Ion channels are macromolecular pores in cell membranes. When they evolvedand what role they may have played in the earliest forms of life we still do notknow, but today ion channels are most obvious as the fundamental excitable ele-ments in the membranes of excitable cells. Ion channels bear the same relation toelectrical signaling in nerve, muscle, and synapse as enzymes bear to metabolism.Although their diversity is less broad than that of enzymes, there are still manytypes of channels working in concert, opening and closing to shape the signalsand responses of the nervous system. Sensitive but potent amplifiers, they detectthe sounds of chamber music and guide the artist’s paintbrush, yet also generatethe violent discharges of the electric eel or the electric ray. They tell the Parameci-um to swim backward after a gentle collision, and they propagate the leaf-closingresponse of the Mimosa plant.

More than 3 billion years ago, primitive replicating forms became enveloped ina lipid film, a bimolecular diffusion barrier that separated the living cell from itsenvironment. Although this lipid membrane had the advantage of retaining vitalcell components, it would have prevented the access of necessary ionized sub-strates and the loss of ionized waste products. Thus new transport mechanismshad to evolve hand-in-hand with the appearance of the membrane. One generalsolution would have been to make pores big enough for all small metabolites topass through, yet small enough to retain macromolecules. Indeed, the outer mem-branes of Gram-negative bacteria and of mitochondria are built on this plan.However, the cytoplasmic membranes of all contemporary organisms follow amore elaborate design, with many, more-selective transport devices handling dif-ferent jobs, often under separate physiological control.

Chapter 1

Introduction

1

How do these membrane transport devices work? Until the 1980s, most ofwhat we knew about them came from physiological flux measurements. Physiol-ogists traditionally divided transport mechanisms into two classes—carriers andpores—largely on the basis of kinetic criteria. For example, the early literaturetried to distinguish carrier from pore on the basis of molecular selectivity, saturat-ing concentration dependence of fluxes, or stoichiometric coupling of the numberof molecules transported. A carrier was viewed as a ferryboat diffusing back andforth across the membrane while carrying small molecules bound to stereospecificbinding sites, whereas a pore was viewed as a narrow, water-filled tunnel, perme-able to those few ions and molecules small enough to fit through the hole.

The moving-ferryboat view of a carrier is no longer considered valid becausethe numerous carrier devices that have been purified from membranes and clonedare large proteins—too large to diffuse or spin around at the rate needed toaccount for the fluxes they catalyze. Furthermore, their amino acid sequencesshow that the peptide chains of the transport protein already are stably threadedback and forth in a large number of transmembrane segments. The newer view ofcarrier transport is that much smaller motions within the protein leave the macro-molecule fixed in the membrane while exposing the transport binding site(s)alternately to the intracellular and extracellular media. It is not difficult to imagineways to do this, but we must develop new experimental insights before such ideascan be tested. Thus the specific mechanism of transport by such important carrierdevices as the Na+-K+ pump, the Ca2+ pump, Na+-Ca2+ exchanger, Cl–-HCO3

–

exchanger, glucose transporter, the Na+-coupled co- and countertransporters, andso on, remains unknown. Fortunately the first crystal structure of a carrierappeared in 2000, promising major new insights.

In contrast to carrier structure, the water-filled pore view for the other class oftransport mechanisms has been firmly established for ion channels of excitablemembranes. In the years between 1965 and 1980, a valuable interplay betweenstudies of excitable membranes and studies on model pores, such as the grami-cidin channel in lipid bilayers, accelerated the pace of research and greatly sharp-ened our understanding of the transport mechanism. The biggest technicaladvance of this period was the development of methods to resolve the activity ofindividual channel molecules. As we consider much more extensively in laterchapters, this led to the discovery that the rate of passage of ions through oneopen channel—often more than 106 ions per second—is far too high for any mech-anism other than a pore. Subsequently, crystal structures revealed the expectedcontinuous aqueous pathway passing right through the channel molecule. Selec-tivity, saturation, and stoichiometry are no longer the best criteria for distinguish-ing between pore and carrier.

Channels and ions are needed for excitation

Physiologists have long known that ions play a central role in the excitability ofnerve and muscle. In an important series of papers from 1881 to 1887, Sidney

2 Chapter 1

Ringer showed that the solution perfusing a frog heart must contain salts ofsodium, potassium, and calcium mixed in a definite proportion if the heart is tocontinue beating for long. Walther Nernst’s (1888) work with electrical poten-tials arising from the diffusion of electrolytes in solution inspired numerousspeculations of an ionic origin of bioelectric potentials. For example, some sug-gested that the cell is more negative than the surrounding medium becausemetabolizing tissue makes acids, and the resulting protons (positive charge) candiffuse away from the cell more easily than the larger organic anions. NextJulius Bernstein (1902, 1912) correctly proposed that excitable cells are sur-rounded by a membrane selectively permeable to K+ ions at rest and that duringexcitation the membrane permeability to other ions increases. His “membranehypothesis” explained the resting potential of nerve and muscle as a diffusionpotential set up by the tendency of positively charged ions to diffuse from theirhigh concentration in cytoplasm to their low concentration in the extracellularsolution while other ions are held back. During excitation, the internal negativi-ty would be lost transiently as other ions are allowed to diffuse across the mem-brane, effectively short-circuiting the K+ diffusion potential. In the English-lan-guage literature, the words “membrane breakdown” were used to describeBernstein’s view of excitation.

