Membrane Biophysics

125
Fundamental Principles of Membrane Biophysics David Njus Department of Biological Sciences Wayne State University © D. Njus, 2000

description

fgsdgdsfgfgfrewregs

Transcript of Membrane Biophysics

Page 1: Membrane Biophysics

Fundamental Principlesof Membrane Biophysics

David Njus

Department of Biological SciencesWayne State University

© D. Njus, 2000

Page 2: Membrane Biophysics

Table of Contents

Chapter 1. Biological Membranes

Chapter 2. Thermodynamics of Micelle Formation

Chapter 3. The Fluid Mosaic Membrane

Chapter 4. Membrane Electrostatics

Chapter 5. Specific and Non-Specific Binding

Chapter 6. Permeability and Conductance

Chapter 7. Permeability and Conductance of Electrolytes

Chapter 8. Channels and Excitable Membranes

Chapter 9. Active Transport

Chapter 10. Facilitated Diffusion

Chapter 11. Coupled Transport

Chapter 12. Energy Coupling

Chapter 13. Epithelial Transport

Appendices

I. Glossary of Symbols

II. Abbreviations

III. Fundamental Constants

IV. Conversion Factors

V. Mathematical Formulae

Page 3: Membrane Biophysics

Glossary of Symbols

A areaC molar concentration, capacitancec velocity of lightcmc critical micelle concentrationD diffusion coefficientd derivativeE energy, reduction potentialE electric fielde electronic chargeF forceF Faraday constantf frictional coefficientf fugacityG Gibbs free energyg conductance, acceleration of gravityH enthalpyh Planck’s constantI currentJ flowKi equilibrium constant for reaction iKp partition coefficientk Boltzmann constantki rate constant for reaction im mass, aggregation numberN Avogadro's numbern amount in molesP permeability coefficient, pressure, powerQ heatq chargeR gas constant, resistancer radiusS entropyT absolute temperaturet timeu mobilityV volumev velocityW workX mole fraction

x distanceZ collision factorz valence

γ activity coefficient∆ differenceδ dipole momentε dielectric constantεo permittivity of vacuumη viscosity, efficiencyλ wavelength of lightµ electrochemical potentialρ densityσ reflection coefficientψ electrical potential

Page 4: Membrane Biophysics

Abbreviations

Å Angstromsatm atmospheres°C degrees centigradecal caloriescoul coulombsD DebyesDa Daltonseq equivalentsesu electrostatic unitsF farads (coul/volt)g gramsj joules°K degrees Kelvinl litersM moles/literm metersmin minutesmol molesS Siemenssec secondsV volts

k kilo 103

c centi 10-2m milli 10-3µ micro 10-6n nano 10-9p pico 10-12

f femto 10-15

a atto 10-18

Page 5: Membrane Biophysics

Fundamental Constants

Gas constant R = 8.3144 j.°K-1.mol-11.9872 cal.°K-1.mol-18.3144 x 107 ergs.°K-1.mol-10.082054 l.atm.°K-1.mol-1

Boltzmann constant k = 1.38044 x 10-16 erg.°K-1Avogadro's Number N = 6.0230 x 1023 molecules.mole-1Ice point To = 273.15 °KFaraday constant F = 96,490 coul.eq-1Permittivity of vacuum εo = 8.854 x 10-12 coul.m-1.volt-1Molar volume, ideal gas, 0°C, 1 atm Vo = 22.4138 l.mol-1Electronic charge e = 4.80286 x 10-10 esu

1.602 x 10-19 coulElectron mass me = 9.1083 x 10-28 gVelocity of light c = 2.997930 x 1010 cm.sec-1Standard acceleration of gravity g = 980.665 cm.sec-2Planck's constant h = 6.6252 x 10-27 erg.secVolume conversion factor α = 103 l.m-3Collision factor Z = 1011 M-1.sec-1

Conversion Factors

1 atm = 760 mm = 1.01325 x 106 dyne.cm-21 cal = 4.184000 j1 j = 107 ergs = 1 volt.coul1 erg = 1 dyne.cm1 ev = 1.60206 x 10-12 erg1 l.atm = 24.22 cal1 Å = 10-10 m1 Debye = 10-18 esu.cm.molecule-1 = 2 x 10-6 coul.m.mol-11 kcal/eq = 0.043362 volts

Page 6: Membrane Biophysics

Mathematical Formulae

sinh x = 1/2 (ex - e-x)

sinh x = x + x3/3! + x5/5! + x7/7! + ...

ex = 1 + x + x2/2! + x3/3! + ...

Surface AreaCylinder (minus ends) A = 2πrhSphere A = 4πr2

VolumeCylinder V = πr2hSphere V = (4/3) πr3

Page 7: Membrane Biophysics

Fundamental Principlesof Membrane Biophysics

CHAPTER 1: BIOLOGICAL MEMBRANES

David Njus

Department of Biological SciencesWayne State University

© D. Njus, 2000

Page 8: Membrane Biophysics

Page 1.1

CHAPTER 1: BIOLOGICAL MEMBRANES

Section 1.1. Biological MembranesBiological membranes maintain the spatial organization of life. Membranes

defined the boundaries of the first living cells and still work to shield cellular metabolismfrom changes in the environment. Membranes prevent undesirable agents from enteringcells and keep needed molecules on the inside. They also organize the interior ofeukaryotic cells by separating compartments for specialized purposes. Membranes are notstatic barriers, but active structures. To function effectively, they must selectively passmolecules, ions, and signals from one side to the other.

The strategy underlying biological membrane function is that the best barrierbetween aqueous compartments is a hydrophobic layer. The water-soluble compoundspresent within cells and in their environments are not soluble in the lipid milieu of themembrane and pass slowly or not at all through even a very thin lipid layer. Thismechanism has a number of advantages which life has exploited. First, the lipid bilayer isa natural structure and assembles spontaneously. Second, the structure is flexible andallows for growth and movement as well as for the insertion and operation of proteinmachinery. Finally, the structure has a low dielectric constant giving the membraneelectrical properties which are used in signalling, transport and energy transduction.

To understand how biological membranes function, we will begin by analyzingtheir structure. The structure determines the fundamental properties of fluidity,permeability, and membrane potential. The origin and characterization of these propertieswill be analyzed next. Finally, we will discuss how these properties contribute to thevarious functions of biological membranes: signal transduction, energy transduction, andtransport.

Life, like all other processes in our universe, obeys the laws of physics andchemistry. Consequently, the theoretical framework of physical chemistry providespowerful tools for understanding living systems. Especially important is the requirementimposed by the second law of thermodynamics: processes must result in a net decrease infree energy in order to occur spontaneously. Free energy changes govern all metabolicprocesses, but they are particularly apparent in common phenomena of biologicalmembranes. For example, membrane structure is governed by the distribution ofcompounds between the hydrophobic interior of the membrane and the aqueous spaces oneither side. Free energy also determines the movement of molecules and ions acrossmembranes in response to concentration gradients and membrane potentials. Becausemembranes have a well defined planar geometry, the mathematics is simpler than it mightotherwise be. Thus, to understand in depth the structure and function of biologicalmembranes, it is essential to understand and apply principles of physical chemistry. Thepurpose of this course is to construct a coherent framework to do that.

Section 1.2. The Fluid Mosaic Model of Membrane StructureEarly on, it was recognized that hydrophobic compounds passed more readily than

water-soluble compounds through biological membranes. This, coupled with theidentification of lipids as a major component, led to the notion that biological membraneshave a hydrophobic character. The calculation (erroneous, as it turns out) that the lipid

Page 9: Membrane Biophysics

Page 1.2

content was twice that needed for a single layer of lipid led to the concept (correct, as luckwould have it) of the lipid bilayer (Gorter and Grendel, 1925; Danielli and Davson, 1935).Lipids are amphiphilic compounds with a small hydrophilic headgroup attached to longhydrocarbon chains. In the lipid bilayer, lipids are aligned with the headgroups facing thewater on either surface of the membrane and the hydrophobic hydrocarbons sandwiched inbetween.

The lipid bilayer concept did not establish the location of the protein components ofthe membrane. Originally, for lack of a better site, the proteins were stuck on to themembrane surface. This was not tenable, of course, because proteins are responsible formoving molecules and messages across membranes and they could not perform thosefunctions without being a integral part of the membrane itself. This realization gave rise tothe concept of integral and peripheral proteins. Peripheral proteins are loosely associatedwith the membrane and located on the surface of the lipid bilayer. Integral proteins areinserted into the membrane and pass all the way (or much of the way) across themembrane. Originally, the integral proteins were thought to form a well defined matrixwith the lipid bilayer filling in the spaces in between. In the late 1960's, however, itbecame clear that many proteins are not rigidly fixed in the membrane, but can diffuseacross the surfaces of cells relatively easily and independently. Membranes came to beviewed as fluid structures with proteins and lipids arranged in their thermodynamicallymost favorable structure. Lipids exist in a bilayer and provide the milieu in which theintegral membrane proteins float. The proteins are oriented so that their hydrophobicsurfaces are immersed in the hydrophobic interior of the lipid bilayer. Hydrophilic aminoacids are exposed only in the aqueous regions on either side of the membrane. Theorganization of the membrane is a direct consequence of the partitioning of its components,both lipid and protein, so that hydrophobic regions are kept within the membrane andhydrophilic parts are exposed to the water on either side. Because the components are notheld together by bonds, they are free to diffuse and move independently within the plane ofthe membrane. Singer and Nicolson (1972) captured this view in the fluid mosaic model.

The fluid mosaic model persists as the accepted view of membrane structure. Therecognition of linkages between membrane proteins and components of the cytoskeletalsystem has modified the concept somewhat. The cytoskeleton does impose someconstraints on the distribution of membrane proteins. As we shall see, phase separationalso can create separate domains with different characteristics within membranes.Consequently, the fluid mosaic model does not imply that all the components of aparticular membrane are randomly and homogeneously distributed.

Section 1.3. Classes of Biological MembranesBiological membranes perform many different functions in all kinds of cells. They

can be divided, however, into four classes based on differences in their fundamentalenergetics. These differences arose during the course of evolution as different organismsadopted different strategies to cope with their environments.

The first class includes prokaryotic cell membranes, the inner mitochondrialmembrane and the thylakoid membrane of chloroplasts. These membranes, which share acommon evolutionary origin, do not contain cholesterol. A proton gradient drives thefunctions of these membranes. The proton gradient is generated by a variety of

Page 10: Membrane Biophysics

Page 1.3

mechanisms, but a redox chain is common. These membranes can use the proton gradientto generate ATP using an ATP synthase of the F1F0 type.

The second class consists of plasma membranes of animal cells. These have aNa+/K+ ATPase which pumps Na+ out of and K+ into the cells. The Na+ and K+

concentration gradients created thereby participate in many functions of the membraneincluding transport, excitability, and signalling.

The third class of membranes includes the plasma membranes of plant and fungalcells. These differ from animal cell membranes in that they lack a Na+/K+ ATPase andinstead have a proton-translocating ATPase of the P-type. The proton gradient created bythis ATPase drives transport and other functions of the plant plasma membrane. Thisfundamental difference between plant and animal cell membranes reflects a basicdifference between plant and animal lifestyles. Unlike animal cells, plant cells cannot relyon seawater or a circulatory fluid to provide external Na+, so the substitution of a protonpump for a sodium pump is necessary. Moreover, because plants are immobile, plant cellscan have a rigid cell wall to support the plasma membrane in times of osmotic imbalance.Animal cells, by contrast, must maintain osmotic balance by regulating the concentrationsof internal osmolytes such as Na+ and K+ to balance the external salt concentration.

The fourth class of membranes includes membranes of the vacuolar system. Thiscomprises the membranes of Golgi-derived organelles including lysosomes, endosomes,secretory vesicles in animal cells and peroxisomes, vacuoles and tonoplasts in plants andfungi. These membranes have a proton-translocating ATPase of the V-type. The protongradient created by this ATPase drives processes in the membrane and also makes theinterior of the organelle acidic, a property frequently important in the function of theorganelle.

Section 1.4. Membrane Biosynthesis and AsymmetryAlthough the lipid bilayer is basically a symmetrical structure, natural membranes

are not. The two sides of the membrane differ, so the membrane has a functional polarity.Molecules, ions and signals will be moved one way but not the other. The two sides of themembrane differ because of the way the membrane is synthesized.

In animal and plant cells, the plasma membrane and vacuolar membranes share acommon synthetic pathway. The proteins are inserted into the endoplasmic reticulum. Themembranes are transferred to the Golgi apparatus where carbohydrates are attached andprocessed. The membranes are also sorted and leave the Golgi targeted to their finaldestinations. This process has a number of consequences. First, the proteins are insertedinto the membranes with a defined orientation; they are inserted from the cytoplasmic sideof the membrane. Second, the carbohydrates are attached only to the other side, the interiorof the Golgi. Thus, the carbohydrates face only the interior of organelles and the exteriorof the cell; they do not appear on the cytosolic side of a membrane. This, along with theprocesses of exocytosis and endocytosis, emphasize that the interior of vacuolar organellesis equivalent to the exterior of the cell in terms of membrane polarity.

In bacterial cells, the proteins and lipids are synthesized inside the cell and insertedinto the membrane. The biosynthesis of the mitochondrial inner membrane and thethylakoid membrane are more complex because some of the proteins are synthesized in thecytoplasm and then inserted into the organelle. The end result, however, is that the

Page 11: Membrane Biophysics

Page 1.4

proteins are placed in the membrane with a defined orientation and this gives themembrane polarity.

The structure of a biological membrane is a consequence of both spontaneousassembly and programmed development. The fluid mosaic model emphasizes thespontaneous assembly of the lipid bilayer with the proteins orienting to accommodate theirown hydrophobic and hydrophilic surfaces. At the same time, the membrane isasymmetrical largely because of its history. The structure of each protein is determined bythe amino acid sequence and the direction in which that sequence was inserted into themembrane. The further processing of that protein, particularly the attachment ofcarbohydrates, is also asymmetrical since the relevant enzymes are confined to one side ofthe membrane or the other. The asymmetry is obviously crucial because it givesmembranes a polarity essential for function. Transport occurs in defined directions. Thisin turn creates the membrane potential, a polarity that plays a fundamental role in manymembrane processes.

Section 1.5. SummaryLipids in biological membranes are arranged predominantly in a bilayer structure.

Proteins are oriented so that hydrophobic amino acids are buried inside the membrane andhydrophilic residues are exposed on the aqueous surfaces. Proteins are inserted with adefined orientation and give polarity to the membrane. Four classes of biologicalmembranes may be distinguished on the basis of their different primary ion pumps. All ofthese considerations should be kept in mind as we proceed to analyze the structure andfunction of biological membranes.

References

S.J. Singer and G.L. Nicolson (1972) The fluid mosaic model of the structure of cellmembranes, Science 175, 720-731.

Page 12: Membrane Biophysics

Fundamental Principlesof Membrane Biophysics

CHAPTER 2: THERMODYNAMICS OF MICELLE FORMATION

David Njus

Department of Biological SciencesWayne State University

© D. Njus, 2000

Page 13: Membrane Biophysics

Page 2.1

CHAPTER 2: THERMODYNAMICS OF MICELLE FORMATION

Section 2.1. Properties of WaterWater is a remarkable substance in many ways, and its unusual properties are

crucial in the functioning of biological membranes. First, water molecules dissociate intoH+ and OH-. This property allows H+ to equilibrate among protonatable groups in all of themolecules in a solution. As we shall see, it also makes H+ a convenient ion to use forcreating electrical potential differences across biological membranes.

A second important property of water is its polarity. The dipole moment of the OHbond is 1.51 Debyes and the water molecule itself has a dipole moment of 1.84 Debyes.As a consequence, water has a high dielectric constant (80.37 at 20°C) and polarizes toneutralize electric fields. For this reason, electric fields and associated differences inelectrical potential exist primarily across biological membranes rather than across aqueousregions of cells.

Finally, water molecules participate in hydrogen bonding. The hydrogen bond isprimarily electrostatic (a dipole-dipole interaction) with an energy of 4.5-6 kcal/mol. Thebond is linear with the hydrogen atom situated directly between two electronegative atoms(two oxygen atoms in the case of the H2O-H2O bond). Hydrogen bonding accounts for therelatively low freezing and boiling points of water relative to other compounds withcomparable molecular weights. Most importantly, hydrogen bonding is responsible for thehydrophobic effect.

Because hydrogen bonds are fairly strong, water molecules will orient so as tohydrogen bond even if this orientation restricts their mobility. For example, watermolecules on the surface (an air-water interface) will have fewer other water moleculeswith which to interact than will molecules in the interior of the solution. Molecules on thesurface will nevertheless hydrogen bond to other water molecules, but the smaller numberof possible bonding partners means that they will have a lower entropy than molecules inthe interior. To increase the entropy of the system, water will minimize its surface area.This effect is responsible for the high surface tension of water.

Because nonpolar molecules will not hydrogen bond, they reduce the bondingpossibilities of adjacent water molecules. Thus, just as increasing the surface areadecreases the entropy of an aqueous solution, so does introducing nonpolar molecules.Exclusion of these nonpolar substances from the aqueous solution increases the entropyand decreases the free energy. It is this entropy-driven effect that causes nonpolarcompounds to be excluded from aqueous solutions. It is important to recognize thatnonpolar molecules do not attract each other; they are pushed together because they aremutually excluded from water. This phenomenon is known as the hydrophobic effect.

Page 14: Membrane Biophysics

Page 2.2

Figure 2.1

Page 15: Membrane Biophysics

Page 2.3

Section 2.2. Structures Formed by Amphiphilic MoleculesAmphiphiles are those molecules that are polar on one end and nonpolar on the

other. These promiscuous molecules have affinities for both aqueous and nonpolar phases.At low concentrations, they dissolve in water. At some critical concentration, however,they reach their solubility limit and begin to aggregate into micellar structures. The micellestructure allows the molecule to keep its polar region in the aqueous phase on the surfaceof the micelle and the nonpolar portion in the nonpolar interior of the micelle. The limitingmonomer solubility is called the critical micelle concentration (cmc). At concentrationsbelow the cmc, the amphiphile will exist as monomers. At concentrations above this level,the excess amphiphile will aggregate to form micelles.

Two factors are involved in the spontaneous formation of micelles. First, thehydrophobic effect causes the nonpolar portion of the molecule to be separated from waterand sequestered in the interior of the structure. Second, interactions between the headgroups determine how closely the molecules may be packed. Amphiphiles with a singlehydrocarbon chain, such as dodecyl sulfate, must pack a number of head groups around arelatively small volume of hydrocarbon. This large surface area to volume ratio is achievedby forming a spherical micelle structure. By contrast, amphiphiles with two hydrocarbonchains, such as phospholipids, must pack the same number of headgroups around twice aslarge a volume of hydrocarbon. This smaller surface area to volume ratio is achieved byforming the bilayer structure (Figure 2.2).

Figure 2.2

It should be recognized that the spherical micelle and the planar bilayer are reallytwo extremes of a continuum. Micelles in the shape of oblate spheroids will exhibitintermediate ratios of surface area to volume. Under a particular set of conditions, anamphiphile will form micelles with a particular surface area to volume ratio and thus willform micelles of a particular size. This characteristic of the micelle is described by theaggregation number m, the average number of amphiphile molecules in a single micelle.

The critical micelle concentration and the aggregation number together characterizethe micelle that a particular amphiphile will form under a given set of conditions. We canmake some intuitive generalizations about how these parameters should respond to avariety of changes. Increasing the chain length of the amphiphile should lower the aqueoussolubility and decrease the critical micelle concentration. For similar reasons, amphiphileswith two hydrocarbon chains (phospholipids) should have a lower cmc than those with asingle chain (detergents). Ionic detergents should have a greater water solubility thannonionic detergents and therefore should have a higher critical micelle concentration.Repulsive forces between polar groups should be less for nonionic detergents than for ionicdetergents. Therefore, nonionic detergents should form micelles with smaller surface areasper amphiphile (larger aggregation numbers). Increasing the ionic strength should diminish

Page 16: Membrane Biophysics

Page 2.4

repulsive forces between polar groups of ionic detergents thereby increasing theaggregation number. These characteristics of the cmc and the aggregation number areillustrated by the data in Table I. Deoxycholate, cholate and sodium dodecyl sulfate areionic detergents; Lubrol WX and Triton X-100 are nonionic.

TABLE 2.1

Detergent cmc (mM) m________________________________________________________________Sodium dodecylsulfate 50 mM NaCl 2.3 72

500 mM NaCl 0.51 126Deoxycholate

10 mM NaCl, pH 7.5 4 4300 mM NaCl, pH 7.5 6 29

Cholate 45 2Lubrol WX 0.125 96Triton X-100 0.24 140________________________________________________________________

Section 2.3. The Hydrophobic EffectThe thermodynamics of micelle formation has been analyzed in an elegant fashion

by Tanford (1980). The two factors determining micelle structure, the hydrophobic effectand head group interaction, are each assumed to contribute separately to the free energy ofthe micelle. A summary of this analysis will be presented here.

To understand the contribution of the hydrophobic effect to micelle structure, let usfirst consider the solubility of hydrocarbons in water. The chemical potential of thehydrocarbon in the aqueous phase is

(2.1) µw = µw° + RT ln Xw + RT ln fw

We assume that ln fw = 0 because the hydrocarbon concentration in water is extremelylow. Now consider the chemical potential of the hydrocarbon in a pure hydrocarbon phase:

(2.2) µHC = µHC° + RT ln XHC + RT ln fHC

In pure hydrocarbon, XHC = 1 so ln XHC = ln fHC = 0. When the hydrocarbon partitionsbetween water and the hydrocarbon phase, equilibrium is reached when the chemicalpotentials are equal (µw = µHC). Therefore,

(2.3) µHC° = µw° + RT ln Xw

Since Xw is the saturating concentration (in mole fraction) of the hydrocarbon in water, thefree energy change for transferring a hydrocarbon molecule from water into thehydrocarbon phase can be determined from the compound's water solubility:

Page 17: Membrane Biophysics

Page 2.5

(2.4) µHC° - µw° = RT ln Xw

For a series of n-alkanes, the following empirical relationship is found:

(2.5) µHC° - µw° = -2436 - 884 nc cal/mol

where nc is the number of carbon atoms in the molecule. Of course, as nc increases, thewater solubility of the hydrocarbon decreases. Double bonds increase the water solubility(decrease the hydrophobicity) of hydrocarbons.

We can use a similar analysis to examine the partitioning of hydrocarbon moleculesbetween water and the interior of a micelle. The chemical potential of the hydrocarbonmolecule in a micelle is

(2.6) µmic = µmic° + RT ln Xmic

Setting µmic equal to the chemical potential of the hydrocarbon in water (equation 2.1)allows us to solve for the free energy change for transfer of the hydrocarbon from water tothe interior of the micelle:

(2.7) µmic° - µw° = RT ln Xw - RT ln Xmic = RT ln (Xw/Xmic)

If Xw is the solubility of the hydrocarbon in water, then Xmic can be calculated from theincrease in solubility observed in the presence of micelles. For a series of n-alkanes andmicelles formed from sodium dodecyl sulfate, the following empirical relationship isobserved:

(2.8) µmic° - µw° = -1934 - 771 nc cal/mol

As in the case of the transfer of hydrocarbon from water to the pure hydrocarbon phase, thefree energy change is proportional to the number of carbon atoms in the compound and theenergy contribution of each carbon atom is about 800 cal/mol.

Section 2.4. Thermodynamics of Single-Component MicellesThere is a dynamic tension at work in the micelle structure. The polar groups tend

to repel each other because they have similar charges and dipole moments. Nevertheless,they must remain close enough together to prevent water from gaining access to thehydrophobic interior of the micelle.

Let us consider the chemical potential of an amphiphile in water (µw) and in amicelle of size m (µmic,m).

(2.9) µw = µw° + RT ln Xw + RT ln fwand

(2.10) µmic,m = µmic,m° + (RT/m) ln (Xm/m)

RT ln (Xm/m) is the contribution of the whole micelle to the free energy, so this term isdivided by m to determine the free energy contribution of each molecule of amphiphile.

Page 18: Membrane Biophysics

Page 2.6

Since amphiphiles will equilibrate between the aqueous phase and micelles, these chemicalpotentials must be equal. Therefore,

(2.11) µmic,m° - µw° = RT ln Xw + RT ln fw - (RT/m) ln (Xm/m)

If the aggregation number m is large or the mole fraction of amphiphile in micelles Xm issmall, then the final term can be ignored. This allows us to calculate the cmc knowingµmic,m° - µw°.

(2.12) µmic,m° - µw° = RT ln Xw = RT ln cmc

Alternatively, equation 2.11 can be solved for ln Xm

(2.13) ln Xm = -(m/RT)(µmic,m° - µw°) + m ln Xw + m ln fw + ln m

This allows us to calculate the concentration of amphiphile in micelles of aggregationnumber m at any given aqueous amphiphile concentration Xw. Again, we must knowµmic,m° - µw°.

To analyze µmic,m° - µw°, we will separate this energy into two parts: thatcontributed by the hydrophobic effect (Um° - µw°) and that caused by head grouprepulsion (Wm).

(2.14) µmic,m° - µw° = Um° - µw° + Wm

Tanford determines each of these components semiempirically, although the expressionscan be rationalized to some extent. The contribution of the hydrophobic effect is specifiedas

(2.15) Um° - µw° = -2100 - 700 (nc -2) + 25 (A-21) + Constant

This expression attributes 700 cal/mol to each carbon atom in the hydrocarbon chain inrough agreement with the values given in Section 2.3. The hydrophobic effect is alsoassumed to depend linearly on the surface area of the micelle per molecule of amphiphile(A). A is measured in square angstroms, so each square angstrom of hydrophobic surfacearea per molecule adds about 25 cal/mol to the free energy.

This dependence of the free energy on micelle surface area per molecule is crucialsince it helps determine the size of the micelle. While the volume of the micelle increasesin proportion to the aggregation number m, the surface area increases only in proportion tom2/3. Therefore, larger micelles will have a smaller surface area per molecule and smallermicelles will have a larger surface area per molecule. Under a given set of conditions, anamphiphile molecule will have an optimum surface area, and this will determine the size ofthe micelle that it will form.

