 CACNA1S: hypokalemic periodic paralysis, malignant hyperthermia	Monomer of 4 × 6 TMs ( Fig. 6-20 L )
		NALCN (1): Na + leak	NALCN gene encodes a voltage-insensitive cation channel that mediates a resting Na + leak current in neurons
9 CatSper cation channels of sperm (VGL superfamily member)	Heterotetrameric 6-TM voltage-gated Ca 2+ -selective channels of sperm	CATSPER (4)	Essential for hyperactivation of sperm cell motility; located in sperm tail membrane; activated by high pH; required for male fertility	CATSPER1: spermatogenic failure CATSPER2: deafness-infertility syndrome	Tetramer of 6-TM subunits ( Fig. 6-20 M )
10 Hv voltage-gated proton channels (VGL superfamily member)	Dimeric 4-TM H + channels	HVCN1	4-TM monomer is similar to S1–S4 region of voltage-gated channels; mediates H + efflux from sperm flagellum, innate immune function of neutrophils where H + efflux compensates outward charge movement of electrons via NADPH oxidase in phagocytes; inhibited by Zn 2+		Dimer of 4-TM subunits ( Fig. 6-20 N )
11 Ligand-gated ion channels (pentameric Cys-loop receptor superfamily)	Pentameric nicotinic, cholinergic ionotropic receptors	CHRNA (10): α subunits CHRNB (4): β subunits CHRNG (1): γ subunits CHRND (1): δ subunits CHRNE (1): ε subunits	Na + , K + non-selective cation channels activated by binding of ACh; mediate depolarizing postsynaptic potentials, EPSPs; site of action of nicotine	CHCRNA1, B1, E, D: slow-channel syndromes, fast-channel syndromes CHRNA2, A4, B2: nocturnal frontal lobe epilepsy CHRNA1: antibodies to channel proteins cause myasthenia gravis	Pentamer of 4-TM subunits ( Fig. 6-20 O )
	Pentameric serotonin 5HT 3 ionotropic receptors	HTR3A (1) HTR3B (1) HTR3C (1) HTR3D (1) HTR3E (1)	Na + , K + nonselective, cation channels activated by binding of serotonin; mediate depolarizing postsynaptic potentials, EPSPs		Same as above
	Pentameric GABA A ionotropic receptors	GABRA (6) GABRB (3) GABRD (1) GABRE (1) GABRG (3) GABRP (1) GABRQ (1) GABRR (3)	Cl − -selective anion channels activated by binding of GABA; mediate hyperpolarizing postsynaptic potentials, IPSPs; site of action of benzodiazepines and barbiturates		Same as above
	Pentameric glycine ionotropic receptors	GLRA (4) GLRB (1)	Cl − -selective anion channels activated by binding of glycine; mediate IPSPs	GLRA1, 1B: hyperekplexia or startle disease	Same as above
12 Glutamate-activated cation channels (see Fig. 13-15 and Table 13-2 )	Tetrameric AMPA receptor cation-selective channels	GRIA (4) GRIA1, 2, 3, 4: GluR1, 2, 3, 4 = GluA1, 2, 3, 4	Na + , K + nonselective cation channels activated by binding of glutamate; mediate depolarizing postsynaptic potentials, EPSPs; involved in long-term potentiation of neuronal memory		Tetramer of 3-TM subunits ( Fig. 6-20 P )
	Tetrameric kainate receptor cation-selective channels	GRIK (5) GRIK1: GluR5 = GluK1 GRIK2: GluR6 = GluK2 GRIK3: GluR7 = GluK3 GRIK4: KA1 = GluK4 GRIK5: KA2 = GluK5	Same as above
	Tetrameric NMDA receptor cation-selective channels	GRIN (7) GRIN1: NR1 = GluN1 GRIN2A: NR2A = GluN2A GRIN2B: NR2B = GluN2B GRIN2C: NR2C = GluN2C GRIN2D: NR2D = GluN2D GRIN3A: NR3A = GluN3A GRIN3B: NR3B = GluN3B	Same as above but also permeable to Ca 2+		Same as above
13 Purinergic ligand-gated cation channels	Trimeric P2X receptor cation channels	P2RX (7) P2RX1, 2, 3, 4, 5, 6, 7: P2X1, P2X2, P2X3, P2X4, P2X5, P2X6, P2X7	ATP-activated cation channels permeable to Na + , K + , Ca 2+ ; involved in excitatory synaptic transmission and nociception, regulation of blood clotting; channel activated by synaptic co-release of ATP in catecholamine-containing synaptic vesicles		Trimer of 2-TM subunits ( Fig. 6-20 Q )