During the twentieth century, major cellular roles were discovered for each ofthe cations of Ringer’s solution (Na+, K+, Ca2+), as well as for most of the otherinorganic ions of body fluids, including H+, Mg2+, Cl–, HCO3

–, and HPO42–. The

rate of discovery of new roles for ions in cell physiology has been acceleratingrather than slowing, so the list of ions and their uses will continue to lengthen.Evidently, no major ion has been overlooked in evolution. Each has been assignedseveral specific regulatory, transport, or metabolic tasks. None is purely passivelydistributed across the cell membrane. Each has at least one carrier-like transportdevice coupling its movement to the movement of another ion. Both Na+ and H+

ions have transport devices coupling their “downhill” movements to the “uphill”movements of organic molecules. Na+, K+, H+, and Ca2+ ions are pumped uphillby ATP-driven pumps. Protons are pumped across some membranes by electrontransport chains, and their subsequent downhill flow can drive the phosphoryla-tion of ADP to make ATP. Proton movements, through their effects on intracellu-lar pH, also influence the relative rates of virtually every enzymatic reaction.

All of the ion movements listed above are considered to be mediated by thecarrier class of transport devices and, although they establish the ion gradientsneeded for excitation, they are not themselves part of the excitation process. Read-ers interested in the details of ion pumps or coupled cotransport and exchangedevices can consult books on cell physiology.

Excitation and electrical signaling in the nervous system involve the movementof ions through ion channels. The Na+, K+, Ca2+, and Cl– ions are responsible foralmost all of the action. Each channel may be regarded as an excitable molecule, asit is specifically responsive to some stimulus: a membrane potential change, aneurotransmitter or other chemical stimulus, a mechanical deformation, and so

Introduction 3

on. The channel’s response, called gating, is apparently a simple opening or clos-ing of the pore. The open pore has the important property of selective permeabil-ity, allowing some restricted class of small ions to flow passively down their elec-trochemical activity gradients at a rate that is very high (>106 ions per second)when considered from a molecular viewpoint. We consider the high throughputrate a diagnostic feature distinguishing ion channel mechanisms from those ofother ion transport devices such as the Na+-K+ pump. An additional major featureis a restriction to downhill fluxes not coupled stoichiometrically to the immediateinjection of metabolic energy.

These concepts can be illustrated using the neurotransmitter-sensitive channelsof muscle fibers. At the neuromuscular junction or endplate region of vertebrateskeletal muscle, the nerve axon instructs the muscle fiber when it is time to con-tract. Pulse-like electrical messages called action potentials are sent down themotor nerve from the central nervous system. When they reach the nerve termi-nal, action potentials evoke the release of a chemical signal, the neurotransmitteracetylcholine, which in turn diffuses to the nearby muscle surface and causesacetylcholine-sensitive channels there to open.

Figure 1.1 shows an electrical recording from a tiny patch of muscle membrane.The preparation is actually an embryonic muscle cell in tissue culture withoutnerves, but it still has neurotransmitter-sensitive ion channels that can be openedby applying a low concentration of acetylcholine. In this experiment, ion fluxes inthe channels are detected as electric current flow in the recording circuit, and sincethe recording sensitivity is very high, the opening and closing of one channelappear as clear step changes in the record. Each elementary current step corre-sponds to over 107 ions flowing per second in the open channel. Gating keeps thechannel open for a few milliseconds. Other experiments with substitutions of ionsin the bathing medium show that this type of channel readily passes monovalentcations with diameters up to 6.5 Å (0.65 nm) but does not pass anions.

How do gated ion fluxes through pores make a useful signal for the nervoussystem? For the electrophysiologist, the answer is clear: Ion fluxes are electric cur-rents across the membrane and therefore they have an immediate effect on mem-brane potential. Other voltage-gated channels in the membrane detect the changein membrane potential and in turn become excited. In this way the electricresponse is made regenerative and self-propagating. This explanation doesdescribe how most signals are propagated, but it is circular. Is the ultimate pur-pose of excitation to make electricity so that other channels will be excited andmake electricity? Except in the case of an electric organ, the answer is clearly “no.”Electricity is the means to carry the signal to the point where a nonelectricalresponse is generated. As far as is known, this final transduction always startsthrough a single common pathway: A membrane potential change opens or closesa Ca2+-permeable channel, either on the surface membrane or on an internalmembrane, and a Ca2+ flux into the cytoplasm is altered, causing a change in theinternal free Ca2+ concentration. The ultimate response is then triggered by the

4 Chapter 1

internal Ca2+ ions. This is how the nervous system controls the contraction of amuscle fiber or the secretion of neurotransmitters, neurohormones, digestiveenzymes, and so on. Internal free Ca2+ also controls the gating of some channels,the activities of many enzymes, and the expression of many genes.

Ion channels are found in the membranes of all cells, prokaryotic and eukary-otic. Their known functions include establishing a resting membrane potential,shaping electrical signals, gating the flow of messenger Ca2+ ions, controlling cellvolume, and regulating the net flow of ions and fluids across epithelial cells ofsecretory and resorptive tissues. The emphasis in this book is on well-known chan-nels underlying the action potentials and synaptic potentials of nerve and musclecells. These have long been the focus of traditional membrane biophysics. As thebiophysical methods eventually were applied to study fertilization of eggs, swim-ming of protozoa, glucose-controlled secretion of insulin by pancreatic beta cells, oracetylcholine-induced secretion of epinephrine from chromaffin cells, similar chan-nels were found to play central roles. We must now consider that nerve, muscle,endocrine and secretory glands, white blood cells, platelets, gametes, and protistsall share common membrane mechanisms in their responsiveness to stimuli. Simi-larly, as biophysical methods were applied to transporting epithelia of animals andplants, ion channels were found that participate in moving salts. These channelstoo are ion-selective, gated pores controlled by hormonal influences.

Channels get names

The first naming of ion channels is typically not systematic. Biophysical workattempts to distinguish different components of membrane permeability by theirkinetics, pharmacology, and response to ion substitution. Then a kinetic model is

Introduction 5

1.1 Open–Shut Gating of an Ion Channel Ionic current flowing across a tinypatch of excitable membrane showing eight brief openings (downward currentdeflections) of single ion channels. The membrane patch has been excised from a cul-tured rat myotube and is bathed artificially on both sides by Na+ salt solutions.Approximately 300 nM of the neurotransmitter acetylcholine applied to the extracel-lular membrane face is causing channels to open occasionally. At the –140-mVapplied membrane potential, one open channel passes –6.6 pA, corresponding to aprodigious flow of 4.1 × 107 ions per second through a single pore. T = 23°C. [FromSánchez et al. 1986.]