Having estimated the contribution of the hydrophobic effect to the free energy ofmicelle formation, we must estimate the contribution of the head group interaction (Wm).This may be calculated from pressure vs. area curves measured for monolayers ofamphiphile. If an amphiphile is added to water, it will form a monolayer on the surface ofthe water with the polar end in the water and the nonpolar region extending into the air.

Page 19: Membrane Biophysics

Page 2.7

The pressure required to compress this monolayer will depend on the repulsive forcesbetween the head groups (Phg) and on the intrinsic pressure exerted by ideal molecules byvirtue of their kinetic energy (Pke). The latter pressure is

(2.16) Pke = kT/A

where A is the surface area per molecule and k is the Boltzmann constant. This expressionis the two-dimensional correlate of the ideal gas law. The pressure attributable to headgroup interaction is therefore

(2.17) Phg = P - (kT/A)

The work done against this head group repulsion is

(2.18) Wm = - ∫ (P-kT/A) dA

This work function can be evaluated by integrating pressure vs. area curves determinedfrom monolayer compression experiments and corrected for kT/A (Figure 2.2). For thetrimethylammonium and sulfate head groups, the work functions as evaluated by Tanfordare

(2.19) Wm = 1.51 x 105/A - 8.3 x 104/A2 - 2.4 x 107/A3

(2.20) Wm = 1.07 x 105/A + 5.4 x 105/A2 - 3.6 x 107/A3

1008060400

10

20

30

40

50�

A (Å )2

Ptotal

hgP

P (e

rg/c

m

)2

Figure 2.3

We now have semiempirical expressions that can be used to calculate the freeenergies involved in the formation of dodecyl sulfate and cetyltrimethylammonium

Page 20: Membrane Biophysics

Page 2.8

micelles. Table II compares experimental results to results of Tanford's semiempiricalcalculations based on equations 2.12 and 2.13. If head group interactions are calculatedtheoretically (Debye-Huckel theory), agreement with experimental results is not as good.

TABLE 2.2 m cmc (M)_________________________________________________________Experimental

N+(CH3)3 59 0.0066OSO3

- 95 0.0015Semiempirical (Monolayer data)

N+(CH3)3 60 0.0062OSO3

- 93 0.0018Theoretical (Debye-Huckel)

OSO3- 39 0.0042

_________________________________________________________

The hydrophobic effect and the head group interaction have both been cast asfunctions of the micelle surface area per molecule (equations 2.15, 2.19 and 2.20). It isinstructive to plot these energies as functions of the surface area (Figure 2.4). This showsthat there is a molecular surface area which minimizes the total free energy. The surfacearea giving this minimum free energy determines the aggregation number of the micelle. Alarger surface area per molecule implies smaller micelles. A smaller surface area permolecule implies larger micelles with the bilayer structure being the limiting case(infinitely large micelle). The minimum free energy itself is RT ln cmc (equation 2.12).

This plot illustrates effects of changes in the energies. For example, decreasing thework function by increasing the ionic strength will lower the cmc and increase theaggregation number (Figure 2.5).

Page 21: Membrane Biophysics

Page 2.9

-10

0

10050

µ (k

cal/m

ol)

A (Å )2

Figure 2.5

U° - µ°m w

W (low ionic strength)m

W (high ionic strength)m

mic,mU° - µ°w

Figure 2.4

µ (k

cal/m

ol)

-8

-6

-4

-2

0

2

4

1007550

mic,mU° - µ°wRT ln cmc

A min

A (Å )2smallermicelles

largermicelles

Wm

U° - µ°m w

Page 22: Membrane Biophysics

Page 2.10

Cetyltrimethylammonium (CTAB)

Lauryl Sulfate (Dodecyl sulfate)

Cholic Acid: X = OHDeoxycholic Acid: X = H

7

OH

CH3

CH3

H

X

CH3

OH

H

H

H

-OOC

-O3S O CH2 (CH2)10 CH3

H3C N+

CH3

CH3

CH2 (CH2)14 CH3

Fig 2.6. Some ionic detergents

Page 23: Membrane Biophysics

Page 2.11

Lubrol W

Triton X-100 (Octylphenoxypolyoxyethanol)

HO (CH2-CH2-O)10 (CH2)7 CH3

HO (CH2-CH2-O)7 (CH2)15 CH3

Fig. 2.7. Some nonionic detergents

References

Stillinger, F.H. (1980) Water revisited, Science 209, 451-457.C. Tanford (1974) Theory of micelle formation in aqueous solutions, J. Phys. Chem. 78,

2469-2479.C. Tanford (1974) Thermodynamics of micelle formation: Prediction of micelle size and

size distribution, Proc. Natl. Acad. Sci. USA 71, 1811-1815.C. Tanford (1977) The hydrophobic effect and the organization of living matter, Science

200, 1012-1018.C. Tanford (1979) Interfacial free energy and the hydrophobic effect, Proc. Natl. Acad.

Sci. USA 76, 4175-4176.C. Tanford (1980) The Hydrophobic Effect, Second Edition, John Wiley & Sons, New

York.

Page 24: Membrane Biophysics

Fundamental Principlesof Membrane Biophysics

CHAPTER 3: THE FLUID MOSAIC MEMBRANE

David Njus

Department of Biological SciencesWayne State University

© D. Njus, 2000

Page 25: Membrane Biophysics

Page 3.1

CHAPTER 3: THE FLUID MOSAIC MEMBRANE

Section 3.1. Characteristics of Lipid BilayersNaturally occurring phospholipids have a very low critical micelle concentration.

For example, the cmc for dipalmitoyl phosphatidylcholine is 4.7 x 10-10 M. Therefore,phospholipids will virtually always form into a bilayer structure. The lipid bilayer has athickness of approximately 40 Å, determined principally by the chain length of the fattyacids in the phospholipids. Each phospholipid molecule occupies a surface area of about70 Å2.

As in a micelle, the surface area occupied by each phospholipid molecule in thelipid bilayer is determined by the balance between head-group interactions and thehydrophobic effect. If the head groups favor greater separation than the hydrocarbonchains will permit, then the hydrocarbon chains may tilt so that they are not alignedperpendicular to the surface of the membrane. This decreases the thickness of the bilayerand increases the cross-sectional area occupied by each fatty-acid chain. If the head-groupstend to pack more tightly than the hydrocarbon chains, the fatty acid chains will be alignedperpendicular to the plane of the membrane and there may also be a force on the bilayerfavoring formation a concave bend.

The hydrocarbon chains in diacyl phospholipids undergo a phase transition from anordered (crystalline) to a disordered (fluid) state. Some phase transition temperatures aregiven in Table 3.1. Increasing unsaturation and decreasing fatty acid chain length lowerthe phase transition temperature. Cholesterol generally causes the phase transition to

Table 3.1. Phase transition temperatures for the transition from ordered to disorderedhydrocarbon chains in diacyl phospholipids in hydrated multilayers

Phospholipid Transition temperature (°C)______________________________________________________________________diC22 phosphatidyl choline 75diC18 phosphatidyl choline 54.9diC16 phosphatidyl choline 41.4diC14 phosphatidyl choline 23.9diC18:1 phosphatidyl choline -22diC16 phosphatidyl serine (low pH) 72diC16 phosphatidyl serine (high pH) 55diC16 phosphatidyl ethanolamine 60diC14 phosphatidyl ethanolamine 49.5______________________________________________________________________

occur over a broader range of temperatures. As described by the fluid mosaic model, thelipids in a biological membrane are typically in a fluid state at physiological temperatures.In the ordered state, the fatty acid chains occupy less volume than in the fluid state. In thecase of phosphatidylcholine, this means that the hydrocarbon chains must tilt to achieveadequate separation of the head groups.

Page 26: Membrane Biophysics

Page 3.2

Bilayers composed of mixed lipids may exhibit phase separation. If the chemicalpotentials of the individual lipids are higher in the mixed phase than in homogeneousphases, then the homogeneous phases will separate out. This can be detected as a spatialseparation of different lipids or as domains having different characteristics (e.g., fluidity).A striking example of this is the ripple phase exhibited by phosphatidylcholine bilayers. Atlow temperatures, the bilayer forms a homogeneous ordered phase and, at hightemperatures, it forms a single fluid phase. At intermediate temperatures, however, thebilayer exhibits a periodic pattern showing an undulating pattern or ripples on its surface.The molecular basis for this is not yet clear, but some inferences can be made. In theintermediate temperature range, phospholipids with tilted chains coexist withphospholipids with extended chains. The differences in chain tilting and bilayer thicknessdiscourage intermixing of phospholipids in different phases and causes the two groups tosegregate (Marden et al., 1984).

Lipids in natural membranes are characteristically distributed asymmetrically in thetwo halves of the bilayer. For example, human red cells are mostly PC on the outside andmostly PE on the inside (Table 3.2). This asymmetry can be maintained becausephospholipids are very slow to redistribute (flip-flop) between the two sides of a bilayer.The original cause of the asymmetry may lie in the biosynthetic history of the membrane,but there also appear to be ATPases that invest cellular energy in the transport ofphospholipid head groups across a variety of natural membranes (Devaux, 1992).

Table 3.2. Distribution of lipids in natural membranes

Outer Monolayer Inner MonoayerMembrane SM PC PE PS SM PC PE PS___________________________________________________________________Human RBC 40 42 10 0 8 13 45 25Rat RBC 42 35 15 0 6 20 40 25Bovine ROS - 10 40 40 - >80 10 10___________________________________________________________________Values are percentages of total lipid in that layer of the membrane.RBC = red blood cell; ROS = rod outer segment; SM = sphingomyelin.

Section 3.2. Model Lipid MembranesBecause phospholipids naturally form bilayer structures (at least at low lipid:water

ratios), artificial membranes can be produced in a number of ways. Phospholipid vesiclesare easily made by sonicating lipids (Huang, 1969), by reverse phase vaporization (Deamerand Bangham, 1976), or by dialyzing away detergent (Milsmann et al., 1978). Sonicationis convenient but produces rather small unilamellar vesicles so the membranes have a highradius of curvature. Reverse phase vaporization produces larger vesicles but the solvent inwhich the lipids are initially dissolved contaminates the preparation and may alterpermeability and other properties of the bilayer. Sonication and reverse phase vaporizationare both rather harsh treatments for proteins, so reconstitution of membrane proteins isgenerally accomplished by some variation of the dialysis method.

Page 27: Membrane Biophysics

Page 3.3

Measuring the electrical properties of membranes requires a planar membraneseparating two aqueous spaces large enough to accommodate electrodes. Planar bilayerscan be made by spreading lipid in solvent over an aperture (Mueller et al., 1963) or byraising two monolayers past the aperture (Montal and Mueller, 1972). This producesartificial membranes with much less total surface area than a liposome suspension, but itdoes permit electrical recording of artificial membrane properties. In the solvent/aperturemethod, the solvent in which the lipid is dissolved moves to the rim of the aperture and thebilayer across the opening thins out to form a "black lipid membrane." Nevertheless, thesolvent can form lenses in the artificial membrane and there is always some question aboutthe influence of remaining solvent on the properties of the membrane. The monolayermethod corrects this. Even using great care, planar membranes formed by either methodare relatively unstable and their short lifetime is an experimental handicap.

To study membrane proteins, liposomes formed by dialysis have proven mostconvenient. The proteins can be solubilized in the detergent and then convenientlyreconstituted. Monitoring effects of these proteins on electrical properties of themembrane is not possible, however, because liposomes are too small to insert electrodes.To overcome this problem a number of new approaches have been tried. First, liposomescontaining the reconstituted protein may be fused into a black lipid membrane (Miller andRacker, 1976). Patch clamping technology has introduced a new approach. A patchpipette can be raised through a lipid monolayer on the surface of a solution and thenlowered back down. A planar bilayer forms across the opening of the patch pipette and thisbilayer will contain proteins dispersed in the lipid monolayer (Tank et al., 1982; Suarez-Isla et al., 1983).

Page 28: Membrane Biophysics

Page 3.4

Section 3.3. Lipid ComponentsFatty acids: Fatty acids have two characteristics that affect the physical properties

of the membrane: chain length and degree of unsaturation. Fatty acids may be identifiedby a common name, by the standard nomenclature, or by the w nomenclature. Accordingto the standard nomenclature, fatty acids are represented as x:y, z1,z2 ... zn where xrepresents the number of carbon atoms, y represents the number of double bonds, and z1,z2... zn represent the carbon atoms preceding double bonds counting from the carboxy end.According to the w nomenclature, fatty acids are represented as x:yωz' where x representsthe number of carbons, y represents the number of double bonds, and ωz' represents theposition of the first double bond counting from the ω carbon (methyl terminus). Forexample, the structure of palmitoleic acid (16:1, 9-cis or 1ω7) is:

CH3-(CH2)4-CH2CH=CH(CH2)7COOH

TABLE 3.3. Some Common Fatty acids

Saturated UnsaturatedNo. of Common Common Nomenclature Carbons Name Name Standard ω___________________________________________________________________10 Capric Palmitoleic 16:1, 9-cis 1ω712 Lauric Oleic 18:1, 9-trans 1ω914 Myristic Vaccenic 18:1, 11-cis 1ω716 Palmitic Linoleic 18:2, 9-cis,12-cis 2ω618 Stearic α-Linolenic 18:3, 9-cis,12-cis,15-cis- 3ω320 Arachidic γ-Linolenic 18:3, 6-cis,9-cis,12-cis- 3ω622 Behenic Arachidonic 20:4, 5,8,11,14 (all cis) 4ω624 Lignoceric___________________________________________________________________

Phospholipids: Phospholipids have the general structure shown below:

O-Headgroup-O-P-O- CH2 O

O CH-O-C-CH2...CH3CH2-O-C-CH2...CH3

O

Page 29: Membrane Biophysics

Page 3.5

The head groups - which differ in charge, polarity and reactivity - give the phospholipidsdifferent characteristics:

Phosphat idic Acid (PA)

Phosphat idylinosit ol (PI)

Phosphat idylser ine (PS)

Phosphat idylet hanolamine (PE)

Phosphat idylcholine (PC or lecit hin)

CH3 N+ CH2CH2

CH3

CH3

O P O

O-

O

NH3+ CH

NH3+ CH2CH2

CH2 O P

O P O

O-

O

O

O-

OCOO-

O P O

O-

O

HO P O

O-

O

CH

CHOHHOHC

HOCH

HOHC CHOH

Page 30: Membrane Biophysics

Page 3.6

Cholesterol: Cholesterol is present in plasma membranes, lysosomes and storagegranules. The cholesterol content of the Golgi complex increases on moving from the cisto the trans cisternae. Cholesterol is not present in bacterial, inner mitochondrial orchloroplast thylakoid membranes.

Cholesterol

2726

25

24

23

2221

20

19

1817

16

1514

13

12

11

109

8

7

65

43

2

1DC

BA

HO

CH3

CH3

CH3

CH3

CH3

Sphingomyelin: The sphingomyelin content is high in lysosomes and storagegranules: rat liver lysosomes, 24%; bovine chromaffin granules,15%; serotonin granulesfrom pig platelets, 24.9%; bovine pituitary neurosecretory vesicles, 21.7%.

Section 3.4. Structure of Membrane ProteinsBecause they are typically hydrophobic, membrane proteins have been notoriously

difficult to study using traditional protein chemistry techniques. The advent of molecularbiology has made it far easier to clone and sequence a membrane protein’s gene than tostudy the protein itself. The usefulness of this approach depends to a large extent on howmuch we can deduce about the structure of the protein from its amino acid sequence.Three-dimensional structures of membrane proteins are difficult to determine, but thosethat have been established seem to follow one of two patterns: the helix bundle whichincludes most integral membrane proteins and the beta barrel which includes some proteinsin the outer membrane of gram negative bacteria and the outer mitochondrial membrane(von Heijne, 1994). The helix bundle type follow the pattern set by bacteriorhodopsin; themembrane-spanning segments are both hydrophobic and alpha-helical. These membrane-spanning segments are separated by hydrophilic domains that are exposed to the aqueousenvironments at either membrane surface. The α-helical nature of the membrane spanningregions can be rationalized. Hydrophobic side chains will tend not to interact with eachother or with lipid components of the membrane. That means that the structure of ahydrophobic domain will be determined primarily by the backbone hydrogen bondingpattern. Accordingly, the a-helix is the expected pattern. The α-helix consists of 3.61amino acids per turn spanning a distance of 1.5 Å per residue. Given a membranethickness of 30-40 Å, a transmembrane a-helix should contain 20-27 amino acid residues.

Page 31: Membrane Biophysics

Page 3.7

To predict which segments in a protein are membrane-spanning regions, it hasbecome common to use a hydropathy index. The most widely used is that described byKyte and Doolittle (1982). A variety of thermodynamic parameters could be used to assigna hydropathy value to each amino acid. The validity of any particular choice may bedebated, but the only important consideration is that some consensus index of hydropathyis established for each amino acid side chain. Kyte and Doolittle based their hydrophathyindex on the water-vapor transfer free energies of the side chains and on the interior-exterior distribution of amino acid side chains. The hydropathy value of a span of aminoacids is then determined by summing the hydropathy indices for each amino acid in thatspan. It is convenient to choose an odd number for the span length so the hydropathy indexcan be associated with the amino acid in the middle of the span. For example, if a span of7 is chosen, the hydropathy value at amino acid 40 is the sum of the hydropathy indices ofamino acids 37-43.

Table 3.4. Kyte-Doolittle Hydropathy Scale

Amino Acid Single Kyte-DoolittleResidue Letter Code Hydropathy Index

________________________________________________________________Isoleucine I 4.5Valine V 4.2Leucine L 3.8Phenylalanine F 2.8Cysteine/cystine C 2.5Methionine M 1.9Alanine A 1.8Glycine G -0.4Threonine T -0.7Tryptophan W -0.9Serine S -0.8Tyrosine Y -1.3Proline P -1.6Histidine H -3.2Glutamic acid E -3.5Glutamine Q -3.5Aspartic acid D -3.5Asparagine N -3.5Lysine K -3.9Arginine R -4.5

________________________________________________________________

Section 3.5. Solubilization and Reconstitution of Membrane ProteinsIntegral membrane proteins, by nature, have hydrophobic surfaces that allow them

to penetrate into the hydrophobic center of lipid bilayers. To get these proteins out of amembrane, therefore, these hydrophobic surfaces must be protected by detergent molecules

Page 32: Membrane Biophysics

Page 3.8

or the protein will denature. The strategy used to solubilize membrane proteins is to adddetergent to the membrane. The detergent intercalates into the lipid bilayer forming amixed micelle with the phospholipids. When enough detergent has entered the membrane,the structure changes from the bilayer favored by phospholipids to the oblate or sphericalmicelles favored by the detergent. As this happens, the bilayer structure breaks down andthe proteins escape into soluble particles consisting of the protein along with detergent andresidual phospholipid. In principle, any detergent can be used to break down the lipidbilayer and solubilize membrane proteins. In practice, however, strong detergents may alsoenter into the protein itself and denature it. Dodecyl sulfate is a good example. It is usefulin gel electrophoresis because it disrupts the secondary and tertiary structure of a proteincausing it to migrate on the gel according to size alone. Dodecyl sulfate definitelysolubilizes membrane proteins, but it also denatures them destroying their structure andactivity.

Many membrane proteins (e.g., channels, transporters) have no assayable functionafter they are solubilized and removed from the membrane. In order to study theirfunctions, they must be reconstituted into some sort of a membrane structure.Reconstitution is typically a matter of simply reversing the solubilization process.Phospholipids are reintroduced, detergent is removed, and the particle in which the proteinis situated changes from a micelle back into a bilayer. The objective here is to remove thedetergent but not the phospholipid. This can be accomplished by dialysis or gel filtration.Because phospholipids have an extremely low critical micelle concentration, the rate atwhich free phospholipids are removed is extremely slow. Obviously, the same will be truefor a detergent with a low cmc, making such a detergent (Triton X-100 is an example)difficult to use in reconstitution. A detergent with a higher cmc can be removed relativelyrapidly, however, permitting reconstitution to occur successfully. The bile acids, cholicacid and deoxycholic acid, have been particularly useful in solubilization and reconstitutionof membrane proteins. They have a high cmc, making it easy to remove them by dialysis.Because they are ionic, their effectiveness as detergents depends on the ionic strength, andthe salt concentration can be manipulated to shift the balance between solubilization andreconstitution. The bile acids also have a structure that is particularly suited to solubilizingmembrane proteins without denaturing them. These molecules have a generally planarstructure with a polar side and a nonpolar side. The nonpolar side protects the hydrophobicprotein surface while the other side faces the water. At the same time, this asymmetricalpolarity does not favor penetration by the detergent into the hydrophobic core of theprotein.

References

D. Deamer and A.D. Bangham (1976) Large volume liposomes by an ether vaporizationmethod, Biochim. Biophys. Acta 443, 629-634.

P.F. Devaux (1992) Protein involvement in transmembrane lipid asymmetry, Ann. Rev.Biophys. Biomol. Struct. 21, 417-439.

C.H. Huang (1969) Studies on phosphatidylcholine vesicles. Formation and physicalcharacteristics, Biochemistry 8, 344-352.

J. Kyte and R.F. Doolittle (1982) A simple method for displaying the hydropathic characterof a protein, J. Mol. Biol. 157, 105-132.

Page 33: Membrane Biophysics

Page 3.9

M. Marder, H.L. Frisch, J.S. Langer, and H.M. McConnell (1984) Theory of theintermediate rippled phase of phospholipid bilayers, Proc. Natl. Acad. Sci. USA 81,6559-6561.

C. Miller and E. Racker (1976) Ca2+-induced fusion of fragmented sarcoplasmic reticulumwith artificial planar bilayers, J. Memb. Biol. 30, 283-300.

M.H.W. Milsmann, R.A. Schwendener and H.G. Weder (1978) The preparation of largesingle bilayer liposomes by a fast and controlled dialysis, Biochim. Biophys. Acta512, 147-155.

M. Montal and P. Mueller (1972) Formation of bimolecular membranes from lipidmonolayers and a study of their electrical properties, Proc. Natl. Acad. Sci. USA 69,3561-3566.

P. Mueller, D.O. Rudin, H.T. Tien, and W.C. Wescott (1963) Methods for the formation ofsingle bimolecular lipid membranes in aqueous solution, J. Phys. Chem 67, 534-535.

J.R. Silvius (1992) Solubilization and functional reconstitution of biomembranecomponents, Ann. Rev. Biophys. Biomol. Struct. 21, 323-348.

B.A. Suarez-Isla, K. Wan, J. Lindstrom, and M. Montal (1983) Single-channel recordingsfrom purified acetylcholine receptors reconstituted in bilayers formed at the tip ofpatch pipets, Biochemistry 22, 2319-2323.

D.W. Tank, C. Miller, and W. Webb (1982) Isolated-patch recording from liposomescontaining functionally reconstituted chloride channels from Torpedo electroplax,Proc. Natl. Acad. Sci. USA 79, 7749-7753.

G. von Heijne (1994) Membrane proteins: From sequence to structure, Ann. Rev. Biophys.Biomol. Struct. 23, 167-192.

Page 34: Membrane Biophysics

Fundamental Principlesof Membrane Biophysics

CHAPTER 4: MEMBRANE ELECTROSTATICS

David Njus

Department of Biological SciencesWayne State University

© D. Njus, 2000

Page 35: Membrane Biophysics

Page 4.1

CHAPTER 4: MEMBRANE ELECTROSTATICS

Section 4.1. Electrostatics in One DimensionTo understand the electrostatics of biological membranes, one must understand the

concepts of charge, electric field and electrical potential. This section, therefore, reviewssome basic concepts of electrostatic theory. The units to be used in this discussion requiresome attention because the choice of units determines the nature of the constants appearingin the equations. Physics textbooks commonly employ cgs units (statcoulombs, ergs, etc.).Most of the practical measurements made in biological research, however, are inrationalized MKS units (coulombs, volts, etc.). For that reason, rationalized MKS unitswill be used throughout this discussion.

Electrostatic forces are so strong that positive and negative charges are generallypaired and overall electroneutrality is strictly observed in nature. If a positive and negativecharge are separated, there will be an attractive force between them. According toCoulomb's Law, this force will be proportional to the absolute magnitudes of the twocharges (q1 and q2) and to the inverse square of the distance between them (x):

(4.1) F = (q1q2)/(4πεox2)

εo, the permittivity of a vacuum, is a constant equal to 8.854 x 10-12 coul/m.volt.Increasing the distance between the two charges will require us to do work, which

(following the usual physical definition) is

(4.2) W = - ∫ F dx

If a charge is located in the vicinity of many other charges, the force on it will be the sumof the forces exerted by the other charges. In such a case, it is often more convenient todefine the electric field at a point x as the force exerted by all of the charges in theneighborhood on a unit charge at that point. Thus, the electric field created by a pointcharge q at a distance x is

(4.3) E(x) = q/4πεox2

The electrical potential ψ(x) is the work that must be done to move a unit charge from areference point to a point x.

(4.4) ψ(x) = - ∫x

ref E dx

Because we are interested in the electrical properties of biological membranes, wecan take advantage of their planar symmetry. All points the same distance from the surfaceof the membrane will have the same electric field and the same electrical potential, so thedimension perpendicular to the membrane surface is the only significant one. We willconsider charge not to be distributed as points in space but to be spread smoothly on planesparallel to the membrane. If positive charge is smoothly distributed on a plane of charge,

Page 36: Membrane Biophysics

Page 4.2

as in Figure 4.1A, it will create an electric field perpendicular to the plane. The electricfield will be proportional to the charge density on the plane (qs):

(4.5) E = qs/2εo

If we have two parallel planes of charge, one with a positive surface charge density(+qs) and one with an equal density of negative charge (-qs), their electric fields will beadditive in the space between them and will cancel in the spaces on either side (Figure4.1B). Between the planes, therefore, the total electric field will be qs/εo; outside of thatspace, the total electric field will be zero. This arrangement is the common textbookexample, the parallel plate capacitor. Because there is an electric field between the twoplates of the parallel plate capacitor, there will be a difference in electrical potential

(4.6) ∆ψ = - ∫0

x E dx = - ∫

0

x (qs/εo) dx = - qsx/εo

This is the potential energy that must be expended to move a unit charge from plate 1 (at 0)to plate 2 (at x).