14 Epithelial sodium channels/degenerins	Heterotrimeric ENaC epithelial amiloride-sensitive Na + channels and homotrimeric ASIC acid-sensing cation channels	SCNN1A (1): α subunit SCNN1B (1): β subunit SCNN1D (1): δ subunit SCNN1G (1): γ subunit ACCN (5)	SCNN1 genes encode amiloride-sensitive Na + -selective channels mediating Na + transport across tight epithelia; ACCN genes encode ASIC cation channels activated by external H + , which are involved in pain sensation in sensory neurons following acidosis	SCNN1A, 1B: pseudohypoaldosteronism 1 SCNN1A, 1B, 1G: bronchiectasis with or without elevated sweat chloride SCNN1B, 1G: Liddle syndrome (hypertension)	Trimer of 2-TM subunits ( Fig. 6-20 R )
15 Cystic fibrosis transmembrane regulator (see Table 5-6 )	CFTR; channel protein contains two internally homologous domains	ABCC7: CFTR Part of the ABC family (49)	Cl − -selective channel coupled to cAMP regulation; Cl − transport pathway in secretory and absorptive epithelia; regulated by ATP binding and hydrolysis at two intracellular nucleotide-binding domains	ABCC7: cystic fibrosis	Monomer of 2 × 6 TMs ( Fig. 6-20 S )
16 ClC chloride channels	Dimeric ClC Cl − channels	CLCN (9) CLCN1, 2, 3, 4, 5, 6, 7: CLC-1, 2, 3, 4, 5, 6, 7 CLCNKA: CLC-K1 CLCNKB: CLC-K2	Cl − -selective, voltage-sensitive anion channels in muscle, neurons, and many other tissues; many ClC channels also function as H + /Cl − exchange transporters in endosomes, synaptic vesicles, and lysosomes; involved in regulation of electrical excitability in skeletal muscle, mediation of Cl − transport in epithelia, regulatory volume decrease	CLCN1: Becker disease, Thomsen disease (congenital myotonia) CLCN2: idiopathic and juvenile epilepsy CLCN5: Dent disease complex, nephrolithiasis CLCN7: osteopetrosis CLCNKA, CLCNKB: Bartter syndromes	Monomer of 14 TMs ( Fig. 6-20 T )
17 CaCC Ca 2+ -activated chloride channels	Anoctamin family of Ca 2+ - and voltage-activated Cl − channels	ANO (10) ANO1, 2, 3, 4, 5, 6, 7, 8, 9, 10: TMEM16A, B, C, D, E, F, G, H, I, J	Present in epithelia, smooth muscle, photoreceptors, olfactory sensory neurons; activated at more than ~1 µM cytosolic Ca 2+ ; involved in Cl − secretion, smooth muscle contraction, amplification of olfactory stimulus		Dimer of 8-TM subunits ( Fig. 6-20 U )
18 IP 3 -activated Ca 2+ channels	Tetrameric IP 3 receptor channels	ITPR (3) ITPR1, 2, 3: IP 3 R1, 2, 3 = InsP3R-1, 2, 3	Intracellular cation channel permeable to Na + , K + , and Ca 2+ ; activated by binding of IP 3 and Ca 2+ ; coupled to receptor activation of PLC and hydrolysis of PIP 2 ; regulated by binding of ATP; mediates excitation-contraction coupling in smooth muscle and participates in intracellular Ca 2+ release and signaling in many cells	ITPR1: spinocerebellar ataxia	Tetramer of 6-TM subunits ( Fig. 6-20 V )
19 RYR Ca 2+ -release channels	Tetrameric ryanodine receptor Ca 2+ -release channels	RYR (3) RYR1, 2, 3: RYR1, 2, 3	Intracellular cation channel permeable to Ca 2+ ; intracellular Ca 2+ -release channel activated by mechanical coupling to Cav channel in skeletal muscle or by plasma membrane Ca 2+ entry in heart and smooth muscle	RYR1: malignant hyperthermia, central core disease, congenital myopathy RYR2: familial arrhythmogenic right ventricular dysplasia, catecholaminergic polymorphic ventricular tachycardia	Tetramer of 4-TM subunits ( Fig. 6-20 W )
20 Orai store-operated Ca 2+ channels	Multimeric Orai Ca 2+ -selective channels	ORAI (3) (also known as ICRAC for Ca 2+ -release activated Ca 2+ current or SOC channels for store-operated Ca 2+ entry)	Plasma membrane, low-conductance Ca 2+ channel predominantly found in nonexcitable cells such as epithelia and lymphocytes; activated via PLC-coupled pathways leading to IP 3 -activated Ca 2+ release from ER; Ca 2+ depletion in ER activates an ER membrane protein (STIM) which activates Orai, resulting in entry of extracellular Ca 2+ ; functions in lymphocyte activation and epithelial secretion	ORAI1: severe combined immunodeficiency syndrome (SCID)	Tetramer of 4-TM subunits ( Fig. 6-20 X )