0

–4

–8

Ioni

c cu

rren

t (pA

)

Channel open

Channel closed

0 100 200 300 400Time (ms)

often made expressing each of the apparent components mathematically. Finally,it is tacitly assumed that each component of the model corresponds to a type ofchannel, and the putative channels are given the same names as the permeabilitycomponents in the original analysis. Thus in their classic analysis of ionic currentsin the squid giant axon, Hodgkin and Huxley (1952d) recognized three differentcomponents of current, which they called sodium, potassium, and leakage. Todaythe names Na channel and K channel are universally accepted for the corre-sponding classes of ion channels in axons.

Thirty years after Hodgkin and Huxley’s work, a new era of channel identifica-tion began. Advances in molecular genetics made it possible to clone individualchannels and eventually to sequence entire genomes. We now can recognize alarge number of channel genes and are approaching the time when we will beable to make comprehensive lists of the complement of channels for organisms atevery level of evolution. Already we are amazed to find far greater numbers ofchannel subtypes than the electrophysiological approach was able to distinguish.For Na, K, and Ca channels alone there are more than 100 identified genes in amammal like the rat, or in the worm Caenorhabditis elegans. Now we face the prob-lem of applying useful names to these channels.

Naming a channel after the most important permeant* ion seems rational butfails when the ions involved are not adequately known, or when no ion is themajor ion. This method is also confusing if many channels use the same ion. Suchproblems classically led to “names” like A, B, C, and so on, for permeability com-ponents in molluscan ganglion cells (Adams, Smith, and Thompson 1980), or qr,si, and x1 in cardiac Purkinje fibers (McAllister et al. 1975). Other approaches weresimply descriptive: Channels were named after inhibitors, as in the amiloride-sen-sitive Na channel; after neurotransmitters, as in glycine receptors; after mutations,as in Shaker; or after diseases associated with a channel defect, as in CFTR (cysticf ibrosis transmembrane regulator). When the cloning of channels began, each lab-oratory invented acronyms for the sequences they obtained (ROMK, GIRK, PN2),and, since several labs might clone the same sequences simultaneously, duplicatenames became abundant.

Such loose nomenclature is not practical, and gradually groups of investigatorshave had to agree on a more systematic approach analogous to that for enzymestaken by the Enzyme Commission. Knowledge of channel sequences allows struc-tural and evolutionary relationships to form the basis for classification. The firststep was a numbering system for mammalian voltage-gated K channels: Kv1.1, 1.2,…, 8.1 (Chandy 1991). The International Union of Pharmacology (IUPHAR) is for-malizing new naming systems for other channel clones, as it has already done formembrane receptors. These consensus names should be used as soon as they areavailable. In addition to these systematic, structurally based names, the post-

6 Chapter 1

*The words “permeant” and “permeable” are sometimes confused. An ion is permeant: capable of per-meating. A channel is permeable: capable of being permeated. In French, a raincoat is an imperméable.

genomic era will offer gene names on the chromosomes of each organism. Thesenames might be less widely used by channel physiologists except when dealing,for example, with specific genetic diseases, where they are completely appropriate.

Channels have families

Biophysicists long recognized that voltage-gated Na, K, and Ca channels havemany functional similarities. Likewise, synaptic channels gated by acetylcholine,glycine, and γ-aminobutyric acid seemed similar. Molecular genetics has confirmedmost of these relationships. The predicted amino acid sequences for channelsreveal strong structural similarities among groups of channels, allowing us to talkabout families of homologous channel proteins that have evolved by processes ofsuccessive gene duplication, mutation, and selection from common ancestral chan-nels. The situation is exactly as has been found for many classes of proteins in biol-ogy. A functionally defined type of channel is not a single structural entity. Allchannels can be expressed in various isoforms coded by different genes or as dif-ferent splice variants that may be selectively expressed in certain cell types and incertain periods of the development and growth of an organism. In evolution, newclasses of channels appear when parts of old ones are recombined with functionaldomains of enzymes and signaling proteins to create new functionality.

Ohm’s law is central

More than in most areas of biology, we see in the study of ion channels how muchcan be learned by applying simple laws of physics. Much of what we know aboution channels is deduced from electrical measurements. Therefore it is essential toremember some rules of electricity before discussing experiments. The remainderof this chapter is a digression on the necessary rules of physics.

To do biophysical experiments well, one must often make sophisticated use ofelectrical ideas; however, as this book is concerned with channels and not with tech-niques of measurement, the essential principles are few. The most important isOhm’s law, a relation between current, voltage, and conductance, which we nowreview.

All matter is made up of charged particles. They are normally present in equalnumbers, so most bodies are electrically neutral. A mole of hydrogen atoms con-tains Avogadro’s number (N = 6.02 × 1023) of protons and the same number of elec-trons. A mole of NaCl contains a mole of Na+ cations and a mole of Cl– anions.Quantity of charge is measured in coulombs (abbreviated C), where the elemen-tary charge, the charge of a proton, is qe = 1.6 × 10–19 C. The charge on Avogadro’snumber of elementary charges is called the Faraday constant: F = Nqe = 6 × 1023 ×1.6 × 10–19 ≈ 105 C/mol. This is the charge on a mole of protons or on a mole ofNa+, K+, or any other monovalent cation. The charge on a mole of Ca2+, Mg2+, andother divalent cations is 2F and the charge on a mole of Cl– ions or other monova-lent anions is –F.