Single plane of charge

A+q s

E =+q s2 ε

οE =

2 εο

-qs

B

E = 0 E = 0

0 x

Parallel plate capacitor

-qs+q s

E =ε

ο

qs

Figure 4.1. Electric fields associated with planes of charge

The preceding equations apply to charges arranged in a vacuum. In the real world,charges will be separated by a medium composed of polarizable molecules. Thesemolecules will tend to align with the electric field and cancel it. The electric field will bediminished in proportion to the polarizability of the medium, expressed as the dielectricconstant ε. Therefore, if the plates of a parallel plate capacitor are separated by a mediumof dielectric constant ε, the electric field between the plates will be

(4.7) E = qs/εεo

Page 37: Membrane Biophysics

Page 4.3

Of course, the electrical potential difference between the plates will be correspondinglysmaller as well:

(4.8) ∆ψ = - qsx/εεo

Water, which has a dielectric constant of 80, is very polarizable. For that reason, chargeseparation across water will tend to create a relatively small electric field and smallelectrical potential differences. By contrast, hydrocarbons such as hexane (ε = 1.89) havea very low dielectric constant. For that reason, charge separation across organic phases(such as the interior of biological membranes) will create large electric fields. It is this lowdielectric constant that makes it possible to create significant electrical potentialdifferences across biological membranes.

Section 4.2. Membrane PotentialSeparation of electrical charges by biological membranes creates an electrical

potential difference across the membrane. This total difference in electrical potential,commonly called the membrane potential, plays a crucial role in many membranefunctions. It is the force that drives ions across the membrane. Because it defines theelectrical energy lost when an ion crosses the membrane, the membrane potential partlydetermines the energy stored in ion concentration gradients. Finally, the electric fieldassociated with the membrane potential acts on dipolar groups in membrane proteins andmay regulate the activities of these proteins. Voltage-dependent channels, for example,open and close in response to changes in the membrane potential.

It is conceptually helpful to divide the charge separation created by biologicalmembranes into three components. First, charges may be separated by moving ions all theway across the membrane from the aqueous medium on one side to the aqueous mediumon the other. This separation of capacitative charge is probably the most importantcomponent of the membrane potential. Second, fixed charge bound to the membranesurface will be neutralized by counterions present in the adjacent aqueous solution.Because these counterions will tend to diffuse away from the membrane surface, there willbe a consequent charge separation. This leads to the surface potential. Finally, because theester linkages between fatty acids and the glycerol backbones of the membrane lipids aredipolar in character, alignment of these dipoles creates a charge separation which gives riseto the dipole potential.

Page 38: Membrane Biophysics

Page 4.4

������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������

����������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������

�������������������������������������������������������������������������������

����������������������������������������������������������������������������������������������������������������������������������

����������������������������������������������������������������������������

����������������������������������������������������������������������������

Outside Membrane Inside

0

∆ψ

surface potential (outside)

surface potential (inside)

dipole potential (outside)

dipole potential (inside)

membrane potential

potential created by capacitative charge

Figure 4.2. Components of the membrane potential

����������������������������������������������������������������������������������������������������������

�����������������������������

�����������������������������

When charge is moved from one side of a biological membrane to the other, weassume that the net charge on one side is equal and opposite to the net charge on the other.This is strictly required to maintain overall electroneutrality. These equal and oppositecharges separated by the low dielectric medium of the lipid membrane form an electricalarrangement like the parallel plate capacitor. We will call the charge transferred across themembrane the capacitative charge qc. The separation of this capacitative charge will createan electric field in the membrane

(4.9) E = qc/Aεεo

and an electrical potential difference across the membrane

(4.10) ∆ψ = qcx/Aεεo

The capacitance C is the ratio of the capacitative charge to the potential difference:

(4.11) C = qc/∆ψ = Aεεo/x

The capacitance, therefore, is proportional to the area of the membrane and inverselyproportional to the thickness. For a biological membrane, the capacitance is typically

Page 39: Membrane Biophysics

Page 4.5

about 1 µF/cm2. This corresponds to a membrane 35 Å thick with a dielectric constant of4.

The total membrane potential will include contributions from the capacitativecharge, from the surface charge on each side of the membrane, and from the surface dipoleon each side of the membrane (Figure 4.2). Because the contributions of surface chargeand surface dipole on one side of the membrane will tend to cancel the contributions ofthose components on the other side, the capacitative charge is generally the primarydeterminant of the total membrane potential.

Section 4.3. Surface PotentialWhen fixed charges are bound to a surface, and the counter ions are dissolved in the

adjacent solution, the counter ions will tend to move away from the surface because ofdiffusion. This creates a charge separation and consequently an electrical potentialdifference called the surface potential. Whereas the relationship between the capacitativecharge and the membrane potential is a simple proportionality (the capacitance), therelationship between the surface charge and the surface potential is quite complex. Themagnitude of the surface potential depends on the amount of fixed surface charge and alsoon the separation between the fixed charge and the diffusable charge. The chargeseparation depends on a dynamic tension with diffusion pushing the counterions away fromthe surface and electrical attraction pulling them toward the surface. The theoreticalanalysis of surface potentials and surface charge was originally developed by Gouy andChapman. Just as the DeBye-Hückel theory describes ion distributions as a function ofdistance from a fixed point charge, the Gouy-Chapman theory describes ion distributions asa function of distance from the membrane surface. The analysis rests on three basicprinciples: the Boltzmann distribution, the Poisson equation, and electroneutrality.

0 x

ψ

������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������

qs q (x)v

ψo

ψ (x)

ψ (∞) = 0

Figure 4.3. The electrical potential as a function of distance from the surface.

Page 40: Membrane Biophysics

Page 4.6

First, we imagine a plane of fixed charge with a surface charge density qs. Adjacentto it is a medium with a space charge density qv(x) where x is the distance from the surface(Figure 4.3). Overall electroneutrality requires that the total space charge be equal andopposite to the sum of the fixed charge qs and the capacitative charge qc where thecapacitative charge is the net charge on the other side of the membrane. As we shall see,the capacitative charge is usually small compared to the surface charge. For simplicity, weshall simply include it as part of qs:

(4.12) - ∫0

∞ qv(x) dx = qs

The space charge at any point x is obtained by summing over all of the ionic speciespresent in the solution:

(4.13) qv(x) = Fα Σi ziCi(x)

For each ion species i, zi is the valence and Ci is the concentration in moles/liter. F is theFaraday constant and α is the constant 103 l/m3. Since the ion concentrations shoulddepend on the electrical potential according to the Boltzmann distribution:

(4.14) Ci(x) = Cio exp(-ziF ψ(x)/RT)

where Cio is the concentration of ion species i far from the membrane (x=∞). TheBoltzmann distribution (eq. 4.14) together with equation 4.13 relates qv(x) to ψ(x).

(4.15) qv(x) = Fα Σi ziCio exp(-ziF ψ(x)/RT)

Another relationship between qv(x) and ψ(x) is provided by Poisson's equation:

(4.16) qv(x)/εεo = -d2ψ/dx2

Poisson's equation can be rationalized by considering the single plane of charge (Figure4.1A). If the plane is assumed to have a thickness dx, the change in the electric field dE/dxupon crossing the plane is qs/εεo. If equation 4.4 is differentiated twice, it is apparent that

(4.17) d2ψ/dx2 = -dE/dx = -qs/εεo

Poisson's equation (eq. 4.16) can be integrated in two ways. First, using eq. 4.12,

(4.18) qs = - ∫0

∞ qv(x) dx = εεo ∫0

∞ (d2ψ/dx2) dx

We define ψ(∞) = 0, so

(4.19) qs = εεo (dψo/dx)

Page 41: Membrane Biophysics

Page 4.7

Second, using eq. 4.15 and 4.16,

(4.20) d2ψ/dx2 = - (Fα/εεo) Σi ziCio exp(-ziFψ/RT)

To integrate equation 4.20, we multiply both sides by 2dψ/dx. We can then integrate thisequation to obtain

(4.21) (dψ/dx)2 = (2RTα/εεo) Σi Cio exp(-ziFψ/RT) + Constant

At x = ∞, ψ = 0 and dψ/dx = 0, so Constant = -(2RTα/εεo) Σi Cio and equation 4.21becomes

(4.22) (dψ/dx)2 = (2RTα/εεo) Σi Cio [exp(-ziFψ/RT) - 1]

Combining this with equation 4.19 gives

(4.23) qs2 = 2RTαεεo (Σi Cio [exp(-ziFψo/RT) - 1])

Equation 4.23 is the general form of the Gouy-Chapman equation relating thesurface charge to the surface potential. A more convenient form is obtained by assumingthat all of the ions in the aqueous phase are univalent. We will let Cio = Co for both anionsand cations. Then equation 4.23 simplifies to

(4.24) qs = (8RTCoαεεo)1/2 [sinh (Fψo/2RT)]

It is apparent that the surface potential depends on the magnitude of the surfacecharge (Figure 4.4) and also on the ionic strength of the aqueous medium (Figure 4.5).Clearly, the counterions necessary to neutralize the fixed charge will have a much greatereffect on the ion concentration near the membrane surface if the ion concentration is low.At low ionic strength, therefore, the concentration gradient will be greater and diffusionforces will be stronger. This will drive the counterions farther from the surface leading to agreater charge separation and a larger potential difference.

Another way of visualizing this is to simplify equation 4.24 by expanding the sinhas a power series and dropping higher order terms (assume ψo is small). Equation 4.24then reduces to

(4.25) qs = (8RTCoαεεo)1/2(Fψo/2RT)

This may be rewritten as

(4.26) qs/εεoψo = (2CoαF2/RTεεo)1/2

The left-hand side of this equation is equal to the inverse of the distance x between platesof a parallel plate capacitor (equation 4.8). A constant κ with units of m-1 is customarilydefined as

Page 42: Membrane Biophysics

Page 4.8

(4.27) κ2 = (αF2/RTεεo) Σi Cio zi2

If all of the ions in the aqueous phase are univalent, then κ is equal to the right-hand side ofequation 4.26. Therefore,

(4.28) 1/x = κ

One may think of 1/κ as a measure of the thickness of the diffuse double layer. That is, thesame surface potential ψo would result if all of the diffusable charge were placed at adistance 1/κ from the surface of the membrane.

In the foregoing analysis, we have incorporated the capacitative charge into thesurface charge. In actual practice, the capacitative charge is usually negligible compared tothe fixed surface charge and contributes little to the surface potential. To illustrate this, wemay note that the capacitative charge will rarely be greater than 10-7 coul/cm2 since thatcharge will create a 100 mV membrane potential given the usual membrane capacitance of1 µF/cm2. Typical surface charge densities are much larger than this (chromaffin granuleshave a surface charge density of -1.38 x 10-6 coul/cm2).

Figure 4.4. The surface potential as a function of the surface charge density, according tothe Gouy-Chapman equation. The curve is for a univalent electrolyte at 10 mMconcentration. The dielectric constant has been taken as 80 and the temperature as 20°C.

Page 43: Membrane Biophysics

Page 4.9

Figure 4.5. The decay of potential from a surface.The surface charge density has been assumed to be 0.0158 coul/m2. The electrolyte isunivalent and the dielectric constant and temperature have been taken as 80 and 20°Crespectively.

Surface charge has a number of interesting effects. Because a negative surfacecharge attracts cations to the membrane surface, it increases the conductance of themembrane to cations (McLaughlin et al., 1970) and enhances the binding of cations to themembrane surface (McLaughlin and Harary, l976). It should be noted that the surfacepotential has a greater affect on the distributions of divalent ions than on distributions ofmonovalent ions. For this reason, high Ca2+ concentrations at the membrane surface,sometimes attributed to binding, may actually be caused by the surface potential. Thesurface potential also changes the pH near the surface of the membrane; this may shift theapparent pK values of protonatable groups and the apparent pH optima of membrane-bound enzymes. Finally, the surface potential may act to repel or attract other surfacesthereby functioning in phenomena such as exocytosis and intercellular communication.

Section 4.4. Dipole PotentialA third kind of charge separation that can be created by biological membranes is

that of the surface dipole. Phospholipids are dipolar in character. These dipoles, alloriented in the same direction, lead to a charge separation, which creates the dipolepotential. The dipole potential may not be a significant component of the membranepotential because the dipoles on opposite surfaces of the membrane are oriented in opposite

Page 44: Membrane Biophysics

Page 4.10

directions and tend to cancel each other. However, as we shall see, the dipole potentialmay be a major factor in determining the ionic permeability of the lipid bilayer.

The dipolar character of phospholipids seems to arise from the ester linkagebetween the fatty acid groups and the glycerol backbone. The head groups of thephospholipids are certainly polar. However, they seem not to contribute appreciably to thedipole potential. This is partly attributable to the fact that the head groups lie on theaqueous surface of the membrane where the dielectric constant of the medium tends toneutralize the dipoles. Moreover, the head groups seem to lie flat on the surface of themembrane so the dipole moment in the direction perpendicular to the membrane surface issmall.

The dipole potential is

(4.29) ∆ψ = D/εεo

where D is the surface dipole density in coul/m. To estimate the potential magnitude ofsurface dipole effects, let us assume that a typical phospholipid has a dipole moment of 1.5Debyes (1.5 x 10-18 esu.cm or 5 x 10-30 coul.m) and occupies a membrane area of 60 Å2.This yields a surface dipole density of 8.34 x 10-12 coul/m and implies a dipole potential of(1000/ε) mV. If we assume that the dielectric constant of membrane in the region of thedipole layer membrane is between 4 and 10, the surface dipole should create a potentialdifference of 100-250 mV. The electric field set up by the capacitative charge may cause the dipoles to orient in adirection perpendicular to the membrane. Thus, the surface dipole may change in responseto a membrane potential. In this way, the surface dipole will affect the electric field nearthe membrane surface and may be important in modulating the response of membrane-bound enzymes to the membrane potential.

References

R.G. Ashcroft, H.G.L. Coster, and J.R. Smith (1981) The molecular organisation ofbimolecular lipid membranes. The dielectric structure of thehydrophilic/hydrophobic interface, Biochim. Biophys. Acta 643, 191-204.

R. Aveyard and D.A. Haydon (1973) An Introduction to the Principles of SurfaceChemistry, Cambridge University Press, Cambridge.

R.F. Flewelling and W.L. Hubbell (1986) Hydrophobic ion interactions within membranes,Biophys. J. 49, 531-540.

R.F. Flewelling and W.L. Hubbell (1986) The membrane dipole potential in a totalmembrane potential model, Biophys. J. 49, 541-552.

D.A. Haydon and S.B. Hladky (1972) Ion transport across thin lipid membranes: A criticaldiscussion of mechanisms in selected systems, Quart. Rev. Biophys. 5, 187-282.

S. McLaughlin (1977) Electrostatic potentials at membrane-solution interfaces, CurrentTopics in Membrane Transport 9, 71-144.

S. McLaughlin and H. Harary (1976) The hydrophobic adsorption of charged molecules tobilayer membranes: A test of the applicability of the Stern equation, Biochemistry15, 1941-1948.

Page 45: Membrane Biophysics

Page 4.11

P.L. Yeagle (1979) Effect of transmembrane electrical potential and micelle geometry onphospholipid head group conformation, Arch. Biochem. Biophys. 198, 501-505.

Page 46: Membrane Biophysics

Fundamental Principlesof Membrane Biophysics

CHAPTER 5: SPECIFIC AND NON-SPECIFIC BINDING

David Njus

Department of Biological SciencesWayne State University

© D. Njus, 2000

Page 47: Membrane Biophysics

Page 5.1

CHAPTER 5: SPECIFIC AND NON-SPECIFIC BINDING

Section 5.1. The Langmuir Adsorption IsothermSmall molecules may bind to membranes either non-specifically or specifically.

Non-specific binding occurs when small amphiphilic molecules adsorb to the surface of thelipid bilayer or to the equivalent hydrophobic/hydrophilic interfaces of membrane proteins.Specific binding occurs when ligands bind to selective binding sites of specific receptors.In either case, similar analyses can be used to describe the binding process.

The simplest description of the relationship between bound and free ligand assumesan equilibrium defined by the law of mass action:

(5.1) L + M = LM

where L is the free ligand, M signifies empty binding sites on the membrane, and LMrepresents ligand bound to the membrane. The equilibrium constant K is

(5.2) K = [L][M]/[LM]

Let us define Lm as the concentration of bound ligand ([LM]) and Lmax as the total numberof binding sites on the membrane ([M]+[LM]). Then the equilibrium constant becomes

(5.3) K = [L](Lmax-Lm)/Lm

Equation 5.3 may be solved to yield an expression for the amount of bound ligandLm

(5.4) Lm = Lmax [L] / ([L] + K)

The amount of bound ligand Lm depends upon the binding constant K, the number ofbinding sites Lmax, and the concentration of ligand [L] at the surface of the membrane(Figure 5.1). As described in Section 5.2, if the ligand is charged, its concentration at thesurface of the membrane [L] will not necessarily equal its concentration in the bulk of theaqueous phase but will be related to it by the surface potential. As described in Section5.3, the binding constant K reflects the change in free energy occurring upon binding.

Equation 5.3 may also be rearranged to yield the Scatchard equation:

(5.5) Lm/[L] = Lmax/K - Lm/K

This form is particularly useful because a plot of Lm/[L] vs. Lm is linear and may be used todefine the parameters Lmax and K (Figure 5.2).

Page 48: Membrane Biophysics

Page 5.2

Lm

Lmax

[L]Lm

KLmax

������������������������������������������������������������������������������������

������������������������������

Lm

Lmax

Lmax2

K

[L]

Figure 5.1 Figure 5.2

Section 5.2. Effect of Surface Potential[L] is the concentration of ligand at the surface of the membrane. As described

earlier, this is related to [L]∞, the concentration far from the membrane, by the Boltzmannequation:

(5.6) [L] = [L]∞ exp (-ziFψo/RT)

Obviously, the known ligand concentration in the bulk of the aqueous phase [L]∞ may beused in place of the ligand concentration at the membrane surface [L] if the ligand isuncharged or if the surface potential ψo is negligible. Since the surface potential dependsupon both the ionic strength of the solution and upon the surface charge density, it can beconsidered negligible if the ionic strength is high or if the surface charge density is low. Ifthe surface potential is greater than a few millivolts, however, [L] will be significantlydifferent from [L]∞. For example, if the surface potential is -18 mV and the ligand is aunivalent anion, [L] will be only 50% of [L]∞, and using [L]∞ in place of [L] willsignificantly change the appearance of the binding curve (Figure 5.3) and the Scatchardplot (Figure 5.4).

A special problem occurs when a charged ligand binds to the membrane to such anextent that the bound ligand itself affects the membrane surface charge. This is generallynot a problem for ligands that bind to receptors since the receptor density (Lmax) is typicallyinsignificant compared to the surface charge density of the membrane. Ligands that adsorbto the membrane, however, may have a significant effect on the surface charge density.This situation, first considered by Stern, has been analyzed more recently by McLaughlinand Harary (1976). If we assume that the surface charge density of the membrane isinitially zero, then [L] = [L]∞ for low values of Lm. As Lm increases, however, the surfacepotential will increase and binding will deviate in the direction expected in the presence ofa surface potential. Thus, the Scatchard plot will appear to curve (Figure 5.4).

Page 49: Membrane Biophysics

Page 5.3

Section 5.3. The Binding ConstantThe binding constant K reflects the standard free energy change upon binding. This

can be seen by recognizing that the free energy change for the reaction in equation 1 is

(5.7) ∆G = ∆G° + RT ln ([LM]/[L][M])

At equilibrium, ∆G = 0 and [LM]/([L][M]) = 1/K. Therefore,

(5.8) ∆G° = RT ln K

For adsorption, the free energy change represents the free energy change for the transfer ofthe ligand from water into the membrane. For receptors, the free energy change carriesadditional significance. In this case, ligand binding not only changes the location of theligand, but it changes the conformation of the receptor to which the ligand binds. Thus, thefree energy of binding includes not just an affinity between the binding site and the proteinbut a change in the conformation of the protein. The conformational change is crucial, ofcourse, because it activates the receptor thereby transmitting the ligand-binding signal.

Section 5.4. Specific vs. Non-Specific BindingOften, it is desirable to study the binding of a ligand to a receptor or other

membrane-bound protein, but the ligand (because it is hydrophilic or amphiphilic) exhibitsconsiderable non-specific binding to the membrane. This may be analyzed as follows:Total binding to the membrane (from eq. 5.4) will be the sum of the specific (sp) and non-specific (ns) contributions:

(5.9) Lm = Lmsp + Lm

ns = Lmaxsp [L]/([L] + Ksp) + Lmax

ns [L]/([L] + Kns)

Let us assume that the affinity of the non-specific binding sites is very weak compared tothe affinity of the specific sites (Kns >> Ksp). To measure specific binding, one uses ligand

Page 50: Membrane Biophysics

Page 5.4

concentrations that range about Ksp, so [L] will be very small relative to the non-specificbinding constant [L] << Kns). Equation 5.9 then reduces to

(5.10) Lm = Lmaxsp [L]/([L] + Ksp) + Lmax

ns [L]/Kns

so non-specific binding will be directly proportional to the ligand concentration over thisrange. This non-specific binding can be measured by adding a high concentration of ligand(Kns >> [L] >> Ksp). Under these conditions, equation 5.9 reduces to

(5.11) Lm = Lmaxsp + Lmax

ns [L] / Kns

Because Lmaxsp << Lmax

ns, the measured binding Lm will all be non-specific. This gives theslope of the non-specific binding line (Lmax

ns / Kns). Then specific binding can becalculated by subtracting non-specific binding from total binding as in Figure 5.5.

References

S. McLaughlin and H. Harary (1976) The hydrophobic adsorption of charged molecules tobilayer membranes: A test of the applicability of the Stern equation, Biochemistry15, 1941-1948.

T.M. DeLorey and R.W. Olsen (1992) γ-Aminobutyric acidA receptor structure andfunction, J. Biol. Chem. 267, 16747-16750.

C.D. Strader, T.M. Fong, M.R. Tota, D. Underwood, and R.A. F. Dixon (1994) Structureand function of G-protein-coupled receptors, Ann. Rev. Biochem. 63, 101-132.

M. Hollmann and S. Heinemann (1994) Cloned glutamate receptors, Ann. Rev. Neurosci.17,

Page 51: Membrane Biophysics

Fundamental Principlesof Membrane Biophysics

CHAPTER 6: PERMEABILITY AND CONDUCTANCE

David Njus

Department of Biological SciencesWayne State University

© D. Njus, 2000

Page 52: Membrane Biophysics

Page 6.1

CHAPTER 6: PERMEABILITY AND CONDUCTANCE

Section 6.1. Formal Analysis of Permeability and ConductanceThe passage of ions and molecules across membranes is a phenomenon often

divided into two parts: permeability and conductance. Conductance describes themovement of electrolytes (ions and charged molecules) across the membrane in response toa membrane potential. Permeability describes the movement of uncharged molecules(nonelectrolytes) as well as the movement of electrolytes in the absence of a membranepotential.

Three different approaches may be used to analyze permeability and conductance:classical thermodynamics, nonequilibrium thermodynamics and statistical mechanics.Strictly speaking, classical thermodynamics and statistical mechanics apply only toequilibrium situations. Nonequilibrium thermodynamics, by contrast, was developed todescribe non-equilibrium phenomena such as permeability and conductance. Many purists,therefore, prefer to discuss permeability and conductance in terms of nonequilibriumthermodynamics. Unfortunately, this approach generally provides little insight into thephysical events occurring as molecules or ions cross biological membranes. For thatreason, we will focus here on the other two approaches: We will see that they both yieldthe same results and provide different insights into the processes involved.

The analysis of permeability and conductance using classical thermodynamics isbased on the concept of the electrochemical potential:

(6.1) µ = µ° + RT ln C + zFψ

At equilibrium, a given ion or molecule must have the same electrochemical potential onboth sides of the membrane. If we use the subscript i to denote one side of the membrane(inside) and the subscript o to denote the other side (outside), then

(6.2) µi° + RT ln Ci + zF ψi = µo° + RT ln Co + zF ψo

Since µi° = µo°, equation 6.2 can be rearranged to yield

(6.3) Ci/Co = exp [-zF(ψi - ψo)/RT]

This is the famous Nernst equation, which gives the equilibrium concentration gradient ofan electrolyte in the presence of a membrane potential. If the molecule is a non-electrolyte(z=0) or if the membrane potential is zero, then equation 6.3 simplifies to

(6.4) Ci = Co

The analysis of permeability and conductance using statistical mechanics is basedon two concepts: the concept of unidirectional fluxes and transition state theory. Thenotion that net flux across the membrane is simply the sum of two individual unidirectionalfluxes was first introduced by Ussing. Transition state theory, championed by Eyring,argues that the rate at which a given transition will occur depends on the probability thatthe ion or molecule has enough energy to cross the transition barrier. The probability of

Page 53: Membrane Biophysics

Page 6.2

having this activation energy is given by the Boltzmann distribution. Let us imagine thatthe activation energy for a particular ion or molecule crossing the membrane is Ea (Figure6.1). Then the rate of influx (Joi) is

(6.5) Joi = Co exp [-(Ea-zF ψo)/RT] x constant

The rate of efflux (Jio) is

(6.6) Jio = Ci exp [-(Ea-zF ψi)/RT] x constant

These two rates must be equal at equilibrium, so

(6.7) Ci exp [-Ea/RT + zF ψi/RT] = Co exp [-Ea/RT + zF ψo/RT]

Equation 6.7 reduces to the Nernst equation (equation 6.3), so the statistical mechanicsapproach predicts the same equilibrium as does the thermodynamics approach.

zF

���������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������

Energy Outside Membrane Inside

��������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������E a

����������������������������������������������������������������������������������������������������������������������������������������

zFoψ

Figure 6.1. Transition-state energy for membrane permeation.