AP, action potential; CNS, central nervous system; ER, endoplasmic reticulum; EPSP, excitatory postsynaptic potential; IPSP, inhibitory postsynaptic potential; PIP ₂ , phosphatidylinositol 4,5-bisphosphate; PLC, phospholipase C; SESAME, seizures, sensorineural deafness, ataxia, mental retardation, and electrolyte imbalance; TM, transmembrane segment.

In general, the resting potential of most vertebrate cells is dominated by high permeability to K ⁺ , which accounts for the observation that the resting V _m is typically close to E _K . The resting permeability to Na ⁺ and Ca ²⁺ is normally very low. Skeletal muscle cells, cardiac cells, and neurons typically have resting membrane potentials ranging from −60 to −90 mV. As discussed in Chapter 7 , excitable cells generate action potentials by transiently increasing Na ⁺ or Ca ²⁺ permeability and thus driving V _m in a positive direction toward E _Na or E _Ca . A few cells, such as vertebrate skeletal muscle fibers, have high permeability to Cl ⁻ , which therefore contributes to the resting V _m . This high permeability also explains why the Cl ⁻ equilibrium potential in skeletal muscle is essentially equivalent to the resting potential (see Table 6-1 ).

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, conductance (G):

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

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 .

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:

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

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:

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.

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

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

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

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 ⁺ :

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

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

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 ):

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

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 ⁺ :

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

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.

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”

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

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

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

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:

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

Thus, the time course of the decay in voltage is

Figure 6-11 C shows that the capacitative current ( I _C ) is zero before the switch is closed, when the voltage is stable at V ₀ . When we close the switch, charge begins to flow rapidly off the capacitor, and the magnitude of I _C is maximal. As the charge on the capacitor gradually falls, the rate at which charge flows off the capacitor gradually falls as well until I _C is zero at “infinite” time. Note, however, that V and I _C relax with the same time constant. N6-16

N6-16

Time Constant of Capacitative Current

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 _C ) by taking the derivative of charge (Q) in Equation 6-16 with respect to time:

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

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

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

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:

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.