Introduction 7

Electrical phenomena arise whenever charges of opposite sign become separat-ed or can move independently. Any net flow of charges is called a current. Cur-rent is measured in amperes (abbreviated A), where one ampere corresponds to asteady flow of one coulomb per second. By the convention established by Ben-jamin Franklin, positive current flows in the direction of movement of positivecharges. Hence if positive and negative electrodes are placed in Ringer’s solution,Na+, K+, and Ca2+ ions will drift toward the negative pole, Cl– ions will drifttoward the positive pole, and an electric current is said to flow through the solu-tion from positive to negative pole. Michael Faraday proposed the names ions,cations, and anions. In his terminology, when anions flow to the positive pole andcations to the negative pole, both are carrying electric current towards the negativepole. The size of the current is determined by two factors: the potential differencebetween the electrodes and the electrical conductance of the solution betweenthem. Potential difference is measured in volts (abbreviated V) and is defined asthe work needed to move a unit test charge in a frictionless manner from onepoint to another. To move a coulomb of charge across a 1-volt potential differencerequires a joule of work. In common usage the words “potential,” “voltage,” and“voltage difference” are used interchangeably to mean potential difference, espe-cially when referring to a membrane.

In physics, the measure of the ease of flow of current between two points iscalled conductance. The conductance between two electrodes in salt water can be

8 Chapter 1

TABLE 1.1 Physical Constants

Avogadro’s number N = 6.0221 × 1023 mol–1

Elementary charge qe = 1.6022 × 10–19 C

Faraday’s constant F = Nqe = 9.6485 × 104 C mol–1

Absolute temperature T(K) = 273.15 + T (°Celsius)

Boltzmann’s constant kB = 1.3807 × 10–23 V C K–1

= 1.3807 × 10–23 J K–1

Gas constant R = 1.987 cal mol–1 K–1

= 8.3145 J mol–1 K–1

= 8.3145 V C mol–1 K–1

Polarizability of free space ε0 = 8.8542 × 10–12 C V–1 m–1

Planck’s constant h = 6.6261 × 10–34 J s

One joule 1 J = 1 kg m2 s–2

= 1 V C = 1 W s

= 0.2389 cal

increased by adding more salt, by switching to a salt whose ions are more mobilein water, or by bringing the electrodes closer together. Conductance can bedecreased by placing a nonconducting obstruction between the electrodes, bymoving them farther apart, or by making the solution between them more vis-cous. Conductance is measured in siemens (abbreviated S and formerly calledmho) and is defined by Ohm’s law in simple conductors:

I = gE (1.1a)

which says that current (I) equals the product of conductance (g) and voltage dif-ference (E) across the conductor. For ion channels, conductance is a natural mea-sure of their ability to allow ions to move, but many electrical discussions areposed instead in terms of the reciprocal of conductance, called resistance (symbol-ized R), which is measured in ohms (abbreviated Ω). Ohm’s law may also be writ-ten in terms of resistance:

E = IR (1.1b)

One can draw an analogy between Ohm’s law for electric current flow and therule for flow of liquids in narrow tubes. In tubes the flow of the liquid (the analogof current) is proportional to the pressure difference (the analog of voltage differ-ence) divided by the frictional resistance.

Homogeneous conducting materials often are characterized by a bulk propertycalled the resistivity, abbreviated ρ. It is the resistance measured by two 1-cm2 elec-trodes applied to opposite sides of a 1-cm cube of the material and has the dimen-sions ohm · centimeter (Ω · cm). Resistivity is useful for calculating resistance of arbi-trary shapes of materials. For example, for a right cylindrical block of length l andcross-sectional area A with electrodes of area A on the end faces, the resistance is

(1.2)

Later in the book we will use this formula to estimate the resistance in a cylindri-cal pore.

Resistivity decreases as salts are added to a solution. Consider the followingapproximate examples at 20°C for solutions of increasing salt content: frogRinger’s solution 80 Ω · cm; mammalian saline 60 Ω · cm; and seawater 20 Ω · cm.Indeed, in sufficiently dilute solutions each added ion gives a known increment tothe overall solution conductance, and the resistivity of electrolyte solutions can bepredicted by calculations from tables of single-ion equivalent conductivities, suchas those in Robinson and Stokes (1965). In saline solutions the resistivity of purephospholipid bilayers is as high as 1015 Ω · cm, because although the physiologi-cal ions can move in lipid, they far prefer an aqueous environment over ahydrophobic one. The electrical conductivity of biological membranes comes notfrom the lipid, but from the ion channels embedded in the lipid.

Rl

A = ρ

Introduction 9

To summarize what we have said so far, when one volt is applied across a 1-Ωresistor or 1-S conductor, a current of one ampere flows; every second, 1/F molesof charge (10.4 µmol) move and one joule (0.24 calorie) of heat is produced. Ohm’slaw plays a central role in membrane biophysics because each ion channel is anelementary conductor spanning the insulating lipid membrane. The total electricalconductance of a membrane is the sum of all these elementary conductances inparallel. It is a measure of how many ion channels are open, how many ions areavailable to go through them, and how easily the ions pass.

The membrane as a capacitor

In addition to containing many conducting channels, the lipid bilayer of biologi-cal membranes separates internal and external conducting solutions by anextremely thin insulating layer. Physicists recognize that such a narrow gapbetween two conductors forms, of necessity, a significant electrical capacitor. Thisconcept may seem less intuitive than the ideas behind Ohm’s law, but it becomesimportant because it determines the number of ions that must move to make elec-trical signals in cell membranes.

To create a potential difference between objects requires only a separation ofcharge: an excess of positive charge on one and an excess of negative charge onthe other. Capacitance (symbolized C) is a measure of how much charge (Q) needsto be transferred from one conductor to another to set up a given potential differ-ence between them and is defined by

(1.3)

The unit of capacitance is the farad (abbreviated F). A 1-F capacitor will becharged to 1 V when +1.0 C of charge is on one conductor and –1.0 C on the other.The passage of current into an ideal capacitor simply removes charge from oneconductor and stores it on the other in a fully reversible manner and withoutevolving heat. The rate of change of the potential under a current IC is obtained bytaking the time derivative of Equation 1.3.