Section 6.2. EquilibriaPermeation and conduction are passive processes and will proceed in the direction

of equilibrium. For electrolytes in the presence of a membrane potential, we have shownthat the equilibrium concentration gradient is given by the Nernst equation (equation 6.3).For non-electrolytes and for electrolytes when the membrane potential is zero, theequilibrium reduces to Co = Ci (equation 6.4). An interesting exception to these rules isfound in the case of weak acids and weak bases. At neutral pH, weak acids and weak basesare predominantly in their charged forms (A- and BH+). These charged species do notpermeate across the hydrophobic barrier presented by biological membranes. The chargedspecies, however, are in equilibrium with uncharged species that will permeate themembrane. Permeation of the uncharged species causes the charged species to reach thefollowing equilibria:

(6.8) [BH+]i/[BH+]o = [H+]i/[H+]o

Page 54: Membrane Biophysics

Page 6.3

(6.9) [A-]i/[A-]o = [H+]o/[H+]i

To understand the logic behind this, let us consider the case of a weak base (Figure 6.2).The uncharged species (B) will reach the equilibrium expected of nonelectrolytes (Bo = Bi).On either side of the membrane, this unprotonated species will be in equilibrium with theprotonated form:

(6.10) K = [B]o[H+] o /[BH+] o = [B]i[H+] i /[BH+] i

Since [B]o = [B]i, equation 6.10 reduces to equation 6.8. A similar analysis may be appliedto weak acids to establish equation 6.9.

�����������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������

Outside Inside

BH BH++

H + B B + H ++

Membrane

Figure 6.2. Permeation of a weak base.

Section 6.3. Permeation of Non-ElectrolytesThe velocity (v) at which a molecule or ion will diffuse is proportional to the

gradient of its electrochemical potential (dµ/dx):

(6.11) v = -(1/Nf) (dµ/dx)

N is Avogadro's number and f is the frictional coefficient. The total rate of flow J is thevelocity multiplied by the concentration C:

(6.12) J = vC = -(C/Nf)(dµ/dx)

For a non-electrolyte, dµ/dx = d(RT ln C)/dx, so

(6.13) J = (-C/Nf)(RT/C)(dC/dx) = -(RT/Nf)(dC/dx)

If we define RT/Nf as the diffusion coefficient D, equation 6.13 reduces to Fick's Law ofdiffusion.

Page 55: Membrane Biophysics

Page 6.4

Membrane permeation involves diffusion across the membrane. To analyze this,we must integrate Fick's Law (equation 6.13) from one side of the membrane to the other(Figure 3).

(6.14) ∫0

d J dx = - ∫

0

d (RT/Nf)(dC/dx)dx

����������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������

Outside Membrane Inside

0 d

���������������������������������������������������������

����������������������������������������������������������

C o

Ci

Cmi

Cmo

Figure 6.3. Concentration profile for steady-state flow across a membrane

This integration, which allows us to determine how the flow depends on the overallconcentration gradient (Ci - Co) across the membrane, is relatively simple. As we shall see(Section 7.1), the corresponding integration for the case of non-electrolytes is much moredifficult.

To integrate equation 6.14, we assume that the flow of non-electrolyte across themembrane reaches a steady state. This means that the flow J must be the same at all pointswithin the membrane so that the concentration profile across the membrane does notchange with time. Steady-state is a natural assumption because it is a stable condition.Suppose that J varied within the membrane so that the flow into a particular region wasgreater than the flow out of that region. The concentration of non-electrolyte in that regionwould then rise. This, in turn, would cause the flow out of the region to increase and theflow into the region to decrease. Thus, the concentration would spontaneously change tomaintain equality between the flows into and out of the region. This assures steady state.We may then integrate equation 6.14 assuming that J is independent of x (constant).

(6.15) Jd = -(RT/Nf)(Cmi-Cmo)

Cmi is the concentration in the membrane at the inner surface and Cmo is the concentrationin the membrane at the outer surface.

We imagine that, at each surface of the membrane, molecules in the aqueous phaseare in equilibrium with molecules in the membrane phase. Therefore, the chemicalpotential in the water phase (µw) must equal the chemical potential in the membrane (µm):

(6.16) µw = µw° + RT ln Cw = µm = µm° + RT ln Cm

Page 56: Membrane Biophysics

Page 6.5

The concentration at the surface of the membrane (Cm) is then

(6.17) Cm = Cw exp [(µw°- µm°)/RT]

An expression for J is obtained by substituting equation 6.17 into equation 6.15 andrearranging:

(6.18) J = -(RT/Nfd){exp [(µw° - µm°)/RT]}(Ci - Co)

The flux is proportional to the concentration difference across the membrane (Ci - Co). It isalso proportional to the permeability coefficient P defined as

(6.19) P = (RT/Nfd){exp [(µw° - µm°)/RT]}

The permeability coefficient includes the term RT/Nf, which is the diffusion coefficient ofthe molecule in the lipid phase of the membrane. This term will depend on the size of themolecule. The term exp[(µw°-µm°)/RT] describes the partitioning of the molecule betweenwater and the membrane and will depend on the lipid solubility of the molecule. SinceCm/Cw is the membrane:water partition coefficient (Kp), equation 6.17 implies that

(6.20) Kp = exp [(µw° - µm°)/RT]

A comparison of equations 6.19 and 6.20 shows that P and Kp should be linearly related.Walter and Gutknecht (1984) tested this prediction in a study of the permeability of lipidbilayers to a series of carboxylic acids. After correcting for unstirred layer effects andassuming that only the protonated form of the carboxylic acid permeates, they found thatthe permeability coefficient is related to the hexadecane:water partition coefficient (Kp') asfollows:

(6.21) log P = 0.90 log Kp' + 0.87

The observed slope of 0.90 is close to the predicted slope of 1.0. Moreover, the free energychange for the transfer of the carboxylic acid from water into the membrane (µm°-µw°) canbe determined from either the permeability coefficient P or the partition coefficient Kp.The incremental change in the free energy per methylene group for the series of carboxylicacids (acetic, propionic, butyric, hexanoic) is -898 ± 159 cal/mole determined from thepartition coefficient and -764 ± 54 cal/mole determined from the permeability coefficient.These numbers agree very well with the energies described in the discussion on micelleformation (Section 2.4).

Section 6.4. Unstirred LayersWhen a compound diffuses from one side of a membrane to the other, the

membrane may be the principal barrier to flow but not the only barrier. Passage of themolecule across the membrane may also be slowed by diffusion across the aqueous layersadjacent to either surface of the membrane. These so-called unstirred layers may range inthickness from 1 µm to 500 µm (Remember that the membrane itself is only 4 x 10-3 µm

Page 57: Membrane Biophysics

Page 6.6

thick). The unstirred layer effect will generally be most prominent for relatively nonpolarcompounds. For these compounds, the permeability coefficient will be large and diffusionacross the membrane itself will be relatively fast. Diffusion across the aqueous layers,therefore, may be partially rate-limiting. For compounds that are quite water soluble,permeation across the membrane will be slow, and diffusion across the unstirred layers willhave relatively less effect.

To consider the effect of unstirred layers, imagine that a compound has a membranepermeability coefficient P and an aqueous diffusion constant D. Let the unstirred layershave thicknesses, di and do (Figure 6.4), let the bulk concentrations of the compound be Ci(inside) and Co (outside) and let the concentrations at the surface of the membrane be Cmiand Cmo. The flow through the membrane then is

(6.22) Jm = P (Cmi - Cmo)

The flow through the unstirred layers will be

(6.23) Ji = (D/di) (Ci - Cmi)

(6.24) Jo = (D/do) (Cmo - Co)

At steady-state, the flows must all be the same (Jm = Ji = Jo = J). Therefore, equations 6.22,6.23 and 6.24 can be transformed to

(6.25) J/P = Cmi - Cmo

(6.26) Jdi/D = Ci - Cmi

(6.27) Jdo/D = Cmo - Co

By summing these three equations, we obtain

(6.28) J (1/P + di/D + do/D) = Ci - Co�������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������

�������������������������������������������������������������������������������������

���������������������������������������������������

��������������������������������������������������������������������

���������������������������������������������������������������������

Unstirred Layer

Unstirred Layer

CiCmi

Co

Cmo

od id

Membrane

Page 58: Membrane Biophysics

Page 6.7

Figure 6.4. Permeation and diffusion through unstirred layers

Therefore, the effect of unstirred layers is to decrease the permeability so the apparentpermeability coefficient (Papp) is smaller than P:

(6.29) 1/Papp = 1/P + di/D + do/D

References

A. Finkelstein (1976) Water and nonelectrolyte permeability of lipid bilayer membranes, J.Gen. Physiol. 68, 127-135.

A. Finkelstein and A. Cass (1968) Permeability and electrical properties of thin lipidmembranes, J. Gen. Physiol.52, 145s-172s.

A.R. Koch (1970) Transport equations and criteria for active transport, Am. Zool. 10, 331-346.

S. G. Schultz (1980) Basic Principles of Membrane Transport, Cambridge UniversityPress, Cambridge.

A. Walter and J. Gutknecht (1984) Monocarboxylic acid permeation through lipid bilayermembranes, J. Membrane Biol. 77, 255-264.

Page 59: Membrane Biophysics

Fundamental Principlesof Membrane Biophysics

CHAPTER 7: PERMEABILITY AND CONDUCTANCE OF ELECTROLYTES

David Njus

Department of Biological SciencesWayne State University

© D. Njus, 2000

Page 60: Membrane Biophysics

Page 7.1

CHAPTER 7: PERMEABILITY AND CONDUCTANCE OF ELECTROLYTES

Section 7.1. Permeation of ElectrolytesThe permeation of electrolytes may be analyzed using the same approach as is used

for nonelectrolytes (Section 6.3). The flux J is assumed to be proportional to thethermodynamic driving force, the derivative of the electrochemical potential:

C Nf (7.1) J = - RT dC zFC d

Nf dx RT dx +[= -ψ]( )[ µ + RT ln C + zFψo ]( ) d

dx

As before, we must integrate this from one side of the membrane to the other. Inthis case, however, we have two parameters, C and ψ, that will vary as we cross themembrane. We may use the steady-state assumption to define one of these parameters interms of the other, but we will need another equation to define both. We will return to thisproblem later.

First, note that

=][ (zF /RT) ψC e dx

d (7.2) [ ]dC zFC dψ+

dx RT dx(zF /RT) ψ

e

A comparison of equations 7.1 and 7.2 reveals that exp (zFψ/RT) may be used as anintegrating factor so that

(7.3) J exp (zFψ/RT) = - (RT/Nf) d/dx[C exp (zFψ/RT)]

If we make the steady-state assumption (J is constant across the membrane), then we maypartially integrate equation 7.3:

zFψ/RT zFψmi/RT zFψmo/RT

(7.4) J ∫0

d e dx = - (RT/Nf) [Cmi e - Cmo e ]

Cmi and Cmo are the concentrations in the membrane at d and 0 respectively (Figure 7.1).They are related to the concentrations Ci and Co in the bulk aqueous phases by themembrane:water partition coefficient (Kp) and by the surface potential:

(7.5) Cmi = Ci Kp exp[zF(ψi- ψmi)/RT]

(7.6) Cmo = Co Kp exp[zF(ψo- ψmo)/RT]

Therefore,zF /RTi

C eψ

dx = - RTKNf

p izF /RToψ

- C eo(7.7) J ∫0

d zF /RTψe ( )

If we define the electrical potential as zero on the outside (ψo = 0), then

Page 61: Membrane Biophysics

Page 7.2

[ ]( )RTKNfd

poC - C ei

zF /RTψm(7.8) J =∫

0d

d

exp (zF /RT) dxψ

where ψm is the membrane potential. We may note that RTKp/Nfd is the permeabilitycoefficient P (equations 6.19 and 6.20). Therefore,

(7.9) J = PQ[Co - Ci exp (zFψm/RT)]

where Q is defined as

(7.10) Q = d/∫0

d exp (zFψ/RT) dx

������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������

MembraneOutside Inside

���������������������������������������������������

�������������������������������������������������

0 d

��������������������������������������������������

���������������������������������������������������

ψmo

ψmi

C i

ψo

CmoC o

Cmi

ψ i

Figure 7.1. Concentration profile for steady-state electrolyte flow

The problem now is to evaluate Q. To integrate exp (zFψ/RT), we need to defineψ(x). We will use the constant field approximation first proposed by Goldman. A secondpossibility is the assumption of electrical neutrality in the membrane, an approach firstexplored by Planck. Physically, the two approaches are similar. A constant field withinthe membrane implies that there must be electrical neutrality. In terms of formalism,however, the Goldman approach is simpler. The equations obtained by assuming aconstant field (or a linear gradient in electrical potential) are simpler than those obtained bysumming anion and cation concentrations within the membrane and setting the total chargeequal to zero.

The constant field assumption states that the electric field (E = -dψ/dx) is constantthroughout the membrane. Therefore,

(7.11) ψ(x) = - Ex + ψ(0)

If we assume that surface potentials are negligible, then ψ(0) = 0 andψm = ψ(d) = - Ed. Equation 7.10 may then be integrated to give

Page 62: Membrane Biophysics

Page 7.3

exp (zF /RT) - 1ψm

(7.12) Q = zF /RTψ

m

Introducing equation 7.12 into equation 7.9 yields a final expression for the flow J:

exp (zF /RT) - 1ψm((7.13) J = P

zF /RTψm ) ( oC - C ei zF /RTψm )

Equations 7.9 and 7.13 provide us with alternative expressions for determining theflow of electrolytes across biological membranes with the latter equation including theassumption of a constant field. These expressions allow us to calculate the flow ofelectrolyte driven across the membrane by a membrane potential, by a concentrationgradient, or by a combination of these two forces. In Figure 7.2, the membrane potentialand the concentration gradient are used as the axes of a two-dimensional graph. Equations7.9 and 7.13, therefore, allow us to calculate the flow at any point on this graph. Let usexamine some special cases. First, note that both equations reduce to the Nernst equationif the flow J is zero:

(7.14) Ci/Co = exp (-zFψm/RT)

Second, if ψm is zero, Q = 1 and equation 7.9 reduces to Fick's Law:

(7.15) J = P (Co - Ci)

Finally, if there is no concentration gradient (Ci/Co = 1), equation 7.13 reduces to:

(7.16) J = - PCzFψm/RT

Since I = -JFz, equation 7.16 is equivalent to Ohm's Law (I = g ψm) where the conductanceg is PCz2F2/RT.

TABLE I. Permeability Coefficients of Ions

H2O 2 x 10-5 - 2 x 10-2 cm/sec Na+ 8 x 10-9 cm/sec (squid axon) K+ 6 x 10-7 cm/sec (squid axon) H+ 10-5 cm/sec (chromaffin vesicle) Na+ and K+ 10-10 - 10-11 cm/sec (lipid bilayer) Cl- 10-10 cm/sec (phospholipid vesicle)

Page 63: Membrane Biophysics

Page 7.4

z ψ

0

00 1

ln C /Ci oC /Ci o

Ohm's Law

Fick's Law

Nernst Equation

Nernst Equation

Fick's Law

Ohm's Law z ψ

Figure 7.2. Regions of validity of the permeation and conductance equations

The above analysis of electrolyte permeation was developed using theelectrochemical potential. We may also analyze the permeation of electrolytes using thetransition state approach. Let us imagine that the energy of the electrolyte is Eo on theoutside of the membrane, Ei on the inside, and Em in the membrane (Figure 7.3). Theunidirectional flux in the inward direction will be

(7.17) roi = Co exp [-(Em- Eo)/RT] x constant

The unidirectional flux in the outward direction will be

(7.18) rio = Ci exp [-(Em- Ei)/RT] x constant

The net flow J will be the difference between these two fluxes:

(7.19) J = roi - rio = {Co exp [-(Em- Eo)/RT] - Ci exp [-(Em- Ei)/RT]} x constant

Page 64: Membrane Biophysics

Page 7.5

��������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������

Membrane InsideOutsideEnergy

������������������������������������������������������������������������������������������

��������������������������������������������������������������������������������������������������������������������������������E i

oE

mE

Figure 7.3. Energy barrier for permeation of electrolytes

Following our usual convention, we will define the electrical potential as zero on theoutside. Then Eo will simply be the standard free energy of the electrolyte in water (Eo =µw°) and Ei will differ from this by the membrane potential (Ei = µw° + zFψm). Uponintroducing these values for the energies, equation 7.19 reduces to

(7.20) J = exp [-(Em- µw°)/RT] [Co - Ci exp (zFψm/RT)] x constant

If we let PQ = exp [-(Em- µw°)/RT] x constant, then equation 7.20 is the same as the fluxequation derived using the electrochemical potential approach (equation 7.9).

Section 7.2. The Born Charging EquationSeveral factors determine the energy Em of an ion in a membrane. These include 1)

hydrophobic interactions, 2) electrostatic potentials (both surface and dipole), 3) shortrange forces (steric effects), and 4) the Born charging energy. Hydrophobic interactionsand short range forces apply to non-electrolytes as well. Surface and dipole potentialeffects were considered earlier. In this section, we will consider the effect of the Borncharging energy.

The Born charging energy is the energy required to assemble a given amount ofcharge on a particle of a given size. Because this energy is lower in a medium with a highdielectric constant, the Born charging energy is much smaller for an ion in water than foran ion in a hydrocarbon medium. This means that an ion requires much more energy toenter a hydrocarbon phase than to enter an aqueous phase. The Born charging energy,therefore, accounts for the insolubility of ions in hydrocarbon phases and for theimpermeability of biological membranes to ions. Because the Born charging energy isgreater for a localized charge than for an equivalent delocalized charge, ions withdelocalized charge will permeate through biological membranes more easily.

Page 65: Membrane Biophysics

Page 7.6

dqx a

Figure 7.4. Charging a conducting sphere

Imagine that an ion is a conducting sphere of radius a (Figure 7.4). If q is thecharge placed on the sphere and ε is the dielectric constant of the medium, the Borncharging energy is

(7.21) W = q2/8πεεoa

This equation can be derived as follows. Outside of a conducting sphere, the electric fieldcreated by the sphere is the same as the electric field created by a point charge (of equalcharge) located at the center of the sphere. Therefore, the force between a sphere of chargeq' and a charge dq' is given by Coulomb's Law:

(7.22) F = q'dq'/4πεεox2

The work required to move the charge dq onto the sphere from an infinite distance away is

(7.23) dW = - ∫∞

a F dx = - ∫

a (q' dq'/4πεεox2) dx

= q' dq'/4πεεoa

The work required to place the entire charge q on the sphere is then

(7.24) W = ∫ δW = ∫0

q q' dq'/4πεεoa = q2/8πεεoa

The change in the charging energy upon moving the ion from water into ahydrocarbon phase is

(7.25) W = (q2/8πεoa) (1/εhc - 1/εw)

The membrane is not an infinite hydrocarbon phase, but is a thin layer of hydrocarbon withwater on both sides. Therefore, the change in charging energy upon moving the ion fromwater into a membrane is somewhat smaller than the change shown in equation 7.25. Thework done in moving a charge q a distance x into a membrane of thickness d has beenapproximated by Flewelling and Hubbell (1986):

Page 66: Membrane Biophysics

Page 7.7

( )( ) ]( )( )[ x

d1 - - 1.2(7.26) W =2 q

8aπεo

2

ε hc

1 a 2x

a d

Section 7.3. The Goldman-Hodgkin-Katz EquationIf ions are in equilibrium across a membrane, then the membrane potential will be

given by the Nernst equation. This is rarely the case, however. Generally, ions are inconstant flux (transport and permeation) and the capacitative charge is determined by thesteady-state distribution of ions. The membrane potential can nevertheless be determinedfrom this steady-state distribution using the Goldman-Hodgkin-Katz equation. At steady-state, the net charge flux will be zero.

(7.27) 0 = Σcations zjFJj + Σanions zjFJj

The fluxes Jj are defined by equation 7.13. If we impose the simplifying assumption thatall of the ions are univalent, then

(7.28) 0 = ΣcationsFPj{(Fψm/RT)/[exp(Fψm/RT)-1]}{Coj - Cijexp(Fψm/RT)}- ΣanionsFPj{(-Fψm/RT)/[exp(-Fψm/RT)-1]}{Coj - Cijexp(-Fψm/RT)}

Dividing by Fψm/RT and rearranging the exponentials in the anion term yields

(7.29) 0 = Σcations FPj{1/[exp (Fψm/RT) - 1] }{ Coj - Cij exp (Fψm/RT) }- Σanions FPj{ (-1/[1 - exp (Fψm/RT)] }{ Coj exp (Fψm/RT) - Cij }

Upon dividing by F{1/[exp (Fψm/RT) - 1] }, we obtain

(7.30) 0 = Σcations Pj{Coj - Cij exp (Fψm/RT)}- Σanions Pj{Coj exp (Fψm/RT) - Cij}

Then solving for the exponential term gives

(7.31) exp (Fψ /RT) =Σ cations anionsj ojP C + Σ j ijP C

Σ cations anionsj ijP C + Σ j ojP Cm

Therefore, the membrane potential is defined by the ion concentrations and permeabilitiesas follows:

(7.32) ψ = (RT/F) lnm

Σ cations anionsj ojP C + Σ j ijP C

Σ cations anionsj ijP C + Σ j ojP C

This is the Goldman-Hodgkin-Katz equation.

Page 67: Membrane Biophysics

Page 7.8

References

R.F. Flewelling and W.L. Hubbell (1986) Hydrophobic ion interactions within membranes,Biophys. J. 49, 531-540.

R.F. Flewelling and W.L. Hubbell (1986) The membrane dipole potential in a totalmembrane potential model, Biophys. J. 49, 541-552.

A.L. Hodgkin and A.F. Huxley (1952) A quantitative description of membrane current andits application to conduction and excitation in nerve, J. Physiol. 117, 500-544.

A.R. Koch (1970) Transport equations and criteria for active transport, Am. Zoologist 10,331-346.

R.I. Macey (1978) Mathematical models of membrane transport processes, in MembranePhysiology (T.E. Andreoli, J.F. Hoffman, and D.D. Fanestil, eds.), Plenum, NewYork, pp. 125-146.

A. Parsegian (1969) Energy of an ion crossing a low dielectric membrane: Solutions tofour relevant electrostatic problems, Nature 221, 844-846.

Page 68: Membrane Biophysics

Fundamental Principlesof Membrane Biophysics

CHAPTER 8: CHANNELS AND EXCITABLE MEMBRANES

David Njus

Department of Biological SciencesWayne State University

© D. Njus, 2000

Page 69: Membrane Biophysics

Page 8.1

CHAPTER 8: CHANNELS AND EXCITABLE MEMBRANES

Section 8.1. Channel-forming AntibioticsIon channels are needed to conduct ions across biological membranes because ions,

particularly cations, do not permeate readily across the lipid bilayer. The structuralsimplicity required to create an ion-conducting channel across a biological membrane isexemplified by channel-forming antibiotics, such as gramicidin and amphotericin B. Theseantibiotics form ungated channels, so they dissipate the ion gradients needed for propermembrane function and cause cells to spend energy in futile ion pumping. Since theseungated channels are lethal, regulation of channel opening or “gating” is obviously animportant feature of natural ion channels.

Gramicidin is a pentadecapeptide consisting of alternating L and D amino acids:

HCO-L-Val-Gly-L-Ala-D-Leu-L-Ala-D-Val-L-Val-D-Val-L-Trp-D-Leu-L-Trp-D-Leu-L-Trp-D-Leu-L-Trp-NHCH2CH2OH

The conductance through gramicidin channels varies as the square of the gramicidinconcentration indicating that the compound functions as a dimer. Indeed, two peptides canbe linked at the formyl groups on their amino terminal ends, and the coupled structure willfunction as a channel. The channel formed by gramicidin is not very selective and has aunitary (single channel) conductance of 5 pS. It is thought that gramicidin forms a helixwith the hydrophobic side chains on the outside and the carbonyl oxygens oriented to theinside. This forms a channel 2 Å in diameter.

Amphotericin B forms a larger channel and increases the permeability of lipidbilayers to water and small electrolytes as well as ions. The conductance depends on the4th-12th power of the concentration suggesting that a number of molecules are required toform the channel. Because amphotericin B is an elongated molecule with a hydrophobicside and a hydrophilic side, it is thought to line the sides of the channel like the staves on abarrel. Amphotericin B requires a sterol for activity, and thus makes channels inmembranes that contain cholesterol.

Section 8.2. Voltage-Gated ChannelsIn recent years, molecular biological techniques have yielded a wealth of

information about the membrane-spanning proteins that form ion channels. Ageneralization that may be emerging is that channels with similar gating mechanisms havesimilar structures. The voltage-gated channels, in particular, have common structuralfeatures despite the fact that they have different ion selectivities and conductances.Voltage-gated channels comprise the S4 superfamily, so named because the proteinsfunction as tetramers having either four subunits or four homologous segments. The K+

channel, for example, is a tetramer with six membrane-spanning regions in each subunit.The Na+ channel is a single large peptide (~260 kDa), but that peptide has fourhomologous segments with 6 membrane-spanning regions in each. Looking down at themembrane, the four segments are arranged at the corners of a square with the ion channelitself passing down through the center between them.

As mentioned, the core of the Na+ channel is formed by a single large α subunit. Inthe eel electroplax, that is the only subunit. The sodium channel from mammalian skeletal

Page 70: Membrane Biophysics

Page 8.2

muscle also contains a β1 subunit (38 kDa), while the sodium channel in mammalian braincontains both a β1 (36 kDa) and a β2 subunit (33 kDa) along with the α subunit.

Voltage-gated potassium channels include the delayed rectifier (DR) channel,which functions in actions potentials in excitable membranes, and the CaK channel, whichis activated by Ca2+ as well as by depolarization. Both are blocked by barium. The CaKchannel has a very high unitary conductance and is specifically blocked by charybdotoxin.

Calcium channels have been classified into a variety of types based on functionalcharacteristics and pharmacology. The well characterized L-type channel is a high-threshold, slow inactivating channel, which is blocked by dihydropyridines. Low-threshold, fast inactivating Ca2+ channels are classed as T-type. Two other high-thresholdchannels (N and P) are distinguished by their sensitivity to peptide toxins. N channels areblocked by ω-conotoxin GVIA while P channels are blocked by ω-agatoxin IVA.Structurally, these channels are thought to be similar. The L-type Ca2+ channel fromskeletal muscle has five subunits: α1 (170 kDa), α2δ (175 kDa), β (52 kDa) and γ (32kDa). The α1 subunit contains binding sites for Ca2+ channel antagonists and is thought tobe the subunit forming the functional Ca2+ channel. The α2δ piece consists of a large α2

subunit linked to the δ subunit by disulfide bonds.

Section 8.3. Ligand-Gated ChannelsExtracellularly activated ligand-gated ion channels seem to fall into three major

groups based on the number of subunits. The nicotinicoid group, represented by thenicotinic acetylcholine receptor, has five homologous subunits arranged in a pentagonalstructure around a central channel. This group includes the cation-conducting nicotinic andserotonin (5HT) receptors and the anion-conducting GABAA, GABAC and glycinereceptors. The best characterized member of the nicotinicoid receptor group is thenicotinic acetylcholine receptor. Structurally, the acetylcholine receptor consists of fivehomologous subunits: 2α, 1β, 1γ and 1 δ. Each subunit has at least four transmembranesegments. The acetylcholine binding sites are located on the α subunits.

The second group of extracellularly activated ligand-gated ion channels, theglutamate-activated cation channels, has four homologous subunits. This group includesthe AMPA, Kainate and NMDA receptors named for ligands (agonists) that specificallyactivate each type. The third group of ligand-gated ion channels, the ATP-gated channels,have three homologous subunits and include the ATP2x and ATP2z receptors.

As ion channels, the ligand-gated channels seem to exhibit less specificity than thevoltage-gated channels. The cation channels do not discriminate between Na+ and K+, sothey drive the membrane potential toward zero (midway between the Na+ and K+

equilibrium potentials). Because this depolarizes the membrane, these receptors are oftencalled excitatory. The anion channels drive the membrane potential toward the Cl-

equilibrium potential (negative inside). Thus, they tend to restore the resting membranepotential and are often called inhibitory.

Section 8.4 Ryanodine and Inositol Tris Phosphate ReceptorsThe ryanodine and inositol trisphosphate receptors are related proteins that form

Ca2+ channels in intracellular membranes. The ryanodine receptor is a very large protein(565 kDa) and is responsible for releasing Ca2+ from the sarcoplasmic reticulum (SR) inmuscle. Most of this protein (the amino-terminal 80%) is cytoplasmic and constitutes a

Page 71: Membrane Biophysics

Page 8.3

"foot" structure. The remaining 20% on the carboxyl end includes 4 to 10 transmembranesegments and presumably creates the ion channel structure. Ryanodine, a plant product,opens this channel. In vivo, however, the channel is gated by cytoplasmic Ca2+. Thus,when L-channels in the T-tubule membranes open and allow Ca2+ to enter the muscle cell,the ryanodine receptor channel in the sarcoplasmic reticulum membrane responds byreleasing more Ca2+ from the SR. Thus, the ryanodine receptor mediates Ca2+-inducedCa2+ release.

A related protein, the inositol-1,4,5-trisphosphate receptor releases Ca2+ from theendoplasmic reticulum in response to the intracellular messenger inositol-1,4,5-trisphosphate (IP3). It has a molecular mass of 260 kDa and, like the ryanodine receptor,consists of a large amino-terminal foot and a smaller carboxyl portion containing 8-10transmembrane segments. Both proteins appear to associate into homotetramers.

TABLE 8.1. Characteristics of Channels

Type Gating Unitary BlockersConductance

Na+ Depolarization 10 pS TetrodotoxinK+ (DR) Depolarization 55 pS Ba2+

K+ (Ca2+) Depolarization/Ca2+ 240 pS Charybdotoxin/Ba2+

Ca2+ (L) High-threshold 9 pS (Ca2+) Dihydropyridines(slow inact.) depolarization 25 pS (Ba2+)

Ca2+ (T) Low-threshold 8 pS (Ba2+)(fast inact.) depolarization 8 pS (Ca2+)

Ca2+ (N) High-threshold 12 pS (Ba2+) ω-conotoxin GVIADepolarization

Ca2+ (P) Depolarization ω-agatoxin IVANicotinic Acetylcholine 90 pS Bungarotoxin

ReceptorRyanodine Ca2+

ReceptorInsP3 Receptor Inositol trisphosphate

Section 8.5. Channel ConductanceAs described in Section 7.1, the flow of an electrolyte across a membrane

(expressed as a current I = -JFz) depends on the membrane potential and the concentrationgradient as described by equation 8.1:

exp (zF /RT) - 1ψm

((8.1) I = -FzPzF /RTψm ) ( oC - C ei

zF /RTψm )

The current carried by ion channels is commonly expressed using a variation on Ohm'sLaw:

(8.2) Ii = gi (ψm - ψi)

Page 72: Membrane Biophysics

Page 8.4

Ii is the current carried by ion i, gi is the conductance of the membrane to that ion, ψm is themembrane potential and ψi is the equilibrium potential for that ion as defined by the Nernstequation (equation 6.3). Adjusting the membrane potential by subtracting ψi insures thatthere will be no current when the membrane potential equals the ion’s equilibriumpotential. Equation 8.2 predicts that a plot of current (Ii) against voltage (ψm), also knownas an IV curve, will be linear. The slope is the conductance gi and the X-intercept is theequilibrium potential ψi.

If a channel is perfectly selective for a particular ion, that channel will exhibit an IVcurve as defined by equation 8.2. Often, however, a channel is not absolutely selective andwill pass different kinds of ions with different conductances. In this case, the currentthrough the channel must be summed over the different ions:

(8.3) I = ∑ [gi (ψm - ψi)] = ψm ∑ gi - ∑ (gi ψ i) = ∑ gi {ψm - ∑ (gi ψ i) / ∑ gi }

For a non-selective channel, Ohm's Law still holds, but the conductance is the sum of theconductances for all of the ions, and the X-intercept is a weighted average of theequilibrium potentials. The X-intercept is called the reversal potential because the currentthrough the channel reverses direction as the membrane potential crosses this value. If thechannel is highly selective for a particular ion, the reversal potential will be close to theequilibrium potential for that ion. Thus, the reversal potential is an indicator of a channel'sion selectivity. Note that the reversal potential of a channel depends on the channel'sselectivity and also on the equilibrium potentials (concentration gradients) of the ions itconducts.

The functioning of individual channels has been illuminated in recent years by thedevelopment of the patch clamp technique. This technique, which permits the observationof currents through single channels, has revealed the opening and closing times fordifferent channel types. It has also enabled measurement of unitary conductances. Theunitary conductance is important because it reflects the restriction imposed by the channel'sselectivity filter. For example, the sodium channel, which has a relatively low unitaryconductance, probably has a relatively long narrow tunnel. It is estimated that this is a pore3 Å x 5 Å in cross section and 10-12 Å in length. The high conductance K channel, bycontrast, probably has a relatively short narrow tunnel. In both cases, it is imagined thatthe channel has large vestibules on one or both sides of the selectivity filter. This permitsfree (and rapid) diffusion through most of the membrane and limits conductance only to theextent necessary to maintain selectivity.

Section 8.6. Channel SelectivityAn important property of ion channels is that they exhibit selectivity for particular

ions over other closely related ions. Eisenman examined the five monovalent metal cations(Cs+, Rb+, K+, Na+, Li+) and noted that, although there are 120 possible selectivitysequences, only 11 sequences are observed. He rationalized this by arranging the elevensequences in order from low field strength to high field strength:

Page 73: Membrane Biophysics

Page 8.5

TABLE 8.2. The Eisenman Selectivity Series for Monovalent Cations

I Cs > Rb > K > Na > Li Low field strengthII Rb > Cs > K > Na > LiIII Rb > K > Cs > Na > LiIV K > Rb > Cs > Na > LiV K > Rb > Na > Cs > LiVI K > Na > Rb > Cs > LiVII Na > K > Rb > Cs > LiVIII Na > K > Rb > Li > CsIX Na > K > Li > Rb > CsX Na > Li > K > Rb > CsXI Li > Na > K > Rb > Cs High field strength

To cross through a channel, an ion must shed its water of hydration and pass a selectivitysite in the channel. This implies that the ion must interact more favorably with theselectivity site than with its water of hydration. At low field strength, the selectivity of thechannel is dominated by the dehydration energy of the ion. Ions that are easily dehydratedpass through the channel more readily than do ions that interact strongly with water. Atlow field strength, therefore, larger ions are favored over smaller ions and the order is inthe direction of decreasing atomic weight. At high field strength, the selectivity of thechannel is dominated by the attraction of the ion for the selectivity site. Those ions thatbind well will be favored over those that bind poorly. At high field strength, therefore,

-∆G

1/rCs+ Rb+ Na+K+ Li+

Figure 8.1. Competition between hydration and selectivity-site bindingin ion channel selectivity.

Hydration energyHigh-field-strength site

Low-field-strength site

Intermediate-field-strength site

Page 74: Membrane Biophysics

Page 8.6

smaller ions are favored over larger ions and the order is in the direction of increasingatomic weight. The sodium channel has selectivity characteristics typical of a high fieldstrength. This high field strength is attributed to one or more carboxylic acid groups thatmay line the tunnel and interact with the passing cations.

TABLE 8.3. Properties of Ions

Ion Atomic Ionic Crystal ∆HhydrationWeight Radius (Å) (kcal/mole)

Li+ 6.94 0.68 121Na+ 23.00 0.98 95K+ 39.10 1.33 76Rb+ 85.47 1.48 69Cs+ 132.91 1.67 62

TABLE 8.4. Selectivity of Channels

Type SelectivityACh Receptor NH4

+ > Cs+ > Rb+ > Na+

Na+ Na+,Li+ > K+ > Rb+ > Cs+

K+ (DR) Tl+ > K+ > Rb+ > NH4+ > Na+ > Li+ >> Cs+

K+ (Ca2+) Tl+ > K+ > Rb+ > NH4+ > Na+, Li+, Cs+

Ca2+ (L) Ca2+ > Sr2+ > Ba2+ > Li+ > Na+ > K+ > Cs+ > Mg2+

Ca2+ (T) Ca2+ ≈ Ba2+

Section 8.7. ExcitabilityAs defined by the Goldman-Hodgkin-Katz equation (Equation 7.32), the membrane

potential is determined by the relative permeability of the membrane to different ions.Because animal cell membranes are relatively more permeable to K+ than to Na+, theresting membrane potential is normally close to the K+ equilibrium potential and isnegative (inside relative to outside). When a nerve or muscle fiber is stimulated andconducts and action potential, the membrane depolarizes (the membrane potential becomesless negative and then positive) and then repolarizes to the resting membrane potential.This happens because Na+ channels open first and then K+ channels open. When Na+

channels open, the membrane becomes more permeable to Na+ than to K+ and themembrane potential shifts toward the Na+ equilibrium potential (or more accurately, thereversal potential of the Na+ channel). When the K+ channels open, the membrane againbecomes more permeable to K+ than to Na+ and the membrane potential shifts back towardthe reversal potential of the K+ channel.

Na+ and K+ channels open and close in the course of an action potential becausetheir opening (gating) depends on the membrane potential. Of course, the membranepotential in turn depends on the opening of the channels making the action potential anautocatalytic event. To analyze this, recognize that the currents across the membrane allcontribute to a change in the capacitative charge:

(8.4) dqc/dt = - ( IK + INa + Il )

Page 75: Membrane Biophysics

Page 8.7

Equation 8.4 includes currents of K+ and Na+ as well as a small residual current (mostly Cl-), which is commonly known as the “leakage current.” Equation 8.4 may be expandedknowing that the membrane potential is related to the capacitative charge by thecapacitance (Equation 4.11), and the currents are given by Ohm’s Law (Equation 8.2):

(8.5) C (dψm/dt) = - gK(ψm-ψK) – gNa(ψm-ψNa) – gl (ψm-ψl)

Equation 8.5 makes it clear that the change in the membrane potential ψm depends on theconductances of the sodium and potassium channels. The sodium and potassium channelsin turn are gated by voltage, so their conductances change depending on the membranepotential. To separate the interdependence of channel conductance and membranepotential, Hodgkin and Huxley used the voltage-clamp technique, in which the membranepotential is fixed or clamped so that Na+ and K+ currents can be recorded at that constantmembrane potential. They found that Na+ and K+ channels remain closed at the restingmembrane potential but their probability of opening increases when the membranepotential is raised. When the membrane is depolarized, Na+ channels open quickly andthen inactivate or close. K+ channels open more slowly and remain open. This means thatdepolarizing a membrane will cause Na+ channels to open first. The inward Na+ currentwill drive the membrane potential in the positive direction toward the Na+ equilibriumpotential. This accelerates the opening of Na+ channels ensuring a strong depolarization.After a brief time, however, the Na+ channels inactivate and the Na+ current stops.Concurrently, K+ channels open, and the outward K+ current drives the membrane potentialback down toward the K+ equilibrium potential. This brings the membrane potential backto the resting value and also causes the K+ channels to close terminating the actionpotential.

From data gathered in their voltage clamp experiments, Hodgkin and Huxley foundempirical relations that express the changes in conductances as functions of the membranepotential. These functions are quite complicated and for our purposes need only berepresented as f1(gK, ψm) and f2(gNa, ψm).

(8.6) dgK/dt = f1(gK, ψm)

(8.7) dgNa/dt = f2(gNa, ψm)

Knowing how the membrane potential depends on conductances (equation 8.5) andhow the conductances depend on membrane potential (equations 8.6 and 8.7), it is possibleto reconstruct the action potential. A membrane action potential is an action potential inwhich the membrane potential changes along the whole length of the nerve fiber at thesame time. Thus, the entire nerve fiber depolarizes and repolarizes simultaneously. Amembrane action potential can be simulated by numerically integrating equations 8.5, 8.6and 8.7. Assume that the membrane potential begins at a value somewhat above theresting potential. Given this membrane potential, calculate the change in gNa, gK and ψm

expected to occur in a brief interval of time (say 10 µsec). After changing the values ofgNa, gK and ψm accordingly, calculate the changes expected in the next brief time intervalby repeating the process. If the initial membrane potential was set to a value above

Page 76: Membrane Biophysics

Page 8.8

threshold, then the membrane potential will rise up in an action potential and then return torest. If the initial membrane potential was set to a value below threshold, then themembrane potential will simply drift back down to its resting value.

The membrane action potential is easy to calculate, but normally an action potentialdoes not occur along an entire nerve fiber simultaneously. Rather, it travels from one endto the other. To analyze this propagating action potential, recognize that the actionpotential travels because charge diffuses down the nerve fiber, depolarizing the membraneahead of the action potential, and causing the action potential to move forward along thefiber. The current along the length of the nerve fiber is given by Ohm’s Law:

(8.8) Ilong = - (1/r ) (dψm/dx)

The longitudinal current is proportional to the change in potential (dψm/dx) andinversely related to the resistance of the nerve fiber r.

The current across the membrane now must equal the change in the current alongthe membrane.

(8.9) Imem = (-1/πD) dIlong/dx

D is the diameter of the nerve fiber. Therefore,

(8.10) Imem = (1/πDr)(d2ψm/dx2) = (1/πDr) (d2ψm/dt2) (dt/dx)2

Recognize that dx/dt is the conduction velocity of the action potential (θ). Then,

(8.11) Imem = (1/ πDrθ2) (d2ψm/dt2) = C (dψm/dt) + gK (ψm - ψK) + gNa (ψm - ψNa +gl(ψm - ψl)

Equation 8.11 may be numerically integrated as for the membrane action potential if thecorrect value for the conduction velocity θ is chosen.

References

T. Begenisich (1987) Molecular properties of ion permeation through sodium channels,Ann. Rev. Biophys. Biophys. Chem. 16, 247-263.

W.A. Catterall (1986) Molecular properties of voltage-sensitive sodium channels, Ann.Rev. Biochem. 55, 953-985.

W.A. Catterall (1988) Structure and function of voltage-sensitive channels, Science 242,50-61.

G. Eisenman and J.A. Dani (1987) An introduction to molecular architecture andpermeability of ion channels, Ann. Rev. Biophys. Biophys. Chem. 16, 205-226.

A.L. Hodgkin and A.F. Huxley (1952) A quantitative description of membrane current andits application to conduction and excitation in nerve, J. Physiol. 117, 500-544.

A.F. Huxley (1959) Ion movements during nerve activity, Ann. N.Y. Acad. Sci. 81, 221-246.

Page 77: Membrane Biophysics

Page 8.9

H.A. Lester (1992) The permeation pathway of neurotransmitter-gated ion channels, Ann.Rev. Biophys. Biomol. Struct. 21, 267-292.

P.S. McPherson and K.P. Campbell (1993) The ryanodine receptor/Ca2+ release channel, J.Biol. Chem. 268, 13765-13768.

R.J. Miller (1992) Voltage-sensitive Ca2+ channels, J. Biol. Chem. 267, 1403-1406.M. Mishina, T. Kurosaki, T. Tobimatsu, Y. Morimoto, M. Noda, T. Yamamoto, M. Terao,

J. Lindstrom, T. Takahashi, M. Kuno and S. Numa (1984) Expression of functionalacetylcholine receptor from cloned cDNA, Nature 307, 604-608.

M. Noda, T. Ikeda, T. Kayano, H. Suzuki, H. Takeshima, M. Kurasaki, H. Takahashi, andS. Numa (1986) Existence of distinct sodium channel messenger RNAs in rat brain,Nature 320, 188-192.

M. Noda, S. Shimizu, T. Tanabe, T. Takai, T. Kayano, T. Ikeda, H. Takahashi, H.Nakayama, Y. Kanaoka, N. Minamino, K. Kangawa, H. Matsuo, M.A. Raftery, T.Hirose, S. Inayama, H. Hayashid, T. Miyata and S. Numa (1984) Primary structureof Electrophorus electricus sodium channel deduced from cDNA sequence, Nature312, 121-127.

M. Noda, H. Takahashi, T. Tanabe, M. Toyosato, S. Kikyotani, Y. Furutani, T. Hirose, H.Takashima, S. Inayama, T. Miyata and S. Numa, (1983) Structural homology ofTorpedo californica acetylcholine receptor subunits, Nature 302, 528-532.

B.M. Olivera, G. Miljanich, J. Ramachandran, and M.E. Adams (1994) Calcium channeldiversity and neurotransmitter release: The ω-conotoxins and ω-agatoxins, Ann.Rev. Biochem. 63, 823-867.

R.W. Tsien, P. Hess, E.W. McCleskey and R.L. Rosenberg (1987) Calcium channels:Mechanisms of selectivity, permeation, and block, Ann. Rev. Biophys. Biophys.Chem. 16, 265-290.

G. Yellen (1987) Permeation in potassium channels: Implications for channel structure,Ann. Rev. Biophys. Biophys. Chem. 16, 227-246.

C.D. Ferris, and S.H. Snyder (1992) Inositol 1,4,5-trisphosphate activated calciumchannels, Ann. Rev. Physiol. 54,

Page 78: Membrane Biophysics

Fundamental Principlesof Membrane Biophysics

CHAPTER 9: ACTIVE TRANSPORT

David Njus

Department of Biological SciencesWayne State University

© D. Njus, 2000

Page 79: Membrane Biophysics

Page 9.1

CHAPTER 9: ACTIVE TRANSPORT

Active transport is usually defined as transport of molecules or ions from a regionof lower to a region of higher concentration (i.e., transport against a concentrationgradient). Here, we will define it in a stricter sense as transport which requires chemical orphotochemical energy. By this definition, only a small number of enzymes are capable ofcatalyzing active transport. With few exceptions, these enzymes transport inorganiccations: Na+, K+, Ca2+ and H+. Active transporters may be divided into four generalclasses: 1) ion-translocating ATPases, 2) H+-transporting electron transfer chains, 3)group-translocating enzymes, and 4) photochemically-driven transporters such asbacteriorhodopsin. It should be recognized that these enzymes do not consume energy;rather they transduce chemical energy into electrochemical potential gradients.

Section 9.1. Ion-translocating ATPasesIon-translocating ATPases derive energy from the hydrolysis of ATP and use this

energy to move ions across the membrane against concentration gradients. The ATPasesfall into two broad classes: the P (or E1E2) type and the F/V/A type. The P-type includethe widely studied Na+/K+ ATPase found in the plasma membrane of animal cells, the Ca2+

ATPases found in sarcoplasmic and endoplasmic reticulum (SERCA) and in the plasmamembrane (PMCA), the H+/K+ ATPase which is involved in acid extrusion, and an H+

ATPase which is found in the plasma membranes of plant and fungal cells. The P-typeATPases are all inhibited by vanadate, a characteristic which seems to be shared byATPases that are phosphorylated in the course of their catalytic cycle. The phosphategroup hydrolyzed from ATP is transiently attached to an aspartyl residue on the enzyme.The P-type ATPases characteristically have a single, large polypeptide chain, whichmediates both ion transport and ATP hydrolysis. In general, the catalytic segment appearsto lie near the middle of this chain with 4 membrane-spanning domains on the amino endand four to six membrane-spanning domains on the carboxyl end. Thus, the aminoterminus and the catalytic site both lie on the cytoplasmic side of the cell membrane.

The P-type ATPases hydrolyze cytosolic ATP and transport the substrate out of thecytosol (out of the cell or into the sarcoplasmic or endoplasmic reticulum). The Na+/K+

and H+/K+ ATPases are exceptions in that they mediate an exchange of ionic substrates.Both enzymes bring K+ into the cytosol as the other ion (Na+ or H+) is transported out.These two enzymes are also unusual in having a second (β) subunit. They are thusheterodimers, and the β subunit is thought to function in K+ transport.

In the catalytic cycle of the Na+/K+ ATPase, 3 Na+ ions bind from the cytosolic sideof the enzyme. ATP hydrolysis phosphorylates the enzyme and initiates a conformationalchange exposing the ion binding sites to the external side of the membrane. The Na+ ionsdissociate and are replaced by 2 K+ ions. Then, the enzyme dephosphorylates and returnsto the original conformation releasing the K+ ions to the inside. It is important that ATPhydrolysis not occur until 3 Na+ ions are bound and that dephosphorylation not occurunless 2 K+ ions are bound. Otherwise, ATP hydrolysis and ion transport would not beproperly coupled, and ATP would be spent inefficiently on suboptimal transport cycles.

The F/V/A-type ATPases include the F (F1FO) ATPases found in mitochondria,chloroplasts and bacterial cells, the vacuolar-type ATPases found in Golgi-derivedorganelles including lysosomes and secretory vesicles in animal cells and vacuoles andtonoplasts in plant and fungal cells, and the A-type ATPases found in archaebacteria.

Page 80: Membrane Biophysics

Page 9.2

These ATPases transport H+ by a rotary mechanism and have a two-part structure includinga membrane stalk and a large head. They differ from the plasma membrane ATPases inbeing insensitive to vanadate and in lacking a stable phosphorylated intermediate. Thesubunit composition of the F, V and A ATPases is also much more complex than that ofthe P-type ATPases. They are characteristically sensitive to N,N'-dicyclohexylcarbodiimide (DCCD), which reacts with carboxyl groups and typicallyinhibits proton-translocating enzymes.

The F or F1/FO class of H+-ATPases are found in mitochondria, chloroplasts andbacterial cells and are, of course, responsible for ATP synthesis rather than ATPhydrolysis. Nevertheless, they should be considered here because they are functionallysimilar to the V and A type ATPases, and they can catalyze active transport of H+ when thefree energy of the reaction favors that direction. These enzymes have a complex subunitcomposition including a hydrophilic F1 component which carries the catalytic site and ahydrophobic FO component which is thought to be the proton channel. The F1 componentincludes five kinds of subunits (α, β, γ, δ, ε) in a stoichiometry of 3:3:1:1:1. Thenucleotide binding sites lie between the α and β subunits. At any given time, these sitesare in the tight, loose or open state depending on the position of the γ subunit. The FOcomponent includes three kinds of subunits (a, b, c) in a stoichiometry of 1:2:~12. DCCDinhibits the F1FO ATPases by reacting with a glutamate residue in the c subunit. The csubunits are relatively small (2 transmembrane segments) and the cluster of c subunitstogether with the γ subunit form an axle around which the α3β3 head rotates. In the ATPhydrolysis mode, the F1FO ATPase is thought to function as follows: ATP hydrolysisdrives the rotation of the α3β3 head group relative to the γ subunit and the c cluster attachedat the base. This rotation causes the three nucleotide binding sites to step alternatelythrough the loose, tight and open conformations. In the process, ATP is bound, hydrolyzedand the products released. Concomitantly, 3 H+ are pumped through the FO component foreach ATP hydrolyzed.

The V or vacuolar ATPases are similar in structure to the F ATPases, although theirnormal function is H+ transport driven by ATP hydrolysis rather than ATP synthesis drivenby H+ movement. The V ATPases differ from the F ATPases in that they are inhibited bythe antibiotic bafilomycin A1 and by N-ethylmaleimide.

ATPases couple a chemical reaction to vectorial movement of an ion. For this towork, three criteria must be met. 1) The energetics must be such that the chemical reactionreleases more free energy than is required for ion translocation. 2) The chemical reactionand ion translocation must be obligatorily coupled processes. One cannot be allowed tooccur in the absence of the other. 3) The enzyme must change its affinity for the ionsduring the translocation cycle.

First, let us consider the energetics. For ion transport, the change in theelectrochemical potential of the transported ion is:

(9.1) ∆µj = µj,final - µj,initial

= µj° + RT ln Cj,final + zjFψfinal - µj° - RT ln Cj,initial - zjFψinitial

= RT ln (Cj,final/Cj,initial) + zjF(ψfinal- ψinitial)

Page 81: Membrane Biophysics

Page 9.3

The total free energy change required for transport is the sum of the electrochemicalpotential differences (∆µj) for each of the transported ions multiplied by the stoichiometrynj for that ion:

(9.2) ∆Gions = Σj nj ∆µj

If we define nj as positive for ions transported from outside to inside and negative for ionstransported from inside to outside, then

(9.3) ∆µj = RT ln (Cj,in/Cj,out) + zjF(ψin- ψout)

This convention simplifies the bookkeeping of directionality and allows us to use themembrane potential (ψin- ψout) without worrying about the sign. Now, for the chemicalreaction, the free energy change is

(9.4) ∆GATP = ∆G°ATP + RT ln ([ADP][Pi]/[ATP])

where ∆G°ATP = -7.3 kcal/mol. Of course, ion transport will proceed as long as

(9.5) ∆GATP + ∆Gions ≤ 0

Equation 9.5 emphasizes that the ATPases catalyze coupled reactions. It would not do foreither ATP hydrolysis or ion movement to occur independently. If ATP hydrolysisoccurred independently of ion movement, then the free energy released by ATP hydrolysiswould be wasted. If ion movement were allowed to proceed in the absence of ATPhydrolysis, then the ions would move in the wrong direction: from higher concentration tolower concentration.