(1.4)

The capacity to store charges arises from their mutual attraction across the gapand by the polarization they develop in the intervening insulating medium. Thecapacitance depends on the dielectric constant of that medium and on the geome-try of the conductors on either side. In a simple capacitor formed by two parallelplates of area A and separated by an insulator of dielectric constant ε and thick-ness d, the capacitance is

(1.5)CA

d = εε0

dEdt

IC

C=

CQE

=

10 Chapter 1

where the natural constant ε0, called the polarizability of free space, is 8.85 × 10–12

CV–1m–1. Cell membranes are parallel-plate capacitors with specific capacitances*near 1.0 µF/cm2 (= 0.01 pF/µm2)—just slightly higher than that of a pure lipidbilayer, 0.8 µF/cm2 (see Cole 1968; Almers 1978). According to Equation 1.5, thismeans that the thickness d of the insulating part of the bilayer is only 23 Å (2.3 nm),assuming that the dielectric constant of hydrocarbon chains is 2.1. Hence the highelectrical capacitance of biological membranes is a direct consequence of their mol-ecular dimensions.

The high capacitance of cell membranes gives a lower limit to how many ions(charges) must move (Equation 1.3) and how rapidly they must move (Equation1.4) to make a given electrical signal. In general, capacitance slows down the volt-age response to any current by a characteristic time τ that depends on the productRC of the capacitance and any effective parallel resistance. For example, supposethat a capacitor is charged up to 1.0 V and then allowed to discharge through aresistor R, as shown in Figure 1.2. The discharge is not instantaneous. From Ohm’slaw, the current in the resistor is I = E/R, which discharges the capacitor at a ratedetermined by Equation 1.4:

(1.4a)dEdt

IC

ERC

–C= =

Introduction 11

*In describing cell membranes, the phrases “specific capacitance,” “specific resistance,” and “specificconductance” refer to electrical properties of a 1-cm2 area of membrane. They are useful for comparingthe properties of different membranes.

(A) CIRCUIT (B) TIME COURSE OF DISCHARGE

0 1 2 3 4

E0—2

E0

E = E0e –t/RC

Time (units of RC)

C++

––I = E –

RR

1.2 Discharge of an RC Circuit The circuit has a resistor anda capacitor connected in parallel; the voltage across the capacitor ismeasured from the two terminals. At zero time the capacitor ischarged to a voltage of E0 and begins to discharge through theresistor. Charge and voltage decay exponentially so that every RCseconds they fall to 1/e, or 0.367…, of their previous value.

The solution of this first-order differential equation is a voltage that decays expo-nentially in time:

(1.6)

where E0 is the starting voltage, t is time in seconds, and exp is the exponentialfunction (power of e, the base of natural logarithms).

For biological membranes the product, RMCM, of membrane resistance andcapacitance is often called the membrane time constant, τM. Using equations likeEquation 1.6, τM can be determined from measurements of the time course ofmembrane potential changes as small steps of current are applied across the mem-brane. For example, in the experiment shown in Figure 1.3, steps of current areapplied from an intracellular microelectrode across the cell membrane of a Para-

E Et

RCE

t =

=

0 0exp – exp –

τ

12 Chapter 1

0(B)

(A)

–20

–40

0.20

–0.2

0 200 400Time (ms)

EM (mV)

IM (nA)

EMIM

RMCM

Outside

Inside

1.3 The Cell Membrane as an RC Circuit An experiment to studymembrane electrical properties of Paramecium. The cell is impaled with twointracellular electrodes. One of them passes steps of current IM across themembrane to an electrode in the bath; the other records the changes of mem-brane potential EM with an amplifier (symbolized as a triangle). On the right,a current of 0.23 nA makes a voltage deflection of 23 mV, corresponding fromOhm’s law to a membrane resistance of 100 MΩ (108 Ω). The exponential timeconstant τM of the rise and fall of the voltage response is approximately 60ms. The responses of this Paramecium to current steps are simpler than forwild-type Paramecium because it contains a genetic mutation of the normalexcitability mechanism. [After Kung and Eckert 1972.]

mecium. The time course of the membrane potential change corresponds to amembrane time constant of 60 ms. Since CM is approximately 1 µF/cm2 in all bio-logical membranes, the measured τM gives a convenient estimate of specific mem-brane resistance. For the Paramecium in the figure, RM is τM/CM or 60,000 Ω · cm2.In different resting cell membranes, τM ranges from 10 µs to 1 s, corresponding toresting RM values of 10 to 106 Ω · cm2. This broad range of specific resistancesshows that the number of ion channels open at rest differs vastly from cell to cell.

Equilibrium potentials and the Nernst equation

All systems are moving toward equilibrium, a state where the tendency for fur-ther change vanishes. At equilibrium, thermal forces balance the other existingforces, and forward and backward fluxes in every microscopic transport mecha-nism and chemical reaction are equal.

We can explain how cells make electrical potentials by considering the problemillustrated in Figure 1.4. Two compartments of a bath are separated by a mem-brane containing pores permeable only to K+ ions. A high concentration of a saltKA (A for anion) is introduced into the left side and a low concentration into theright side. A voltmeter measures the potential across the membrane. In the first

Introduction 13

K+ K+

A– A–

First jiffy

E = EK

K+ K+

A– A–

Equilibrium

– +

––

–––

––

++

+++

++

E = 0

1.4 Diffusion Potentials in Pores A membrane with perfectlyK+-selective pores separates solutions with different concentrations ofa potassium salt, KA. A voltmeter records the potential across theporous membrane. At the moment when the salt solutions are pouredin, there is no membrane potential (E = 0). However, as a few K+ ionsdiffuse from the left side to the right, a potential develops, with theright side becoming positive. Eventually the membrane potentialreaches the Nernst potential for K+ ions (E = EK).

jiffy, the voltmeter reads 0 mV, since both sides are neutral. However, K+ ionsimmediately start diffusing down their concentration gradient into the right-handside, giving that side an excess positive charge and building up an electricalpotential difference across the membrane as charges become separated. The anioncannot cross the membrane, so the charge separation persists. However, the ther-mal “forces” causing net diffusion of K+ to the right are now countered by a grow-ing electrical force tending to oppose the flow of K+. The positive side begins torepel K+ ions. The potential builds up only until it finally reaches an equilibriumvalue, EK, where the electrical force opposing diffusion of K+ ions balances the dif-fusional force, and the system stops changing. The problem is to find a formulafor EK, the equilibrium potential for K+ ions. It is referred to as an equilibriumproblem even though parts of the system, such as the anions A– and the watermolecules (osmotic pressure), are not allowed to equilibrate. We may focus on K+

ions alone and discuss their equilibrium. As we shall see, equilibrium potentialsare the starting point in any description of biological membrane potentials.