To catalyze ion movement against a concentration gradient, the enzyme must takethe ion from a solution in which the ion is present at a lower concentration and release theion into a solution in which the ion is present at a higher concentration. Thus, in additionto moving the ion, the enzyme must change its affinity for the ion in order to functionefficiently. The ATPase must have a high affinity for the ion on the low-concentration sideand a low affinity for the ion on the high-concentration side. Part of the energy released byATP hydrolysis, therefore, must go into changing the enzyme's affinity for the ion.

Section 9.2. Electron-transfer chainsIn mitochondria, chloroplasts and bacterial cells, substrate oxidation is used to

create an H+ gradient. This was first suggested by Peter Mitchell in 1961 as the celebrated"Chemiosmotic Hypothesis." We will discuss some of the details of this energy couplinglater. Here, let us just review the energetics of redox-driven H+-translocation.

In electron transfer reactions, an electron is passed from an electron donor to anelectron acceptor.

Donorred + Acceptorox → Donorox + Acceptorred

The free energy change associated with this reaction may be expressed in the usual way:

Page 82: Membrane Biophysics

Page 9.4

(9.6) ∆Gredox = ∆G° + RT ln ( ) Rather than tabulate ∆G° values for all combinations of electron carriers, however, electronaffinities are catalogued by considering each half- reaction separately:

Donorred → Donorox + n e-

Acceptorox + n e- → Acceptorred

The reduction potential E of the donor is defined as

(9.7) E = E° + (RT/nF) ln {[Donor]ox/[Donor]red}

where n is the number of electrons transferred in the reaction and E° is the midpoint orstandard reduction potential for the electron donor. A similar equation defines thereduction potential of the electron acceptor. The reduction potentials are related to the freeenergy change as

(9.8) ∆Gredox = nF Edonor - nF Eacceptor

= nF (E°donor -E°acceptor) + RT ln ( )Since E is measured in volts, ∆G will be in units of joules/mole.

The best characterized redox-driven proton-pump is cytochrome oxidase, theenzyme catalyzing the transfer of electrons from cytochrome c to oxygen. Mitchelloriginally suggested that cytochrome oxidase accepted electrons from cytochrome cexternally and that it carried the electrons across the membrane to O2. Since the reductionof O2 to H2O takes up two protons, Mitchell hypothesized that this enzyme contributed toproton translocation simply by removing H+ from the mitochondrial matrix. Since E° forcytochrome c is +0.24 volts and E° for O2/H2O is +0.816 volts, this electron transfer stepreleases a lot of energy and one might expect it to be captured to a greater extent. In 1978,Krab and Wikstrom showed that this is the case; cytochrome oxidase not only takes upprotons because of the reduction of oxygen but it also physically transports protons acrossthe membrane.

Cytochrome oxidase contains two hemes (a and a3) and two copper atoms. One Cuand heme a are on the cytoplasmic side of the inner mitochondrial membrane and acceptelectrons from cytochrome c. The other Cu and heme a3 are on the matrix side of themembrane and react with O2. Cytochrome oxidase from bovine heart consists of eightsubunits. Three (I, II and III) are coded by the mitochondrial DNA and the rest are codedby nuclear DNA.

[Donor]ox[Acceptor]red [Donor]red [Acceptor ]ox

[Donor]ox[Acceptor]red [Donor]red [Acceptor ]ox

Page 83: Membrane Biophysics

Page 9.5

Section 9.4. BacteriorhodopsinBacteriorhodopsin is a light-driven proton pump. It is found in Halobacterium

halobium, a bacterium found growing in hot salt springs. In the presence of oxygen,Halobacterium uses respiration as a source of metabolic energy. Under anaerobicconditions, however, Halobacterium uses light energy to produce ATP directly.Halobacterium is not photosynthetic since it does not split H2O (produce O2) or exhibitlight-driven electron flow. Instead, it produces a "purple membrane" which is essentiallypure bacteriorhodopsin: 25% lipid and 75% bacteriorhodopsin. Bacteriorhodopsin pumpsH+ out of the cell in the presence of light. This sets up an H+ gradient, just as respirationwould, and the proton gradient can be used to make ATP in the usual chemiosmotic way.

Bacteriorhodopsin is a particularly well characterized ion pump, because it can beobtained in large quantity and essentially pure. Moreover, it is a rather simple proteinhaving a molecular weight of 26,000. The amino acid sequence has been determined(Ovchinnikov et al., 1979) and its three-dimensional structure elucidated (Henderson andUnwin, 1975). Bacteriorhodopsin consists of seven membrane-spanning domains. Acofactor, retinal, is responsible for light absorption. Retinal is covalently bound to the εamino of a lysine residue via a Schiff's base linkage.

The energy of a quantum of light depends on the wavelength (λ):

(9.10) Elight = hν = hc/λ

where h is Planck's constant and c is the speed of light. For an einstein (mole) of quanta,the energy is

(9.11) ∆Glight = hcN/λ

The value of hcN is 28,592 nm.kcal/einstein.

Section 9.5. Group TranslocationGroup translocation is a transport mechanism in which the substrate is metabolized

as it is transported. The energy released by that metabolism is used to drive transport. Forexample, glucose is brought into some bacterial cells as glucose 6-phosphate.Phosphoenolpyruvate serves as the source of both the energy and the phosphate. There arethree proteins involved: Enzyme I, Enzyme II, and HPr. Enzyme I is a soluble protein thatcatalyzes the phosphorylation of HPr using phosphoenolpyruvate. P-HPr then donates thephosphate to glucose bound to enzyme II. Enzyme II is specific for the sugar. This system,studied by Saul Roseman, is found in E. coli, Salmonella and Staphylococcus.

References

J.P. Abrahams, A.G.W. Leslie, R. Lutter, and J.E. Walker (1994) Structure at 2.8Aresolution of F1- ATPase from bovine heart mitochondria, Nature 370, 621-628.

R. Addison, (1986) Primary structure of the Neurospora plasma membrane H+-ATPasededuced from the gene sequence, J. Biol. Chem. 261, 14896-14901.

Page 84: Membrane Biophysics

Page 9.6

K.B. Axelsen and M.G. Palmgren (1998) Evolution of substrate specificities in the P-typeATPase superfamily, J. Mol. Evol. 46, 84-101.

P.D. Boyer, (1997) The ATP Synthase - a splendid molecular machine, Ann. Rev. Biochem.66, 717-

T. Elston, H. Wang and G. Oster, G. (1998) Energy transduction in ATP synthase, Nature391, 510-513.

S.P.A. Fodor, J.B. Ames, R. Gebhard, E.M.M. van den Berg, W. Stoeckenius, J.Lugtenburg and R.A. Mathies (1988) Chromophore structure in bacteriorhodopsin'sN intermediate: Implications for the proton-pumping mechanism, Biochemistry 27,7097-7101.

M. Forgac (1999) Structure and properties of the vacuolar (H+)-ATPases, J. Biol. Chem.274, 12951-12954.

K.M. Hager, S.M. Mandala, J.W. Davenport, D.W. Speicher, E.J. Benz, Jr. and C.W.Slayman (1986) Amino acid sequence of the plasma membrane ATPase ofNeurospora crassa: Deduction from genomic and cDNA sequences, Proc. Natl.Acad. Sci. USA 83, 7693-7697.

P. Henderson and P.N.T. Unwin (1975) Three-dimensional model for purple membraneobtained by electron microscopy, Nature 257, 28-32.

E. Hilario and J.P. Gogarten (1998) The prokaryote-to-eukaryote transition reflected in theevolution of the V/F/A-ATPase catalytic and proteolipid suunits, J. Mol. Evol. 46,703-715.

P.L. Jorgensen, J.M. Nielsen, J.H. Rasmussen, and P.A. Pedersen (1998) Structure-function relationships of E1-E2 transitions and cation binding in Na, K-pumpprotein, Biochim, Biophys. Acta 1365, 65-70.

P.M. Kane (1999) Vacuolar ATPases: structure, function, assembly and biosynthesis, J.Bioenerg. Biomembr. 31, 1-83.

K. Kinosita, Jr., R. Yasuda, H. Noji, S. Ishiwata and M. Yoshida (1998) F1-ATPase: arotary motor made of a single molecule, Cell 93, 21-24.

K. Krab and M. Wikström (1987) Principles of coupling between electron transfer andproton translocation with special reference to proton-translocation mechanisms incytochrome oxidase, Biochim. Biophys. Acta 895, 25-39.

S. Lutsenko and J.H. Kaplan (1995) Organization of P-type ATPases: Significance ofstructural diversity, Biochemistry 34, 15607-15613.

D.H. MacLennan, C.J. Brandl, B. Korczak and N.M. Green (1985) Amino-acid sequenceof a Ca2++Mg2+-dependent ATPase from rabbit muscle sarcoplasmic reticulum,deduced from its complementary DNA sequence, Nature 316, 696-700.

D.H. MacLennan, W.J. Rice and N.M. Green (1997) The mechanism of Ca2+ transport bysarco(endo)plasmic reticulum Ca2+-ATPases, J. Biol. Chem. 272, 28815-28818.

M. Mohraz, M.V. Simpson, and P.R. Smith (1987) The three-dimensional structure of theNa, K-ATPase from electron microscopy, J. Cell Biol. 105, 1-8.

J.V. Moller, B. Juul and M. LeMaire (1996) Structural organization, ion transport andenergy transduction of P-type ATPases, Biochim. Biophys. Acta 1286, 1-51.

V. Muller, C. Ruppert and T. Lemker (1999) Structure and function of the A1A0-ATPasesfrom methanogenic archaea, J. Bioenerg. 31, 15-27.

N. Nelson and W.R. Harvey (1999) Vacuolar and plasma membrane proton-adenosinetriphosphatases, Physiol. Rev. 79, 361-385.

Page 85: Membrane Biophysics

Page 9.7

D. Njus, P.M. Kelley, and G.J. Harnadek (1986) Bioenergetics of secretory vesicles,Biochim. Biophys. Acta 853, 237-265.

H. Noji, R. Yasuda, M. Yoshida and K. Kinosita, Jr. (1997) Direct observation of therotation of F1-ATPase, Nature 386, 299-302.

Y.A. Ovchinnikov, N.G. Abdulaev, M.Y. Feigina, A.V. Kiselev, and N.A. Lobanov (1979)The structural basis of the functioning of bacteriorhodopsin: An overview, FEBSLett. 100, 219-224.

M.G. Palmgren and K.B. Axelsen (1998) Evolution of P-type ATPases, Biochim. Biophys.Acta 1365, 37-45.

G. Rudnick (1986) ATP-driven H+ pumping into intracellular organelles, Ann. Rev.Physiol. 48, 403-413.

Y. Sambongi, Y. Iko, M. Tanabe, H. Omote, A. Iwamota-Kihara, I. Ueda, T. Yanagida, Y.Wada and M. Futai, M. (1999) Mechanical rotation of the c subunit oligomer inATP synthase (F0F1): direct observation, Science 286, 1722-1724.

B. Schulenberg, R. Aggeler, J. Murray and R.A. Capaldi (1999) The γε-c subunit interfacein the ATP synthase of Escherichia coli: cross linking of the ε subunits to the csubunit ring does not impair enzyme function, that of γ to c subunits leads touncoupling, J. Biol. Chem. 274, 34233-24237.

R. Serrano, M.C. Kielland-Brandt and G.R. Fink (1986) Yeast plasma membrane ATPaseis essential for growth and has homology with (Na++K+), K+- and Ca2+-ATPases,Nature 319, 689-693.

G.E. Shull and J.B. Lingrell (1986) Molecular cloning of the rat stomach (H++K+)-ATPase,J. Biol. Chem. 261, 16788-16791.

G.E. Shull, L.K. Lane and J.B. Lingrel (1986) Amino-acid sequence of the beta-subunit ofthe (Na++K+)ATPase deduced from a cDNA, Nature 321, 429-431.

G.E. Shull, A. Schwartz and J.B. Lingrel (1985) Amino-acid sequence of the catalyticsubunit of the (Na++K+)ATPase deduced from a complementary DNA, Nature 316,691-695.

D. Stock, A.G.W. Leslie, and J.E. Walker (1999) Molecular architecture of the rotarymotor in ATP synthase, Science 286, 1700-1705.

C. Toyoshima, M. Nakasako, H. Nomura and N. Ogawa (2000) Crystal structure of thecalcium pump of sarcoplasmic reticulum at 2.6 Å resolution, Nature 405, 647-655.

J. Weber and A.E. Senior (1997) Catalytic mechanism of F1-ATPase, Biochim. Biophys.Acta 1319, 19-58.

Page 86: Membrane Biophysics

Fundamental Principlesof Membrane Biophysics

CHAPTER 10: FACILITATED DIFFUSION

David Njus

Department of Biological SciencesWayne State University

© D. Njus, 2000

Page 87: Membrane Biophysics

Page 10.1

CHAPTER 10: FACILITATED DIFFUSION

As we have discussed, some molecules and ions cross biological membranes bysimple diffusion through the hydrophobic milieu and some pass through specific channels.The passage of other molecules and ions across biological membranes is facilitated bymeans of transporters. Except in those few cases in which the transporters draw uponchemical or electromagnetic energy to transport ions against concentration gradients, thisfacilitated transport is known as facilitated diffusion.

Section 10.1. IonophoresOf the agents which mediate facilitated diffusion, the simplest are antibiotics

known as ionophores, hydrophobic compounds which can complex an ion and carry itacross a lipid bilayer. Ionophores are traditionally classified in two categories: neutralionophores such as valinomycin and carboxylic ionophores such as nigericin. To these twocategories, we should add a third: those agents that carry hydrogen ions across biologicalmembranes. H+ carriers are commonly called uncouplers of oxidative phosphorylation (orsimply uncouplers) because of their effect on mitochondrial respiration. Because theireffect is more precisely to transport protons across membranes, we will refer to them hereas protonophores.

The protonophores are weak acids capable of permeating the lipid bilayer in their

2,4-Dinitrophenol(DNP)

Carbonylcyanide p-trifluoromethoxyphenylhydrazone (FCCP)

Carbonylcyanide m-chlorophenylhydrazone (CCCP)

5-Chloro-3-tert-butyl-2'-chloro-4'-nitrosalicylanilide(S-13)

NO2

OH

OCF3NHC N

CNO2N

N C

NHC N

CN

N C

ClC NHO

ClCCH3

NO2

OH

Cl

H3C

CH3

Figure 10.1. Structures of some protonophores

anionic form. Of course, they can also pass across the membrane in their uncharged(protonated) form, so they carry H+ across the membrane and bring H+ concentrations on

Page 88: Membrane Biophysics

Page 10.2

the two sides of the membrane to electrochemical equilibrium. In terms of structure, theprotonophores are generally aromatic compounds that are both hydrophobic and capable ofdistributing the negative charge over a number of atoms in the molecule. Structures ofsome protonophores are shown in Figure 10.1.

Neutral ionophores generally have a cyclic structure so they are effectivelyhydrophobic doughnuts capable of complexing a cation in the hole. Valinomycin, forexample, is a ring composed of three units each of D-hydroxyisovalerate, L-valine, L-lactate and D-valine linked by alternating peptide and ester bonds (Figure 10.2).Valinomycin complexes K+, Rb+ or Cs+, but Na+ and Li+ are too small to be effectivelybound. The selectivity for K+ is 10,000 times greater than the selectivity for Na+.Valinomycin will cross the membrane either with or without a bound ion so it carriescharge across the membrane. Ion transport mediated by valinomycin, therefore, dependsupon the membrane potential. Moreover, valinomycin will create a membrane potential bytransporting capacitative charge.

Dicyclohexyl-18-crown-6Enniatin B

Valinomycin

N

OO

O

O

O NO

O

O

N

O

O O

O

O

OO

OO

O

ONN

O O

OO

N

O

O

O

O

N

O

O

OO

O NO

N

Figure 10.2. Some neutral ionophores

Unlike the neutral ionophores, the carboxylic ionophores have a linear structurewith a carboxyl group on one end and one or two hydroxyls on the other (Figure 10.3).They cyclize by head-to-tail hydrogen bonding and will cross the membrane with thecarboxyl group either protonated or complexed to an ion. Nigericin, for example, willcross the membrane carrying either H+ or K+. It functions, therefore, as a K+/H+ exchanger.Because nigericin does not carry a net charge across the membrane, transport is notaffected by the membrane potential nor does it contribute to the creation of a membranepotential.

Page 89: Membrane Biophysics

Page 10.3

Section 10.2. TransportersIntegral membrane proteins which catalyze facilitated diffusion differ from

ionophores in some important ways. First, the proteins span the membrane creating apathway for facilitated diffusion. They do not diffuse across the membrane as doionophores. Although transporters are sometimes called carriers, the latter term suggests aprotein that binds to a substrate and moves with it through a medium, not a protein throughwhich the substrate moves. Therefore, the term carrier may be applied to ionophores butshould not be used for transporter proteins.

A second important characteristic of transporters is that they are inserted into themembrane with a fixed orientation. Therefore, whereas the binding process for ionophoresis the same on both sides of the membrane, transporters may be asymmetric. In particular,the binding constants observed on the two sides of the membrane may be quite different.In fact, for transporters to achieve maximum efficiency, the transporter should have abinding constant higher than the expected substrate concentration on one side of themembrane and lower than the expected concentration on the other side. This will allow thetransporter to bind the substrate on one side and release it on the other.

Finally, transporters customarily facilitate coupled transport of two or moresubstrates. That is, like nigericin, transporters facilitate either exchange diffusion of two ormore substrates or codiffusion of two or more substrates. For example, the sodium/hexosetransporter, commonly found in the plasma membrane of animal cells, mediatescodiffusion of Na+ and a hexose molecule (glucose) into the cell (Figure 10.4). As we shallsee, this coupled transport is perhaps the most important mechanism for transportingsubstances across biological membranes.

Page 90: Membrane Biophysics

Page 10.4

������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������

H

Na+

MembraneOutside Inside

Figure 10.4. Codiffusion mediated by the sodium/hexose transporter

Section 10.3. Equilibria of Facilitated DiffusionThe ionophores and transporters mediating facilitated diffusion will reach

equilibrium when the free energy change for the processes they mediate is zero. Fortransport processes

(10.1) ∆G = Σi ni ∆µi = Σ ni (µiin - µiout)

= Σi ni [RT ln (Cin/Cout) + ziF ψm]

ni is the stoichiometry for transport of substrate i and is positive if i is transported inwardand negative if i is transported outward.

For protonophores, which carry only H+,

(10.2) ∆G = RT ln ([H+]in/[H+]out) + F ψm

We obtain the equilibrium reached by protonophores by setting ∆G = 0. Not surprisingly,this yields the Nernst equation:

(10.3) ψm = (RT/F) ln ([H+]out/[H+]in)

For nigericin, which exchanges K+ and H+,

(10.4) ∆G = RT ln ([H+]in/[H+]out) + Fψm - RT ln ([K+]in/[K+]out) - Fψm

By setting ∆G = 0, we obtain

Page 91: Membrane Biophysics

Page 10.5

(10.5) [H+]in/[H+]out = [K+]in/[K+]out

Nigericin will reach equilibrium when the [H+] and [K+] gradients are proportional.For the sodium/hexose transporter,

(10.6) ∆G = RT ln ([H]in/[H]out) + RT ln ([Na+]in/[Na+]out) + F ψm

Upon setting ∆G = 0, we obtain the equilibrium condition:

(10.7) [H]in/[H]out = [Na+]out/[Na+]in exp (-F ψm /RT)

Section 10.4. Kinetics of Facilitated DiffusionThe kinetics of facilitated diffusion may be analyzed by considering the simple

system illustrated in Figure 10.5. Let us imagine that this facilitated diffusion is mediatedby an ionophore so that we may make the following assumptions:

1. The rate constants for transmembrane movement of the liganded ionophore (k1)are the same in both directions.

2. The rate constants for transmembrane movement of the empty ionophore (k2) arethe same in both directions.

3. The binding constants (K) on both sides of the membrane are the same.Finally, we will make the important assumption of rapid equilibrium. That is, we

will assume that the rate limiting steps are the ones involving transmembrane movementand that the binding processes on the two surfaces of the membrane occur relativelyquickly. This allows us to assume that the binding reactions are always at equilibrium.Consequently,

(10.8) K = To Co/TCo = Ti Ci/TCi

The rate at which the total internal transporter concentration changes is

(10.9) d(TCi+Ti)/dt = k1TCo + k2To - k1TCi - k2Ti

If we assume that the transporter distribution reaches a steady-state, then d(TCi+Ti)/dt = 0,so

(10.10) 0 = TCo (k1 + k2K/Co) - TCi (k1 + k2K/Ci)

Multiplying both sides of equation 10.10 by KCi + 2CiCo + KCo gives

(10.11) 0 = TCo[(k1Co + k2K)(Ci+K) + (K+Co)(k1Ci + k2KCi/Co)]

- TCi[(k1Ci + k2K)(Co+K) + (K+Ci)(k1Co + k2KCo/Ci)]

If we let T be the total amount of transporter in the membrane, then

(10.12) T = TCo + To + TCi + Ti = TCo (1 + K/Co) + TCi (1 + K/Ci)

Page 92: Membrane Biophysics

Page 10.6

Multiplying both sides of equation 10.12 by k2K(Co - Ci) gives

(10.13) k2KT(Co-Ci) = TCo(k2K-k2KCi/Co)(K+Co)

- TCi(k2K-k2KCo/Ci)(Ci+K)

Upon adding equations 10.11 and 10.13, we obtain

(10.14) k2KT(Co-Ci) = (TCo-TCi)[(k1Co + k2K)(Ci+K)+(K+Co)(k1Ci+k2K)]

Note that the flow of substrate J = k1 (TCo-TCi) so

(10.15) J =

This expression gives us the steady-state flux in terms of three parameters: k1T, K, and theratio k1/k2. If we restrict our consideration to the initial velocity of transport (Ci = 0), thenequation 10.15 reduces to

(10.16) J = (k1k2CoT)/(k1Co + 2k2K + k2Co)

Equation 10.16 may be rewritten as

(10.17) 1/J = (k1+k2)/k1k2T + 2K/k1CoT

If we define

(10.18) Jmax = k1k2T/(k1 + k2)

(10.19) Km = 2Kk2/(k1 + k2)

then

(10.20) 1/J = 1/Jmax + Km/JmaxCo

Initial velocity of transport, therefore, can be described by only two parameters, Jmax andKm. As in enzyme kinetics, Jmax is the velocity at infinite substrate concentration; Km is thesubstrate concentration giving one-half the maximum velocity. Note that the Km measuredfor initial velocity of transport is equal to K, the binding constant for the substrate, only ifthe rate constants (k1 and k2) are about equal.

k1k2KT(Co-Ci)

(k1Co + k2K)(Ci+K) + (k1Ci + k2K)(Co+K)

Page 93: Membrane Biophysics

Page 10.7

�����������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������

Co

To

TC o

Ti

TC i

Ci

Membrane InsideOutside

Figure 10.5. A simple model of facilitated diffusion

Section 10.5. Facilitated Diffusion Mediated by IonophoresMcLaughlin and his colleagues have analyzed the kinetics of H+ transport mediated

by the ionophores FCCP and CCCP (Benz and McLaughlin, 1983; Kasianowicz et al.,1984). In the absence of a membrane potential, the model illustrated in Figure 10.5 appliesat least in the short term. Values for the rate constants k1 and k2 and the binding constant(expressed as pK) are given below:

FCCP CCCP k1 10,000 sec-1 12,000 sec-1

k2 700 sec-1 175 sec-1

pK 6.0 5.9

First, note that protonation reactions occur on a time scale of 1011 sec-1 so thetransmembrane rates (k1 and k2) are clearly limiting. Second, note that the rate constantsfor FCCP are larger than those for CCCP consistent with the more potent protonophoricactivity of FCCP. The simple model shown in Figure 10.5 applies to protonophores withtwo restrictions. First, it applies only in the absence of a membrane potential since amembrane potential will change the rate constants for the movement of the anionic species.In fact, because the membrane potential will facilitate the movement of the anionic speciesin one direction and slow its movement in the other, k2 will not be the same in bothdirections. This introduces an additional parameter. Second, the model applies only forshort time periods. Because the ionophore slowly partitions between the aqueous andmembrane phases, the membrane-bound concentration may not remain constant overlonger time periods.

ReferencesR. Benz and S. McLaughlin (1983) The molecular mechanism of action of the proton

ionophore FCCP (carbonylcyanide p-trifluoromethoxy phenylhydrazone), Biophys.J. 41, 381-398.

Page 94: Membrane Biophysics

Page 10.8

J. Kasianowicz, R. Benz and S. McLaughlin (1984) The kinetic mechanism by whichCCCP (carbonyl cyanide m-chlorophenylhydrazone) transports protons acrossmembranes, J. Membrane Biol. 82, 179-190.

R.I. Macey (1978) Mathematical models of membrane transport processes, in MembranePhysiology (T.E. Andreoli, J.F. Hoffman, and D.D. Fanestil, eds.), Plenum, NewYork, pp. 125-146.

S.G.A. McLaughlin and J.P. Dilger (1980) Transport of protons across membranes byweak acids, Physiol. Rev. 60, 825-863.

B.C. Pressman (1973) Properties of ionophores with broad range cation selectivity, Fed.Proc. 32, 1698-1703.

B.C. Pressman (1976) Biological applications of ionophores, Ann. Rev. Biochem. 45, 501-530.

B.C. Pressman and M. Fahim (1982) Pharmacology and toxicology of the monovalentcarboxylic ionophores, Ann. Rev. Pharmacol. Toxicol. 22, 465-490.