A physicist would begin the problem with the Boltzmann equation of statisti-cal mechanics, which gives the relative probabilities at equilibrium of finding aparticle in state 1 or in state 2 if the energy difference between these states is u2 – u1:

(1.7)

Here kB is Boltzmann’s constant and T is absolute temperature on the Kelvin scale.This equation conveniently describes the equilibrium distribution of particles inforce fields. Qualitatively it says that at equilibrium a particle spends less time instates of higher energy than in states of lower energy. For example, the moleculesin Earth’s atmosphere are attracted by Earth’s gravitational field, and Equation 1.7correctly predicts that the probability of finding O2 molecules at the top of Mt.Everest is only one-third that of finding them at sea level. The O2 molecules arefree to move, but the statistical balance of thermal and gravitational forces favorstheir spending more time at sea level than at the tops of mountains.

For our purposes, Equation 1.7 can be recast into a slightly more chemical formby changing from probabilities p to concentrations c, and from single-particleenergies u to molar energies U:

(1.8)

where R is the gas constant (R = kBN). Finally, taking natural logarithms of bothsides and rearranging gives

(1.9)U U RTcc1 2

2

1– ln =

cc

U URT

2

1

2 1 =

exp –

–

pp

u uk T

2

1

2 1 B

=

exp ––

14 Chapter 1

Now we have a useful equilibrium relation between concentration ratios andenergy differences. In our problem, U1 – U2 is the molar electrical energy differ-ence of the permeant ion due to a membrane potential difference E1 – E2. If weconsider a mole of an arbitrary ion S with charge zS, then U1 – U2 becomes zSF(E1 – E2). Substituting into Equation 1.9 shows that the equilibrium potential ESis a function of the concentration ratio and the valence:

(1.10)

This well-known relationship is called the Nernst equation (Nernst 1888).Before discussing the meaning of the equation, let us note as an aside that the

equilibrium potential ES can be derived in other, equivalent ways. A chemistwould probably think in terms of thermodynamics, using the principle of J. W.Gibbs that the electrochemical potential of ion S is the same on both sides at equi-librium, or equivalently that the work of transfer of a tiny quantity of S from side2 to side 1 has to be zero. This work comprises two terms: the work of concentrat-ing the ions as they cross, –RT ln(c2/c1), plus all other energy changes, U1 – U2,which in this case is only the electrical term. These considerations lead at once toEquations 1.9 and 1.10. Thermodynamics would also point out that because allsolutions are at least slightly nonideal (unlike ideal gases), one should use ionactivities rather than ion concentrations (see, e.g., Moore 1972). This book refers tothe symbol [S] as the concentration of S while recognizing that careful quantitativework requires consideration of activities instead.

According to the Nernst equation, ionic equilibrium potentials vary linearlywith the absolute temperature and logarithmically with the ion concentrationratio. As would be expected from our discussion of Figure 1.4, equilibrium poten-tials change sign if the charge of the ion is reversed or if the direction of the gradi-ent is reversed, and they fall to zero when there is no gradient. To correspond tothe physiological convention, we now define side 1 as inside (intracellular), side 2as outside (extracellular), and all membrane potentials to be measured inside minusoutside. Hence if the inside of a cell is –90 mV relative to the outside, the mem-brane potential is –90 mV. We can write the equilibrium potentials for K+ and theother biologically relevant ions:

(1.11a)

(1.11b)

(1.11c)ERT

FCao

i

Ca

Ca=

[ ][ ]2

ln

ERTFNa

o

i

Na

Na=

[ ][ ]ln

ERTFK

o

i

K

K=

[ ][ ]ln

E E ERTz FS

S

S

S= =

[ ][ ]1 2

2

1

– ln

Introduction 15

(1.11d)

The subscripts o and i stand for outside and inside, respectively. The meaning ofthe equilibrium potential values EK, ENa, and so on, can be stated in two ways.Using EK as an example: (1) If the pores in a membrane are permeable only to K+

ions, a few ions would move and the membrane potential will change to EK; and(2) if the membrane potential is held somehow at EK, there will be no net flux ofK+ ions through K+-selective pores.

How large are the equilibrium potentials for living cells? Table 1.2 lists valuesof the factor RT/F in the Nernst equation; also given are values of 2.303(RT/F) forcalculations with log10 instead of ln as follows:

(1.11e)

From Table 1.2, at 20°C an e-fold (e ≈ 2.718) K+ concentration ratio across the mem-brane corresponds to EK = –25.3 mV, a 10-fold ratio corresponds to EK = –58.2 mV,and a 100-fold ratio corresponds to EK = –58.2 × 2 = –116.4 mV. Table 1.3 lists theactual concentrations of some ions in mammalian skeletal muscle and their calcu-lated equilibrium potentials ranging from –98 to +128 mV. EK and ECl are negativenumbers, and ENa and ECa are positive. We can see that EK sets the negative limitand ECa the positive limit of membrane potentials that can be achieved by open-ing ion-selective pores in the muscle membrane. All excitable cells have negativeresting potentials because at rest they have far more open K+-selective channels(and in muscle, Cl–-selective channels as well) than Na+- or Ca2+-selective ones.