Page 95: Membrane Biophysics

Fundamental Principlesof Membrane Biophysics

CHAPTER 11: COUPLED TRANSPORT

David Njus

Department of Biological SciencesWayne State University

© D. Njus, 2000

Page 96: Membrane Biophysics

Page 11.1

CHAPTER 11: COUPLED TRANSPORT

In 1961, Peter Mitchell (1961) proposed that the H+ concentration gradient drivesthe synthesis of ATP. In the same year, Crane (1965) showed that the sodiumconcentration gradient drives sugar and amino acid transport across epithelial cell plasmamembranes. In the years since, it has become evident that ion concentration gradients arean important energy store in all cells. Ion pumps are therefore crucial transducers thattransform chemical-bond energy into electrochemical potential energy. Coupled transport,in turn, uses the electrochemical potential of one substrate to drive the transport of asecond.

Section 11.1. The Master Pump ConceptIt is obviously possible to imagine all sorts of combinations of coupled transport.

Fortunately, this picture may be greatly simplified by applying the concept of a masterpump. According to this concept, each cellular membrane possesses one master pumpwhich uses chemical energy to transport one or two inorganic ions. This pump createstransmembrane gradients in the electrochemical potential of the transported ions across themembrane in which it is located. Other ions and molecules are transported across thatmembrane by coupling their movement to the movement of the transported ion(s).

It is evident that a given pump can store electrochemical potential energy onlyacross the membrane in which it is located. Thus, the master pump must be thepredominant active transport enzyme in the membrane. For example, the Na+/K+ ATPase,which is located in the plasma membrane of animal cells, can store energy only across theplasmalemma. Accordingly, transport of other ions and molecules across animal cellplasma membranes is likely to be coupled to Na+ or K+. Similarly, transport acrossbacterial cell, chloroplast thylakoid and mitochondrial inner membranes is likely to becoupled to the H+ gradients generated by the redox chains in those membranes. In Golgi-derived organelles in both plant and animal cells, the master pump is the V-type H+-translocating ATPase, while, in plant and fungal cell membranes, the master pump appearsto be the P-type H+-translocating ATPase.

A master pump must have three attributes: 1) high capacity, 2) high efficiency, and3) low dissipation. Low dissipation is probably the reason that pumps almost exclusivelytransport the relatively impermeant inorganic cations. The exception is H+ which, becauseof its small size, is relatively permeant. The saving feature of H+ is that the rate ofdissipation (leakage current) is proportional to the concentration and H+ is present atexceedingly low concentrations. Thus, because of low concentration, H+ gradients exhibita low dissipation.

High capacity is the requirement that the ion gradient involve concentrations thatare relatively large compared to the concentrations of the compounds that are to betransported. Thus, if the master pump is to drive uptake of glucose and other metabolitespresent at millimolar concentrations, the ion gradient established by the pump should be anorder of magnitude or two greater than this so the ion gradient is not greatly perturbed bythe secondary transport systems. For example, the Na+ concentration outside animal cellsis usually on the order of 100 mM while that inside is on the order of 10 mM.Consequently, the Na+ gradient has a high capacity and is suitable for the transport ofmillimolar concentrations of metabolites. By contrast, the Ca2+ concentration is on theorder of 5 mM outside animal cells and 10-7 M inside. While the concentration gradient is

Page 97: Membrane Biophysics

Page 11.2

large, the capacity of the system is low. Ca2+ concentration gradients would be greatlyperturbed by the transport of millimolar concentrations of metabolites. The H+ ion isanomalous because it is normally present intracellularly at very low concentrations (10-7

M). Although this is necessary to reduce dissipation of the gradient, it might be expectedto adversely affect capacity. H+, however, is unusual in that its concentration is bufferedand the capacity of the system is determined not by the absolute H+ concentration but bythe buffering capacity. Because the cytosol is well buffered, the capacity of the H+ gradientis in the millimolar range eventhough the H+ concentration is micromolar or less.

The concept of a master pump is significant because it suggests a likely mechanismfor all transport systems in a given membrane once the master pump has been identified.The master pump concept raises some interesting questions. First, why is transportcoupled to a single master pump? A number of possible reasons can be imagined.Coupled transport may be more efficient than independent pumps. Ion gradients generallystore smaller packets of energy than ATP. By choosing the proper coupling stoichiometry,the energy required for transport can be matched more precisely, thereby using energy moreefficiently. Coupled transporters may be inherently more efficient than pumps. Anindication that this may be so is provided by bacteria. Under anaerobic conditions, somecells produce ATPases for transporting amino acids. However, under aerobic conditions,the cells dispense with the amino acid pumps and transport amino acids by couplingtransport to the H+ gradient. Coupling transport to a single master pump may also serve acontrol function. This is particularly evident in the case of sequential coupled transport(tertiary and quaternary transport). For example, the mitochondrion does not need toimport citrate as a substrate for the TCA cycle if no phosphate is available forphosphorylation. Because citrate transport is coupled to the phosphate gradient, citratetransport automatically stops when phosphate is depleted.

There are two reasons why a transport system might not be coupled to the masterpump. First, if the transport system has a high capacity itself, it may adversely affect theion gradients established by the master pump. Second, if the transported substrate serves aregulatory function, then it may be desirable to control its concentration separately.Exceptions to the master pump concept may exist for both of these reasons. The K+/H+

ATPase is a possible example of an exception for the reason of high capacity. The ATPasemust transport large quantities of H+ into the lumen of the gut and kidney. If H+ movementwere coupled to the Na+/K+ ATPase, it might radically affect the magnitude of the Na+

gradient. The Ca2+ ATPase may be an exception for the reason of independent regulation.The Ca2+ concentration serves a regulatory function in animal cells, so its concentrationshould not be dependent on the size of the Na+ gradient.

Ion pumps may catalyze a net transport of charge. Because this increases chargeseparation across the membrane, it is termed electrogenic. Coupled transport may alsoproduce a net transfer of charge (non-neutral exchanges). These transport systems arenormally driven by the membrane potential, however. Therefore, these transporters tend todissipate rather than create a membrane potential. For that reason, they should be termedelectromotive or electrically dissipative rather than electrogenic.

Section 11.2. Animal Cell Plasma MembraneAnimal cells, which find themselves in a medium that is high in sodium, can use

the sodium gradient for transport. Since no other membrane can rely on a high sodiumcapacity, it is not surprising that the Na+/K+ ATPase is limited to the plasma membrane of

Page 98: Membrane Biophysics

Page 11.3

animal cells. The Na+/K+ ATPase transports both Na+ and K+ and, in principle, the K+

gradient could also be used for transport. In this regard, it is significant that the membranepotential in animal cells is usually negative inside. This electrical potential adds to thechemical potential gradient for Na+ but nearly cancels the potential gradient for K+. Thusthe Na+/K+ ATPase puts the chemical energy into the Na+ gradient rather than the K+

gradient. Since K+ ions can contribute little energy for transport, it is not surprising that K+

is usually not involved in coupled transport across the plasma membrane. For example, inthe squid axon, the membrane potential is -90 mV, the K+ equilibrium potential is -102mV, and the Na+ equilibrium potential is +50 mV. A K+ ion can contribute only 12 mV ofenergy to transport whereas a Na+ ion can contribute +140 mV. This illustrates how theenergy stored by the Na+/K+ ATPase is put mainly into the Na+ ions and not into K+.

ATP

ADP + P

3 Na

2K+i

+ Na+

Glucose

Figure 11.1. Coupled transport across the animal cell plasma membrane.

In animal cell plasma membranes, a glucose transporter uses the Na+ gradient tobring glucose into the cell. It does this by mediating the cotransport of 1 glucose moleculeand 1 Na+ ion. The energetics of this coupled transport may be analyzed by applying thesame logic used earlier. The total free energy change is the sum of the electrochemicalpotential changes of all transported species (equation 10.1):

(11.1) ∆G = Σi ni ∆µi = Σ ni (µiin - µiout)

= Σi ni [RT ln (Cin/Cout) + ziF ψm]

Both glucose (G) and Na+ are transported from outside to inside, so they may both beassigned a stoichiometry of +1. Then, at equilibrium,

(11.2) ∆G = RT ln ([G]in/[G]out) + RT ln ([Na+]in/[Na+]out) + F ψm = 0

Consequently, the glucose concentration gradient will depend on the sodium gradient andon the membrane potential as

(11.3) [G]in/[G]out = {[Na+]out/[Na+]in } exp (- F ψm/RT)

Page 99: Membrane Biophysics

Page 11.4

Section 11.3. Inner Mitochondrial MembraneThe electron transfer chain in the mitochondrial inner membrane uses energy

derived from substrate oxidation to pump H+ out across the membrane. This gradient, ofcourse, is used to drive the synthesis of ATP. It is also used to transport many metabolitesacross the inner mitochondrial membrane. ATP and ADP cross the inner mitochondrialmembrane via a transporter catalyzing ATP/ADP exchange. ATP carries one morenegative charge than ADP, so the exchange is electrically dissipative. The membranepotential (inside negative) generated by H+ transport, will drive ATP/ADP exchange in thedirection of ATP efflux and ADP influx.

(11.4) ∆ψ = RT/F ln { [ATP]out[ADP]in/[ATP]in[ADP]out }

PO4- enters mitochondria via PO4

-/OH- exchange. This is an electroneutral exchange but itwill respond to the H+ concentration gradient by virtue of the inverse relationship betweenH+ and OH- concentrations. Consequently, PO4

- will accumulate in mitochondria becauseof the higher internal OH- concentration.

(11.5) [PO4-]in/[PO4

-]out = [OH-]in/[OH-]out = [H+]out/[H+]in

The electrochemical potential of one proton (∆µH+) is

(11.6) ∆µ H+ = ∆ψ + RT/F ln {[H+]in/[H+]out}

From equations 11.4 - 11.6, it should be apparent that the electrochemical potential of oneproton drives the uptake of ADP and phosphate into the mitochrondrial matrix and theexport of ATP from the matrix. The electrical part of ∆µ H+ drives ADP uptake and ATPexport, while PO4

- uptake is driven by the chemical part of ∆µH+.

OH-

Pi

H+ATP 4-

ADP3-

NAD+

+ H O2

NADH+ 1/2 O2H+

Figure 11.2. Coupled transport across the inner mitochondrial membrane.

Page 100: Membrane Biophysics

Page 11.5

Section 11.4. Secretory Vesicle MembranesSecretory vesicle membranes have a vacuolar ATPase that pumps H+ into the

intravesicular space. Consequently, we would expect transport across secretory vesiclemembranes to be coupled to the H+ gradient. Catecholamines (epinephrine, norepinephrineand dopamine) are indeed transported across the secretory vesicle membrane via acatecholamine/proton exchange. The stoichiometry of this exchange is 2 protons perprotonated amine so the equilibrium catecholamine gradient is

(11.7) [RNH3+]in/[RNH3

+]out = {[H+]in/H+]out}2 exp (F∆ψ/RT)

The equilibrium gradient depends on the square of the H+ gradient because two protons areexchanged for each amine. Because the charge on the amine cancels the charge on one ofthe two protons, there is a net movement of only one charge in the exchange. For thisreason, the catecholamine gradient depends on the membrane potential only to the firstpower.

H +

ATP

ADP+ Pi

2 H +

+RNH 3

Figure 11.3. Coupled transport across secretory vesicle membranes.

Section 11.5. Some Coupled Transporters

Transporter Substrates InhibitorsSodium/glucose transporter Hexose/Na+ cotransportVesicular monoamine transporter monoamine:H+ exchange Reserpine

(Major facilitator superfamily)Serotonin transporter Serotonin/Na+ cotransport FluoxetineAnion exchange (Band 3) protein Cl-/HCO3

- exchange Stilbenedisulfonates

Sodium/proton antiporter Na+/H+ exchange AmilorideCa2+/Na+ antiporter 3 Na+/Ca2+ exchangeMitochondrial nucleotide transporter ATP4-/ADP3- exchange AtractylosideCation/chloride symporter Na+/K+/2Cl- cotransport Furosemide

For a much longer listing, go to www-biology.ucsd.edu/~msaier/transport

Page 101: Membrane Biophysics

Page 11.6

References

Bell, G.I., C.F. Burant, J. Takeda, and G.W. Gould (1993) Structure and function ofmammalian facilitative sugar transporters, J. Biol. Chem. 268, 19161-19164.

Crane, R.K., Miller, D., and Bihler, I. (1961) in Symposium on Membrane Transport andMetabolism (A. Kleinzeller and A. Kotyk, ed.), Academic Press, London, p. 439.

Crane, R.K. (1965) Na+-dependent transport in the intestine and other animal tissues, Fed.Proc. 24, 1000-1006.

Mitchell, P. (1961) Coupling of phosphorylation to electron and hydrogen transfer by achemi-osmotic type of mechanism, Nature 191, 144-148.

Schuldiner, S. (1994) A molecular glimpse of vesicular monoamine transporters, J.Neurochem. 62, 2067-2078.

West, I.C. (1980) Energy coupling in secondary active transport, Biochim. Biophys. Acta604, 91-126.

Page 102: Membrane Biophysics

Fundamental Principlesof Membrane Biophysics

CHAPTER 12: ENERGY COUPLING

David Njus

Department of Biological SciencesWayne State University

© D. Njus, 2000

Page 103: Membrane Biophysics

Page 12.1

CHAPTER 12: ENERGY COUPLING

Section 12.1. Thermodynamic EfficiencyBiological systems, like other machinery, often use the energy released by one

process to drive another. Maximizing the efficiency of this energy coupling is obviously ofgreat importance. The thermodynamic definition of efficiency η is

(12.1) η = [output power]/[input power] = - [JoutputXoutput]/]JinputXinput]

where J is the flow and X is the force driving that flow. Conservation of energy requiresthat output power not exceed input power:

(12.2) JinXin ≥ - JoutXout

Consequently, 1 ≥ η ≥ 0.The flows are related by the stoichiometry of coupling n:

(12.3) n = Jout/Jin

Combining equations 12.1 and 12.3 gives

(12.4) η = - n[Xout/Xin]

Maximum efficiency is attained when η = 1, but this is the equilibrium condition (JinXin = -JoutXout). Therefore, no net flow will be produced. Consequently, productive energycoupling with maximum efficiency requires that η be close to but not equal to one, and thatthe coupled flows be close to but not at equilibrium.

Section 12.2. Characteristics of Biological Redox ReactionsAmong the most important phenomena mediated by biological membranes are

transport processes that are linked to redox reactions. These participate in numerouscellular phenomena including the vital bioenergetic functions of mitochondria andchloroplasts. To understand these processes, it is necessary to consider both the equilibriaand the kinetics of redox reactions.

Redox reactions involve the transfer of n reducing equivalents (either e- or H) froma donor to an acceptor:

Donorred + Acceptorox → Donorox + Acceptorred

As discussed in Chapter 9 (equation 9.8), the free energy change for this reaction is

(12.5) ∆Gredox = nF Edonor - nF Eacceptor

= nF (E°donor -E°acceptor) + RT ln ( )[Donor]ox[Acceptor]red [Donor]red [Acceptor ]ox

Page 104: Membrane Biophysics

Page 12.2

The free energy change will be less than zero when Edonor < Eacceptor. Therefore, reducingequivalents will be transferred from carriers with lower reduction potentials to carriers withhigher reduction potentials and this will generally be in the direction of increasing E°.

At equilibrium, ∆Gredox = 0 and the ratio of products to reactants is equal to theequilibrium constant K12. Therefore,

(12.6) K12 = exp{(E°acceptor - E°donor)(nF/RT)}

The mechanisms of biological redox reactions have significant implications,because biology uses both hydrogen atom carriers and electron carriers and exploits thedifference between them. Although it is common to speak of electron transfer chains andelectron transfer reactions in biology, typical “outer-sphere” electron transfer reactions arerare. Outer-sphere electron transfer reactions occur spontaneously at rates determined onlyby the relative reduction potentials of the donor and acceptor and by the intrinsic reactivityof each reactant. The Marcus theory for electron transfer reactions in solution defines therate constant for electron transfer from compound 1 to compound 2 (k12) in terms of self-exchange rate constants for each compound (k11 and k22) and the equilibrium constant forthe reaction:

(12.7) k12 = {k11k22K12f12}1/2

f12 is a collision factor which may be defined as

(12.8) log (f12) = (log K12)2/(4 log {k11k22/Z2})

where Z = 1011 M-1s-1. In practice, f12 is usually about equal to 1. A significant implicationof the Marcus theory is that electron transfer is an intrinsically non-specific process andwill occur most rapidly between the lowest-potential electron donor and the highest-potential electron acceptor. Electron transfer, therefore, is not compatible with thesequential redox reactions of a redox chain. In fact, electron carriers tend to be avoided inbiological systems because their reactions are uncontrolled.

Instead, biological redox reactions employ organic compounds that donate or accepthydrogen atoms. These compounds do not react spontaneously so their redox reactions canbe controlled by enzymes. The enzymes that react with these organic hydrogen-atomcarriers, however, are generally metalloproteins. The metal ligand will accept and donateonly electrons, not hydrogen atoms. The active site of the metalloenzyme, therefore, mustprovide its substrate (the hydrogen-atom carrier) with a way to exchange a proton as wellas an electron. We will call this phenomenon concerted proton/electron transfer.

Concerted proton/electron transfer implies that biological redox reactions willnaturally release or consume H+. The generation of proton gradients, therefore, is a naturalconsequence of the vectorial organization of these reactions. Because electron transferreactions are avoided in biology, terminology such as “electron transfer chain” ismisleading. Instead, we will refer to the respiratory chain in mitochondria or to redoxchains.

Page 105: Membrane Biophysics

Page 12.3

Section 12.3. The Secretory-Vesicle Ascorbic Acid-Regenerating ChainA good, simple example of a biological redox chain is the ascorbic acid-

regenerating system in secretory vesicles. The function of this system is to providereducing equivalents for redox reactions occurring inside the secretory vesicles. Tworeactions, those catalyzed by dopamine β-monooxygenase (DβM) and peptidylglycine α-amidating monooxygenase (PAM), are especially significant. DβM converts dopamine tonorepinephrine and mediates catecholamine biosynthesis in adrenal chromaffin cells, inperipheral sympathetic nerve endings and in noradrenergic neurons in the central nervoussystem. PAM and an associated lyase lead to amidation of the carboxyl termini of manypeptide hormones including vasopressin, oxytocin, VIP, neuropeptide Y, α-MSH,substance P, calcitonin and gastrin.

The monooxygenases both incorporate one atom from O2 into the substrate andreduce the second oxygen atom to H2O. The reducing equivalents are provided byintravesicular ascorbic acid (vitamin C) which in turn is oxidized to the radical anion,semidehydroascorbate. The intravesicular ascorbate is recycled by importing reducingequivalents across the vesicle membrane through cytochrome b561 (Figure 12.1).Cytochrome b561, which spans the secretory-vesicle membrane, is reduced in turn bycytosolic ascorbic acid. This transport of electrons into the vesicles is driven by both themembrane potential (interior positive) and the pH gradient (interior acidic) generated by aV-type H+-translocating ATPase in the vesicle membrane. The pH-gradient favors inwardelectron flow because the midpoint potential of ascorbate is pH-dependent being higher atthe lower internal pH. Thus, contrasting with the mitochondrial respiratory chain in whichH+ is transported using energy from redox reactions, the secretory-vesicle system drives theredox reaction using energy from the proton gradient.

Figure 12.1. Mechanism of ascorbic acid regeneration in secretory vesicles. 1)semidehydroascorbate reductase; 2) cytochrome b561, 3) dopamine β-monooxygenase, 4)H+-translocating ATPase.

Page 106: Membrane Biophysics

Page 12.4

Figure 12.2. Interconversion of ascorbate species. Protonation reactions are shownhorizontally and electron-transfer reactions are shown vertically.

KH

-e-+e-

+e- -e--e-

+H2O

-H+

+H+

+H+

-H+-H+

+H+

-H2O

+e-

..AH

. .-

AH-

A

AH2 A=

A-

AH2O

pK1 pK2

pKr

E°1 E°2

E°3

+H-H

O O

O

H

OH

O O

O--O

H

COH

COHH2

HO O

O-HO

H

COH

COHH2

HO O

OHHO

H

COH

COHH2

H

OH

O O

OO

H

COH

COHH2

HO O

OHO

H

COH

COHH2

H

O O

OO

H

COH

COHH2

H

OHH

H

HOH

Page 107: Membrane Biophysics

Page 12.5

To understand how this system works, it is important to understand that ascorbicacid functions as a donor of single hydrogen atoms. At physiological pH, ascorbic acidexists predominantly as a monoanion (AH-). Similarly, the oxidized form occurs as aradical anion (A-) because this form is stabilized by its capacity to distribute the unpairedelectron over a number of atoms. The fully oxidized form, dehydroascorbate, is notnormally formed because that requires formation of a very unfavorable reactionintermediate (A). Consequently, cytochrome b561 must react with ascorbic acid and itsradical anion by exchanging the equivalent of a single hydrogen atom. Because the metalligand (Fe) will only accept an electron, there must be a mechanism to facilitate ascorbatedeprotonation. This implies that cytochrome b561 reacts withascorbate/semidehydroascorbate by concerted proton/electron transfer. Because thereaction involves only one reducing equivalent and causes no other chemical changes, it isan especially simple example of this kind of redox reaction.

The distinction between concerted H+/e- transfer and e- transfer is illustrated bycomparing cytochrome b561 with cytochrome c. The rate at which ascorbate reducescytochrome c is relatively slow at neutral pH, strongly pH-dependent, and does not saturateat high ascorbate concentrations. By contrast, the rate of cytochrome b561 reduction isfaster, only weakly pH dependent, and saturates at ascorbate concentrations over about 1mM. This implies that cytochrome c is reduced by the ascorbate dianion (A=) by outersphere electron transfer whereas cytochrome b561 is reduced by the ascorbate monoanion byconcerted H+/e- transfer.

A probable but unproven implication of concerted H+/e- transfer is that the substratebinding site is sufficient to make the substrate reactive. By creating a mechanism forproton transfer, the binding site:substrate complex becomes capable of engaging in electrontransfer reactions. This means that metalloproteins can be thought of as two (or more)separate electron transfer centers: the metal and the bound substrate(s). The boundsubstrate will normally exchange electrons with the metal because that pathway isavailable, but the bound substrate may also react with other redox centers that may happento be in the vicinity. Consequently electrons may occasionally go astray, for example,reducing O2 to superoxide (O2-).

Cytochrome b561 reacts with ascorbate on both sides of the membrane and thereforemust possess two ascorbate binding sites. The site on the cytoplasmic side probablycontains a histidine residue, because histidine modification inhibits reduction ofcytochrome b561 by external ascorbate. Cytochrome b561 has recently been cloned andsequenced and has a molecular weight of 30,061. Hydropathy analysis suggests that theprotein has six transmembrane domains with both amino and carboxyl termini being on thecytoplasmic side of the membrane. The protein contains seven histidine residues, two ofwhich act as ligands to the heme. The other five are candidates for the ascorbate bindingsites which mediate concerted H+/e- transfer.

Section 12.4. The Mitochondrial Respiratory ChainIn mitochondria, a pair of reducing equivalents are transferred from NADH or

succinate through the respiratory chain to molecular oxygen. Peter Mitchell championedthe hypothesis that the flow of reducing equivalents down the respiratory chain resulted inthe transfer of protons out across the inner mitochondrial membrane and that the protongradient generated thereby provided the energy for ATP synthesis. To analyze therespiratory chain and the manner in which it transports H+, it is convenient to divide the

Page 108: Membrane Biophysics

Page 12.6

chain into three segments: the NADH oxidase/CoQ reductase, CoQ oxidase/cytochrome creductase, and cytochrome c oxidase. Let us consider each of these components in turn.

TABLE 12.1. Electron Carriers in the Mitochondrial Electron Transfer Chain

Redox Pair n E° (mV)NADPH/NADP+ 2 -324NAPH/NAD+ 2 -320FMNH2/FMN 2 -219FADH2/FAD 2 -219Fe/S centers 1/center -30 to -300Succinate/Fumarate 2 -31Ubiquinone/Ubihydroquinone 2 +9

Ubiquinone/Ubisemihydroquinone 1 -240Ubisemihydroquinone/Ubihydroquinone 1 +258

Cytochrome b566 1 -30Cytochrome b562 1 +30Cytochrome c1 1 +230Cytochrome c 1 +260Cytochrome a/CuA 1 +280 to +350Cytochrome a3/CuB 1 +260 to +9002 H2O/O2 4 +816

NADH oxidase/CoQ reductase oxidizes NADH and reduces coenzyme Q(ubiquinone). Thus it takes the reducing equivalents from a reduction potential of -320 mVthrough a series of iron-sulfur centers to a potential of about -30 mV. In the process, eachpair of reducing equivalents causes 4 H+ to be transferred across the inner mitochondrialmembrane. The H+/e- stoichiometry was measured in a clever experiment by Rottenbergand Gutman (1977). A proton gradient can drive a reverse flow of reducing equivalentsfrom succinate to NAD+ if the flow of reducing equivalents down the respiratory chain isblocked. Rottenberg and Gutman inhibited the flow of reducing equivalents throughcytochrome oxidase using cyanide, and used ATP to generate a proton gradient via the F1FoATPase. The free energy derived from ATP hydrolysis is

(12.9) ∆GATP = ∆G°ATP + 1.38 log {[ADP][Pi]/[ATP]}

The free energy required to drive reverse redox flow is

(12.10)∆Gox = ∆G°ox + 1.38 log {[NADH][fumarate]/[NAD+][succinate]}

If the two processes are allowed to come to equilibrium, ∆Gox + n ∆GATP = 0, where n isthe stoichiometry of coupling. By measuring the equilibrium concentrations of the variousreactants, Rottenberg and Gutman could calculate ∆Gox and ∆GATP and then determine n.They found a stoichiometry of 4/3. Since the F1Fo ATPase transports 3H+/ATP, thissegment of the respiratory chain must transport 4 H+ per pair of reducing equivalents.