ERTF

RTFK

o

i

o

i

K

K

K

K=

[ ][ ] =

[ ][ ]ln . log2 303 10

ERTFCl

i

o

Cl

Cl=

[ ][ ]ln

16 Chapter 1

TABLE 1.2 Values of RT/F (or kBT/qe)

Temperature RT/F 2.303 RT/F(°C) (mV) (mV)

0 23.54 54.20

5 23.97 55.19

10 24.40 56.18

15 24.83 57.17

20 25.26 58.17

25 25.69 59.16

30 26.12 60.15

35 26.55 61.14

37 26.73 61.54

Current-voltage relations of channels

Biophysicists like to represent the properties of membranes and channels by sim-ple electrical circuit diagrams that have equivalent electrical properties to themembrane. We have discussed the membrane as a capacitor and the channel as aconductor. But if we were to try to test Ohm’s law on the membrane of Figure 1.4,we would immediately recognize a deviation: Current in the pores goes to zero atEK, not at 0 mV. The physical chemist would say, “Yes, you have a concentrationgradient, so Ohm’s law doesn’t work.” The biophysicist would then suggest thata gradient is like a battery with an electromotive force (emf) in series with theresistor (see Figure 1.5). The modified current-voltage law then becomes

Introduction 17

TABLE 1.3 Free Ion Concentrations and Equilibrium Potentialsfor Mammalian Skeletal Muscle

Extracellular Intracellular Equilibriumconcentration concentration [Ion]o potentiala

Ion (mM) (mM) [Ion]i (mV)

Na+ 145 12 12 +67

K+ 4 155 0.026 –98

Ca2+ 1.5 100 nM 15,000 +129

Cl– 123 4.2b 29b –90b

a Calculated from Equation 1.11 at 37°C.b Calculated assuming a –90-mV resting potential for the muscle membrane and that Cl– ions are at equilibrium at rest.

EK

CM

gK

(A) EQUIVALENT CIRCUIT (B) INTERPRETATION

1.5 Two Views of a K+-Selective Membrane In electricalexperiments the membrane acts like an equivalent circuit with twobranches. The conductive branch with an electromotive force of EKsuggests a K+-selective aqueous diffusion path, a pore. The capacitivebranch suggests a thin insulator, the lipid bilayer.

IK = gK(E – EK) (1.12)

The electromotive force in the pore is EK and the net driving force on K+ ions isnow E – EK and not E. This modification, introduced by Hodgkin and Huxley, is,like Ohm’s law itself, empirical and requires experimental test in each situation.To a first approximation this linear law is often excellent, but many pores areknown to have nonlinear current-voltage relations when open. As we shall seelater, some curvature is predicted by explicit calculations of the electrodiffusion ofions in pores, particularly when there is a higher concentration of permeant ion onone side of the membrane than on the other, or when the structure of the channelis asymmetrical. Indeed, both situations are usually true.

Consider now how simple current-voltage measurements can be used to gaininformation on ion channels. Figure 1.6 gives examples of hypothetical observa-tions and their interpretation in terms of electrical equivalent circuits. Figure 1.6Ashows three linear I–E curves. They pass through the origin, so no battery isrequired in the equivalent circuit. We can deduce that the channels are nonselec-tive or that there is no effective ion gradient. The slopes of the successive I–E rela-tions increase in the ratio 1:2:3, so the equivalent conductance, and hence thenumber of open channels, must increase correspondingly. Thus conductances givea useful measure of how many channels are open in a given area of membrane.

Figure 1.6B shows two I–E relations of equal slope but with different zero-cur-rent potentials. The corresponding equivalent circuits have equal conductancesbut different electromotive forces in their batteries. This situation could arise fromdifferent channels with different ion selectivities, or from the same channel bathedon the two sides by different concentrations of its permeant ions. Hence zero-cur-rent potentials are useful in studies of selectivity.

Figure 1.6C shows the effect of a voltage-dependent channel opening. This isharder and needs to be analyzed in several steps. Since the I–E relations do notpass through the origin, we know again that there is an electromotive force in

18 Chapter 1

1.6 Current-Voltage Relations of Membranes Measured I–E relations can beinterpreted in terms of electrical equivalent circuits and the modified form of Ohm’slaw (Equation 1.12) that takes into account the electromotive force in the pores. Fourhypothetical conditions are shown. (A) Membranes with 1, 2, and 3 pores open giveI–E relations with relative slopes of 1, 2, and 3. (B) Pores with negative or positiveelectromotive forces give I–E relations with negative or positive zero-current poten-tials. (C) Pores that step abruptly from a low-conductance state to a high-conduc-tance state (see inset graph of g versus E) give I–E relations consisting of two linesegments. (D) Pores with smoothly voltage-dependent probability of being open(see inset graph of average g versus E) give curved I–E relations. The dashed lines,corresponding to a constant high conductance, are the same I–E relations as in (B).However, when the pores close at negative potentials, lowering g, the currentdecreases correspondingly from its maximal value.

Introduction 19

3

3g

2

2g

1

g

E

21

I 3

(A)

EE1 E2

2

I 1

(B) 1

g

E1

2

g

E2

1

E1

g(E)

E

E2

g

E1

2

E2

g(E)

(C)

E1 E2

1

2

2

1

1

E1

E1

g(E)

2

E2

E2

E

g

g(E)

a b

(D)

EE2E1

2a2b

1b

1a

I

these channels. Both a negative emf, E1, and a positive emf, E2, are illustrated, asin Figure 1.6B. Unlike Figure 1.6B, however, these I–E relations are not singlestraight lines. This tells us that the membrane conductance changes with voltage,a property called rectification in electric circuit theory. In biological membranes,strong rectification usually means that the ion channels carrying current are openat some membrane potentials and shut at others. We can imagine a voltage-gatedswitch that opens and closes the channels. In this example, the conductance islow at very negative membrane potentials and suddenly steps up to a higherlevel as the potential is made less negative. More channels have opened. Thelow- and high-conductance segments of the I–E relation are each linear andextrapolate back to a zero-current point corresponding to the emf of the channelswhen open.