Page 109: Membrane Biophysics

Page 12.7

The CoQ oxidase/cytochrome c reductase segment of the respiratory chaintransports H+ across the inner mitochondrial membrane through a process that PeterMitchell termed the Q Cycle. This mechanism is diagrammed in Figure 12.3. It dependsupon two reactions occurring on opposite sides of the inner mitochondrial membrane. Onthe matrix side, ubiquinone is reduced by two reducing equivalents, one taken fromcytochrome b562 and the other taken from an iron sulfur center:

Q + cyt b562 red +FeS red + 2 H+ → QH2 + cyt b562 ox + FeS ox

The reduced quinone is uncharged and hydrophobic and will diffuse to the other side of themembrane where it reduces cytochrome c1 and cytochrome b566:

QH2 + cyt c1 ox + cyt b566 ox → Q + cyt c1 red + cyt b566 red + 2 H+

FeS

Q

QH 2QH 2

Q

Cyt c

Cyt c 1

Cyt b566 Cyt b5622 H + 2 H +

MatrixIntermembrane Space

Figure 12.3. The Q Cycle

Page 110: Membrane Biophysics

Page 12.8

Figure 12.4. Interconversion of quinone species. Protonation reactions are shownhorizontally and electron-transfer reactions are shown vertically.

. -.

E°3

E°2E°1

pKr

pK2pK1

Q-

Q=QH2

Q

QH-

QH. .

+e-

+H+

-H+ -H+

+H+

+H+

-H+

-e- -e-+e-

+e- -e-

OH

OH

O-

OH

R

CH3

O

CH3OOH

CH3O

R

CH3

R

R

CH3

O

CH3

CH3OO

CH3O

O-

O-R

CH3

CH3O

CH3O

CH3O

CH3O

R

CH3

O

CH3OO

CH3O

CH3O

CH3O

Page 111: Membrane Biophysics

Page 12.9

This results in the net transfer of 2 H+ across the membrane. Because one of the tworeducing equivalents is recycled by cytochromes b562 and b566, a pair of reducingequivalents passing completely through the Q cycle will cause 4 H+ to be pumped outacross the mitochondrial inner membrane.

Note that the quinone is a carrier of two hydrogen atoms and would be expected toreact with the metalloproteins on both sides of the membrane by concerted proton/electrontransfer. The requirement for a specific active site for each reaction is indicated by the factthat the reactions are blocked by different compounds; myxothiazol inhibits quinoneoxidation on the outside and antimycin inhibits quinone reduction on the inside.

Figure 12.5. Dioxygen reactions of cytochrome oxidase.

Cytochrome oxidase is the final segment in the mitochondrial respiratory chain. Itis linked to cytochrome c1 by cytochrome c, a soluble protein found in the space betweenthe inner and outer mitochrondrial membranes. Because cytochrome c is soluble, itintroduces an experimentally accessible break in the respiratory chain and isolates thecytochrome oxidase segment. Cytochrome oxidase is a multicenter protein with two hemegroups and two copper atoms. Heme a and CuA are located on the cytosolic side of themembrane and act as the primary acceptors of electrons from cytochrome c. Heme a3 andCuB lie on the matrix side of the membrane and function in the four-equivalent reaction by

Page 112: Membrane Biophysics

Page 12.10

which O2 is reduced to 2 H2O. Cytochrome oxidase catalyzes a series of electron transferreactions and only the reduction of O2 involves H+. Consequently, the cytochrome oxidaseredox reactions do not naturally lead to formation of a proton gradient. Instead,cytochrome oxidase actually functions as a proton pump. Redox energy is captured andused to transport H+ across the inner mitochondrial membrane. H+ transport is believed tobe linked specifically to the flow of electrons through the heme a/CuA region ofcytochrome oxidase.

Having now considered the entire mitochondrial respiratory chain, it is possible toassess the theoretical energy yield of oxidative phosphorylation. Assuming that the ATPsynthase has a stoichiometry of 3 H+/ATP and that 1 H+/ATP is consumed for transport ofATP, ADP, and inorganic phosphate, the following stoichiometries hold:

Segment of electron transfer chain H+/2e- P/2e-

NADH/succinate 4 1.0Succinate/cytochrome c 4 1.0Cytochrome c/O2 2-4 0.5-1.0

The passage of two reducing equivalents down the entire mitochondrial respiratory chainwill cause 1 atom of O to be reduced to H2O. The theoretical P/O ratio (molecules of ATPformed per atom of O reduced) is 2.5 - 3.0. According to long-held dogma, the theoreticalP/O ratio is 3 but this was established before it was recognized that the ratio did not have tobe an integer. The observed P/O ratio is typically between 2 and 3.

The observed P/O ratio is expected to be less than the theoretical value because ofH+ leakage or poor coupling. The degree of coupling is indicated by the phenomenon ofrespiratory control. In the absence of ADP, mitochondria consume O2 at a slow butmeasurable rate. This so-called state 4 rate is the rate at which reducing equivalents maybe transferred down the respiratory chain in the absence of ATP synthesis. Presumably,this redox flow generates a large ∆µH+ and the energy needed to pump H+ against thissubstantial gradient slows the flow of reducing equivalents. Another way to look at this isto recognize that, in the steady state, electron flow may pump protons only as fast as theprotons come back across the membrane. When ADP is added, the rate of O2 consumptionincreases dramatically. This so-called state 3 rate is faster because ∆µH+ is dissipated byATP synthesis. Thus, electron flow may pump protons as fast as H+ are consumed by theATP synthase (F1Fo ATPase). The respiratory control ratio is the ratio of the rate of O2consumption in state 3 to the rate of O2 consumption in state 4. Generally this ratio liesbetween 3 and 15. A high ratio indicates tight coupling between electron flow and ATPsynthesis. A low ratio signifies poor coupling. Complete uncoupling (for example, byadding an uncoupler) will accelerate state 4 respiration to the rate of state 3 respiration andwill cause the respiratory control ratio to drop to 1.

Section 12.5. The Redox Chain of Photosynthetic BacteriaThe redox chain in purple photosynthetic bacteria is interesting for a number of

reasons. First, it combines features of the mitochondrial respiratory chain and ofphotosystem II of the photosynthetic redox chain. As such, it provides some insight intothe evolutionary origins of biological redox chains. Second, the reaction center ofRhodopseudomonas has been crystallized and its three-dimensional structure has been

Page 113: Membrane Biophysics

Page 12.11

solved. It was one of the first membrane structures for which this was accomplished, andfor it Diesenhofer and Michel shared the 1988 Nobel Prize.

The redox chain incorporates a reaction center which takes reducing equivalentsfrom cytochrome c2 (a soluble protein), invests energy from light, and then reducesubiquinone. Reducing equivalents from ubiquinone return to cytochrome c2 by passingthrough b and c cytochromes. In the process, protons are pumped across the membrane.The proton gradient can be used to make ATP as in mitochondria or chloroplasts.Alternatively, it can be used to drive reverse electron flow from succinate to NADH.Cytochrome c2 can pass reducing equivalents to the reaction center or to a cytochromeoxidase which reduces O2. Thus, the redox chain can create a proton gradient by a cyclicphotosynthetic pathway in the light or it can create a proton gradient by oxidizing succinateor NADH aerobically in the dark.

The photosynthetic reaction center incorporates four molecules ofbacteriochlorophyll, two of pheophytin and two of ubiquinone. The structure has two foldsymmetry. A bacteriochlorophyll dimer near the outer surface of the membrane donates anelectron upon absorbing light. The electron passes crosses the membrane via abacteriochlorophyll and pheophytin molecule on one side of the reaction center. Thepigment molecules on the other side apparently do not function in the photosyntheticprocess. The electron finally passes to a quinone on the inside of the membrane. Thisquinone (QA) is reduced to the semiquinone but apparently does not become fully reduced.QA passes the electron to QB, accepts a second electron from the reaction center, and passesit to QB to form the fully reduced dihydroquinone. The dihydroquinone (QH2B) thendissociates from the reaction center and is replaced by an oxidized quinone from thequinone pool.

The quinone reduction results in proton uptake and the nature of the proton uptakemechanism has received some attention. Note that the oxidized quinone Q may be reducedto the semiquinone by electron transfer, because the radical species have a pK nearneutrality so both Q- and QH. are formed readily. The second reduction step must formQH2 because the pK for ionization of the species is greater than 12. Consequently, wemight imagine that QH2 will form by reduction of QH. by concerted proton/electrontransfer.

Page 114: Membrane Biophysics

Page 12.12

NADHSuccinate

Q

Cyt b

Cyt c 1

Cyt c

Cyt a/a3

1/2 O2

FeS

Mitochondrial Respiratory Chain

Photosynthetic Redox Chain

PQ

Cyt f

Fd

P680 700P

H O2

Y

Cyt b559

PC

NADP+

X

Photosystem II Photosystem I

PQ = plastoquinonePC = plastocyaninFd = ferredoxin

Y

BChl

Q

Cyt b

Succinate

NAD+

1/2 O2

Cyt a/a3

Cyt c 1

Cyt c 2

Rhodopseudomonas Redox Chain

Q = UbiquinoneBChl = bacteriochlorophyll

hνhν

Page 115: Membrane Biophysics

Page 12.13

Section 12.6 The Photosynthetic Redox ChainThe thylakoid membranes in chloroplasts accomplish the following so-called light

reactions of photosynthesis:

H2O + NADP+ + hν → 1/2 O2 + NADPH + H+

2 ADP + 2 Pi → 2 ATP

The free energy of NADP+ reduction is about 40 kcal/mole and that of ATP synthesis isabout 10 kcal/mole. The energy obtainable from a single photon of red light is about 40kcal/mole. The efficiency of energy conversion in photosynthesis is about 40%, but thismeans that the energy from four photons of light are needed to drive one pair of reducingequivalents through the redox chain. Each electron must receive the energy from twophotons and this is accomplished by placing two photosystems in the redox chain.

The light energy is collected by antenna pigments in the chloroplast and transferredto the reaction centers. To make photosynthesis proceed efficiently, it is important thatlight be distributed evenly between the two photosystems so that they run at the samespeed. Plants accomplish this by using the redox state of the quinone pool to regulate thedistribution of light energy between the two photosystems. If the quinone pool is tooreduced, then PS I is operating too slowly and more light needs to be diverted to it.Conversely, if the quinone pool is too oxidized, then more light must be shunted to PS II.This is accomplished by phosphorylation control of light harvesting protein (Allen et al.,1981). If the quinone pool is reduced, a kinase is activated increasing phosphorylation ofthe LHCP. This directs more of the light energy to PS I. If the quinone pool becomes toooxidized, the kinase is inactivated and the LHCP is dephosphorylated by a phosphatase. Agreater fraction of the absorbed quanta are then shunted to PS II.

Photosystem II is quite similar to the reaction center of bacterial photosynthesis inRhodopseudomonas. Consequently, the solution of the structure of the bacterial reactioncenter has also illuminated the mechanism of photosystem II.

References

J.F. Allen, J. Bennett, K.E. Steinback and C.J. Arntzen (1981) Chloroplast proteinphosphorylation couples plastoquinone redox state to distribution of excitationenergy between photosystems, Nature 291, 25-29.

P. Brzezinski (1996) Internal electron-transfer reactions in cytochrome c oxidase,Biochemistry 35, 5611-5615.

S.I. Chan and P.M. Li (1990) Cytochrome c oxidase: Understanding Nature's design of aproton pump, Biochemistry 29, 1-12.

K. Krab and M. Wikstrom (1987) Principles of coupling between electron transfer andproton translocation with special reference to proton translocation mechanisms incytochrome oxidase, Biochim. Biophys. Acta 895, 25-39.

R.A. Marcus and N. Sutin (1985) Electron transfers in chemistry and biology, Biochim.Biophys. Acta 811, 265-322.

H. Michel and J. Deisenhofer (1988) Relevance of the photosynthetic reaction center frompurple bacteria to the structure of photosystem II, Biochemistry 27, 1-7.

Page 116: Membrane Biophysics

Page 12.14

P. Mitchell (1961) Coupling of phosphorylation to electron and hydrogen transfer by achemi-osmotic type of mechanism, Nature 191, 144-148.

P. Mitchell (1975) Protonmotive redox mechanism of the cytochrome b-c1 complex in therespiratory chain: Protonmotive ubiquinone cycle, FEBS Lett. 56, 1-6.

P. Mitchell (1976) Possible molecular mechanisms of the protonmotive function ofcytochrome systems, J. Theoret. Biol. 62, 327-367.

P. Mitchell (1979) Keilin's respiratory chain concept and its chemiosmotic consequences,Science 206, 1148-1159.

D. Njus and P.M. Kelley (1993) The secretory-vesicle ascorbate-regenerating system: Achain of concerted H+/e- transfer reactions, Biochim. Biophys. Acta 1144, 235-248.

M.Y. Okamura and G. Feher (1992) Proton transfer in reaction centers from photosyntheticbacteria, Ann. Rev. Biochem. 61, 861-896.

H. Rottenberg and M. Gutman (1977) Control of the rate of reverse electron transport insubmitochondrial particles by the free energy, Biochemistry 16, 3220-3227.

B.L. Trumpower and R.B. Gennis (1994) Energy transduction by cytochrome complexes inmitochondrial and bacterial respiration: The enzymology of coupling electrontransfer reactions to transmembrane proton translocation, Ann. Rev. Biochem. 63,675-716.

I.C. West, P. Mitchell and P.R. Rich (1988) Electron conduction between b cytochromes ofthe mitochondrial respiratory chain in the presence of antimycin plus myxothiazol,Biochim. Biophys. Acta 933, 35-41.

J.R. Winkler, B.G. Malmstrom, and H.B. Gray (1995) Rapid electron injection intomultisite metalloproteins: intramolecular electron transfer in cytochrome oxidase,Biophys. Chem. 54, 199-209.

Page 117: Membrane Biophysics

Fundamental Principlesof Membrane Biophysics

CHAPTER 13: EPITHELIAL TRANSPORT

David Njus

Department of Biological SciencesWayne State University

© D. Njus, 2000

Page 118: Membrane Biophysics

i

Page 13.11

CHAPTER 13: EPITHELIAL TRANSPORT

Section 13.1. Topology and General Principles of EpitheliaEpithelia, such as the lining of the gastrointestinal tract and the nephrons of the kidney,

are layers of cells across which ions and metabolites must be transported. Transport acrossepithelial layers follows a common pattern with different functions achieved using differentcombinations of transporters. It is possible, therefore, to consider a general mechanism and toestablish conventions for defining the various parts and parameters of the system.

The epithelium separates two compartments: the serosal side (blood) and the mucosalside (lumen). The epithelial cells are polarized, so membranes with different characteristics faceeach side. The basolateral membrane faces the serosal side and the apical (or brush border)membrane faces the mucosal side. The Na+/K+ ATPase is found in the basolateral membrane butnot in the apical membrane. Usually, the basolateral membrane is permeable to K+ (it has abarium-sensitive K+ channel) but not to Na+. The apical membrane, by contrast, is permeable toNa+ but not to K+. The K+ current creates a membrane potential (negative inside) across thebasolateral membrane. The inward Na+ current dissipates the membrane potential across theapical membrane, so the potential on the serosal side is generally positive relative to the mucosalside. This transepithelial potential (ψtep)drives current through the paracellular space.

The transepithelial potential is a measure of the tightness of the epithelium. Tightepithelia have a greater resistance and therefore can have a greater transepithelial potential.Examples are the frog skin, urinary bladder, colon and distal nephron. Leaky epithelia have alesser resistance and a smaller transepithelial potential. Epithelia of the small intestine and gallbladder are examples.

Many kinds of transporters are found in epithelial cells, but those involved in transport ofthe principal ions (Na+, K+, Cl-, H+, Ca2+) are common and relatively few in number. A listing ofthese and their diagnostic inhibitors follows.

Table 13.1. Common Ion Transport Activities in Cells

Transporter Membrane InhibitorsNa+/K+ ATPase Basolateral OuabainK+ channel Either BariumNa+/Ca2+ exchanger BasolateralNa+/H+ exchanger Apical AmilorideNa+/K+/2Cl- cotransporter Either Furosemide, BumetanideCl-/HCO3

- exchanger Apical SITS, DIDSCl- channel Apical

Section 13.2. Analysis of a Tight EpitheliumThe frog skin is the classical example of a tight epithelium. It transports Na+ inward

across the skin. This is achieved by an apical membrane that is permeable only to Na+ and abasolateral membrane that is permeable only to K+. To analyze the electrical properties of thissystem, we define the lumen as the reference potential. The intracellular potential ψapi is then the

Page 119: Membrane Biophysics

i

Page 13.22

membrane potential across the apical membrane. The potential on the serosal side is thetransepithelial potential (ψtep). The membrane potential across the basolateral membrane is then

(13.1) ψbaso = ψapi - ψtep

ADP + Pi

K+

ATP

3 Na+

2 K+

Na+

Serosal (Blood)

Mucosal (Lumen)

Basolateral Membrane

Apical Membrane

ψ = 0ψtep

ψapi

I api

I baso

I para

Figure 13.1. Frog skin

The apical membrane is permeable to Na+ but not K+, so the current across the apical membraneis

(13.2) Iapi = (ψapi - ψNa)/RNa

ψNa is the Na+ equilibrium potential across the apical membrane, and RNa is the resistance to theNa+ current across the apical membrane.

The basolateral membrane is permeable to K+ but not Na+, so the current across thebasolateral membrane is

(13.3) Ibaso = - (ψbaso - ψK)/RK = - (ψapi - ψtep - ψK)/RK

ψK is the basolateral K+ equilibrium potential and RK is the basolateral resistance to the K+

current.The current through the paracellular space is

Page 120: Membrane Biophysics

i

Page 13.33

(13.4) Ipara = ψtep/Rpara

At steady state, these currents must all be equal, so

(13.5) Iapi = Ibaso = - Ipara

Upon solving this series of equations, it is apparent that

(13.6) ψapi = ψNa - (ψNa - ψK)RNa/(Rpara+RNa+RK)

(13.7) ψtep = (ψNa - ψK)Rpara/(Rpara+RNa+RK)

This shows that ψapi lies between the apical Na+ equilibrium potential (ψNa) and the basolateralK+ equilibrium potential (ψK). If RNa = 0, then Na+ will be in equilibrium across the apicalmembrane and ψapi = ψNa. At the other extreme, if RK = Rpara = 0, then K+ will be in equilibriumacross the basolateral membrane, and ψbaso = ψK. The transepithelial potential will be zero, soψapi = ψbaso = ψK.

The transepithelial potential (ψtep) is a little smaller than the difference between the apicalNa+ and basolateral K+ equilibrium potentials (ψNa - ψK).

Section 13.3. GI Epithelial TransportThe small intestine functions to transfer nutrients and other important metabolites into the

blood. The intestinal epithelium is leaky, so the transepithelial potential is small. Thismaximizes the membrane potential across the apical membrane (the brush border membrane) andthereby optimizes the energy that Na+ ions can contribute to coupled transport across thatmembrane. As an example of transport across the epithelium of the small intestine, hexoses(such as glucose) are carried across the apical membrane into the epithelial cells by Na+-linkedtransporters. Glucose transport across the basolateral membrane occurs by facilitated transportand is driven only by the sugar's concentration gradient. Note that a Na+-linked transporterwould not function to transport glucose out of the cell into the blood because the Na+ gradientopposes transport in that direction.

Page 121: Membrane Biophysics

i

Page 13.44

ADP + Pi

K+

ATP

3 Na+

2 K+

Hexose

Hexose

Na+

Serosal (Blood)

Mucosal (Lumen)

Basolateral Membrane

Apical Membrane

+10 mV -40 mV 0 mV

Figure 13.2. Sugar transport in the small intestine

The oxyntic cells of the gastric mucosa are responsible for secreting HCl to acidify gastricjuice in the stomach. Because gastric juice is extremely acidic, H+ and Cl- are moved againstlarge concentration gradients, and this requires considerable energy. Estimating the H+ and Cl-

concentrations in gastric juice and blood as shown below, the transepithelial ∆µ for H+ and Cl-

may be calculated:

Serosal Concentration Mucosal Concentration ∆µH+ 3.4 x 10-8 M (pH 7.4) 150 mM 8.4 kcal/equivCl- 50 mM 150 mM 1.1 kcal/equiv

The energy for H+ and Cl- transport is satisfied by a K+/H+ ATPase located in the apicalmembrane. This enzyme actively pumps H+ against the large H+ concentration gradient acrossthe apical membrane. The concomitant K+ influx neutralizes charge movement. The K+ gradientis then used to drive Cl- into the lumen via an electroneutral K+/Cl- cotransport. Cl- influx acrossthe basolateral membrane is driven by the Na+ gradient generated by the Na+/K+ ATPase. Na+

influx is coupled to Cl- influx via parallel transporters: a Cl-/HCO3- exchanger and a Na+/H+

exchanger.

Page 122: Membrane Biophysics

i

Page 13.55

K+

ADP + Pi3 Na+

2 K+

Cl-

Serosal (Blood)

Mucosal (Lumen)

Basolateral Membrane

Apical Membrane

ATP

ADP + Pi

K +

H+

Cl-

K+Cl-

Na+

H+

HCO3-

H2O

CO2

ATP

Figure 13.3. Gastric mucosa

Section 13.4. Renal Epithelial TransportIn the kidney, the proximal tubule is responsible for reabsorption of nutrients and other

important metabolites. Consequently, like the small intestine, the renal proximal tubule is aleaky epithelium. This maximizes the energy stored in the Na+ gradient across the apicalmembrane and optimizes the efficiency of Na+-linked uptake of amino acids, sugars, phosphateand sulfate. The proximal tubule also functions in the reabsorption of Na+ and Cl-. This isaccomplished by a Na+/H+ exchanger and an anion exchanger in the apical membrane.Acidification of the lumen by Na+/H+ exchange facilitates reabsorption of HCO3

- as CO2. Na+

and Cl- uptake causes H2O to be reabsorbed by osmosis. Amiloride, which blocks the Na+/H+

exchanger, inhibits reabsorption of Na+ and HCO3-. It also blocks the concomitant uptake of

water and consequently functions as a diuretic.

Page 123: Membrane Biophysics

i

Page 13.66

ADP + Pi

K+

ATP

3 Na+

2 K+

Na+

Serosal (Blood)

Mucosal (Lumen)

Basolateral Membrane

Apical Membrane

H +HCO 3

-

Cl -

CO 2

H O2

Figure 13.4. Renal proximal tubule

ADP + Pi

K+

ATP

3 Na+

2 K+

Serosal (Blood)

Mucosal (Lumen)

Basolateral Membrane

Apical Membrane

Cl -

Na+

K+

2 Cl -

Figure 13.5. Thick ascending limb - Loop of Henle

In the thick ascending limb of the loop of Henle, NaCl reabsorption must be achievedwithout loss of water by osmosis. Na+ and Cl- are taken up from the lumen by the Na+/K+/2Cl-

Page 124: Membrane Biophysics

i

Page 13.77

cotransporter which is driven by the Na+ and Cl- gradients. The Na+ concentration in theepithelial cells is kept low by the Na+/K+ ATPase. The Cl- concentration is kept low by a Cl-

channel in the basolateral membrane. Cl- efflux through this channel is driven by the membranepotential. A K+ channel is located in the apical membrane rather than in the basolateralmembrane. This reduces the K+ gradient across the apical membrane and prevents it frominterfering with the proper functioning of the cotransporter. The K+ channel can also cause thetransepithelial potential to be positive on the lumen side. This arrangement results in the activetransport of NaCl across the epithelium and can make the urine hypotonic relative to the blood.The loop diuretic furosemide (Lasix) inhibits NaCl reabsorption by blocking the Na+/K+/2Cl-

cotransporter. Because it blocks K+ reuptake from the lumen, furosemide leads to K+ loss.Amiloride, which blocks water reabsorption by acting on Na+/H+ exchange, does not do this andis known as a K+-sparing diuretic.

Section 13.5. Airway EpitheliaIn airways, mucus production in the lumen depends on H2O secretion by the epithelium.

H2O secretion occurs by osmosis consequent to Cl- secretion. Cl- secretion occurs via aNa+/K+/2Cl- cotransporter in the basolateral membrane and a Cl- channel in the apical membrane.

ADP + Pi

K+

ATP

3 Na+

2 K+

Serosal(Blood)

Mucosal(Lumen)

BasolateralMembrane

ApicalMembrane

Na +

K+

2 Cl -

Cl -

Figure 13.6. Airway Epithelium

The Na+/K+ ATPase drives Cl- influx across the basolateral membrane, and Cl- flux across theapical membrane then occurs passively. The symptoms of cystic fibrosis are caused by a defect

Page 125: Membrane Biophysics

i

Page 13.88

in regulation of the Cl- channel resulting in inadequate H2O secretion making secretionscharacteristically salty and too viscous.

Airway epithelia and the epithelia in the nephrons of the kidney are structured in similarways, yet airway epithelia function to secrete Cl- while the epithelia in the kidney take up Cl-.This illustrates how the general structure of an epithelium can be adapted to achieve differentfunctions simply by incorporating appropriate combinations of transporters in the apical orbasolateral membranes.

ReferencesW.B. Reeves and T.E. Andreoli (1992) Renal epithelial chloride channels, Ann. Rev. Physiol. 54,

29-J.R. Riordan (1993) The cystic fibrosis transmembrane conductance regulator, Ann. Rev. Physiol.

55, 609-630.B. Thorens (1993) Facilitated glucose transporters in epithelial cells, Ann. Rev. Physiol. 55, 591-

608.E.J. Weinman and S. Shenolikar, (1993) Regulation of the renal brush border membrane Na+-

exchanger, Ann. Rev. Physiol. 55, 289-304.M.J. Welsh (1987) Electrolyte transport by airway epithelia, Physiol. Rev. 67,1143-1184.M.J. Welsh, M.P. Anderson, D.P. Rich, H.A. Berger, G.M. Denning, L.S. Ostedgaard, D.N.

Sheppard, S.H. Cheng, R.J. Gregory and A.E. Smith, (1992) Cystic fibrosis transmembraneconductance regulator: A chloride channel with novel regulation, Neuron 8, 821-829.

C.S. Wingo and B.D. Cain (1993) The renal H-K-ATPase: Physiological significance and role inpotassium homeostasis, Ann. Rev. Physiol. 55,323-347.

E.M. Wright (1993) The intestinal Na+/glucose cotransporter, Ann. Rev. Physiol. 55, 575-589.