Figure 1.6C corresponds to measurements on a system with a sharp voltagethreshold for opening of ion channels. Real voltage-gated channels subject to ther-mal agitation cannot measure the membrane potential this precisely and the volt-age dependence of their opening is necessarily less abrupt, as in Figure 1.6D. Thecase illustrated in Figure 1.6D may seem quite difficult, but because it correspondsclosely to practical observations, it is worth working through. First note that thereis no ionic current at membrane potentials more negative than E1. Therefore theconductance there is zero, and the channels must be closed. Positive to 0 mV, theI–E relations are steep, straight lines like those in Figure 1.6B. Here the conduc-tance is high, and the channels must be open. In the intermediate voltage range,between E1 and 0 mV, the current is smaller than expected from the maximal con-ductance (dashed lines), and hence only some of the channels are open.

To determine how many channels are open at each voltage, we should calculatethe ionic conductance at each potential. When this is done using the modifiedform of Ohm’s law (Equation 1.12) and the appropriate channel electromotiveforce, E1 or E2, one derives the conductance-voltage (g–E) relations shown in theinset. Over a narrow voltage range the conductance changes smoothly from fullyoff to fully on. As a first approximation, this continuous conductance-voltage rela-tion reflects the steep voltage dependence of the open probability of a populationof channels.* We can think of this channel as being electrically excitable, a voltage-gated pore.

The I–E relations in Figure 1.6 are representative of observations made daily inelectrophysiological studies of ion channels. Examples will appear in Chapter 2.Interested readers will want to work out for themselves how voltage-dependentchannel opening accounts for the results by re-sketching each I–E relation and cal-culating the corresponding conductance-voltage relation point-by-point fromEquation 1.12.

20 Chapter 1

*Some nonlinearities may be due to other factors, including an intrinsic nonlinearity of the I–E curve fora single open channel, discussed above.

Ion selectivity

It is essential for electrical excitability that different ion channels be selective fordifferent ions. However, no channel is perfectly selective. Thus the Na channel ofaxons is fairly permeable to NH4

+ ions and even slightly permeable to K+ ions.How can we determine ion selectivity from electrical measurements? The simplestway is to measure the electromotive force or zero-current potential for the channelwith, say, ion A+ on the outside and B+ on the inside. This is called a biionicpotential. Suppose that A+ and B+ have the same valence. If no other permeantion is present, the permeability ratio PA/PB is defined by the equation

(1.13)

where the zero-current potential is often called the reversal potential (Erev), sincethat is the potential around which the current reverses sign.

Equation 1.13 resembles the Nernst equation, but with two ions. It expresses animportant and simple idea: The permeability of a channel for A is said to be equalto that for B if you need the same concentration of A+ on one side as B on the otherto get zero electromotive force, and the permeability for A+ is said to be half thatfor B when you need two concentration units of A on one side and one concentra-tion unit of B on the other to get zero electromotive force. Equation 1.13 is the sim-plest form of an expression derived from diffusion theory by Goldman (1943) andHodgkin and Katz (1949). Unlike the Nernst equation, such expressions describe asteady-state interdiffusion of ions away from equilibrium. Even at this zero-currentpotential, there is a net flux of ion A in one direction and ion B in the other. There-fore, the simplifying rules of equilibrium cannot be applied, and the derivationmust make assumptions about the structure of the channel.

Signaling requires only small ion fluxes

To close this chapter, we exercise our electrical knowledge by reconsidering theexperiment in Figure 1.4 using biologically realistic numbers and the electricalequivalent circuit in Figure 1.5. Suppose that the membrane contains K+-selectivepores that contribute 20 pS (20 × 10–12 siemens) of electrical conductance apiece.*If an average of 0.5 pore is open per square micrometer, the specific membraneconductance is

gM –8 0.5 20 10 S m

10 cm m 1 mS cm= × × =

– //

/12 2

2 22µ

µ

ERTzF

P

PrevA o

B i

A

B=

[ ][ ]ln

Introduction 21

*Most biological ion channels have an electrical conductance in the range of 1 to 150 pS.

Then the specific membrane resistance is RM = 1/gM = 1000 Ω · cm2, and the mem-brane time constant for CM = 1 µF/cm2 would be τM = RMCM = 1 ms. Suppose thatthe concentration ratio of KA salt across the membrane is 52:1 (inside:outside) sothat EK is 58.2 log10 (1/52) = –100 mV. Now what happens immediately after thesalt solutions are introduced and K+ ions start to diffuse? The voltmeter reports amembrane potential changing from 0 mV to –100 mV (EK) along an exponentialtime course with a time constant of 1 ms (τM):

E = [1 – exp(–t/1 ms)] · (–100 mV)

After a few milliseconds the system reaches equilibrium and an excess charge of Q= ECM = 10–7 C/cm2, all carried by K+ ions, has been separated across the mem-brane. This amounts to an outward movement of Q/F = 10–12 mol of K+ ions percm2 of membrane—a tiny amount that would alter the original 52-fold gradientvery little. Hence our calculation shows that full-sized electrical signals can begenerated rapidly even with relatively few pores per unit area and with onlyminute ion fluxes.

Notice that the size of the needed ion flux depends on the surface area of the cell,whereas the effect of the flux on internal ion concentrations depends on the volumeof the cell. In a giant cell (a 1000-µm-diameter squid axon) the surface-to-volumeratio is the lowest, and electrical signaling with a 110-mV action potential changesthe available ion concentration gradient by only 1 part in 105. On the other hand,in the smallest cells (a 0.1-µm axon or dendrite), the surface-to-volume ratio is 104

times higher, and a single action potential might move as much as 10% of thestored-up ions.

Having reviewed some essential rules of physics, we may now embark on theexperimental study of ion channels.

22 Chapter 1