fundamental principles of membrane biophysics

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Fundamental Principlesof Membrane Biophysics

David Njus

Department of Biological SciencesWayne State University

D. Njus, 2000
ftp://glove.2y.net/ftp://glove.2y.net/ftp://glove.2y.net/


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F u n d a m e n t a l P r in c ip l e s

o f M e m b r a n e B io p h y s ic s

David Njus

Depart men t of Biologica l SciencesWayne Stat e University

D. Njus , 2000


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Tab l e o f Con t en t s

Cha pter 1. Biological Membr an es

Cha pt er 2. Ther modyna mics of Micelle Form at ion

Cha pter 3. The Fluid Mosaic Membra ne

Chapter 4. Membrane Electrostatics

Cha pt er 5. Specific an d Non-Specific Bindin g

Chapt er 6. Permeability and Conductance

Cha pter 7. Perm eability an d Condu cta nce of Electr olytes

Chapter 8. Chan nels an d Excitable Membranes

Cha pter 9. Active Tra nsport

Cha pter 10. Fa cilita ted Diffusion

Cha pter 11. Coupled Tran sport

Chapter 12. Energy Coupling

Chapt er 13. Epithelial Tran sport

A p p e n d i c e s

I. Glossary of Symbols

II. Abbreviations

III. Fundamental Constants

IV. Conversion Factors

V. Mathematical Formulae


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Glossa r y o f Symb ols

A areaC molar concentration, capacitancec velocity of light

cmc 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 Plancks constantI currentJ flowKi equilibrium constant for reaction iKp partition coefficientk Boltzmann constantki rate constant for reaction im mass, aggregation numberN Avogadro's number

n amount in molesP permeability coefficient, pressure, powerQ heatq chargeR gas constant, resistancer radiusS entropyT absolute temperaturet timeu mobilityV volume

v velocityW workX mole fraction

x distanceZ collision factorz valence

activity coefficient difference dipole moment dielectric constanto permittivity of vacuum viscosity, efficiency wavelength of light electrochemical potential density reflection coefficient electrical potential


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A b b r e v i a t i o n s

An gstrom sa tm a tmospheresC degrees cent igrade

ca l ca lor iescoul coulombsD DebyesDa Daltonseq equ iva len t sesu electrostat ic unitsF fa rads (coul/volt )g gramsj joulesK degrees Kelvin

l lit ersM moles /lit erm metersm in m in u tesmol molesS Siemenssec secondsV volts

k k ilo 103

c cent i 10-2

m milli 10-3

micro 10-6

n nano 10-9

p pico 10-12

f femto 10-15

a a t to 10-18


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F u n d a m e n t a l C on s t a n t s

Gas constan t R = 8.3144 j.K-1.mol-1

1.9872 cal.K-1.mol-1

8.3144 x 107 ergs.K-1.mol-1

0.082054 l.a tm.K-1.mol-1

Boltzmann constant k = 1.38044 x 10-16 er g.K-1

Avogadro's Number N = 6.0230 x 1023 molecules.mole-1

Ice poin t To = 273.15 K

Faraday constant F = 96,490 coul.eq-1

Perm ittivity of vacuum o = 8.854 x 10-12 coul.m-1.volt -1

Molar volume, ideal gas, 0C, 1 a tm Vo = 22.4138 l.mol-1

Electronic charge e = 4.80286 x 10-10 esu

1.602 x 10-19

coulElect ron mass m e = 9.1083 x 10-28 g

Velocity of light c = 2.997930 x 1010 cm.sec-1

Standard accele ra t ion of gravity g = 980.665 cm.sec-2

Planck's constant h = 6.6252 x 10-27 er g.sec

Volume conversion factor = 103 l.m-3

Collision factor Z = 1011 M-1.sec-1

C o n v e r s i o n F a c t o r s

1 at m = 760 mm = 1.01325 x 106 dyne.cm-2

1 cal = 4.184000 j

1 j = 107 ergs = 1 volt .coul

1 erg = 1 dyne.cm

1 ev = 1.60206 x 10-12 er g1 l.atm = 24.22 cal

1 = 10-10 m

1 Debye = 10-18 esu .cm.molecule-1 = 2 x 10-6 coul.m.mol-11 kcal/eq = 0.043362 volts


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M a t h e m a t i c a l F o r m u l a e

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

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

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

Su rface AreaCylinder (minus ends) A = 2rhSphere A = 4r2

Volume

Cylinder V = r2hSphere V = (4/3) r3


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F u n d a m e n t a l P r in c ip l es


CHAPTER 1: BIOLOGICAL MEMBRANES

David Njus

Depa rt men t of Biologica l SciencesWayne Stat e University

D. Njus, 2000


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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 provides

powerful 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. The

purpose 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


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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 in

between.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


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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 membrane

including 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 exterior

of 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


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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 their

own 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, Science175, 720-731.


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CHAPTER 2: THERMODYNAMICS OF MICELLE FORMATION

David Njus


D. Njus, 2000


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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 20C) 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 in

the 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.


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Page 2.2

Figure 2.1


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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 micelle

structure 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 2.2It should be recognized that the spherical micelle and the planar bilayer are really

two 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 a

variety 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


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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 2.1Detergent cmc (mM) m________________________________________________________________Sodium dodecylsulfate

50 mM NaCl 2.3 72500 mM NaCl 0.51 126

Deoxycholate10 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 the

hydrocarbon in the aqueous phase is

(2.1) w = w + 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) HC = HC + 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 chemical

potentials are equal (w = HC). Therefore,

(2.3) HC = w + 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:


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(2.4) HC - w = RT ln Xw

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

(2.5) HC - w = -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) mic = mic + RT ln Xmic

Setting mic 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) mic - w = 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) mic - w = -1934 - 771 nc cal/mol

As in the case of the transfer of hydrocarbon from water to the pure hydrocarbon phase, the

free 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 (w) and in amicelle of size m (mic,m).

(2.9) w = w + RT ln Xw + RT ln fwand

(2.10) mic,m = mic,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.


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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 ( )2

Ptotal

hgPP(erg/cm)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


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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 the

aggregation number (Figure 2.5).


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Page 2.9

-10

0

10050

(kcal/mol)

A ( )2

Figure 2.5

U - m w

W (low ionic strength)m

W (high ionic strength)m

mic,mU - w

Figure 2.4

(kcal/mol)

-8

-6

-4

-2

0

2

4

1007550

mic,mU - wRT ln cmc

A min

A ( )2smallermicelles

largermicelles

Wm

U - m w


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Page 2.10

Cetyltrimethylammonium (CTAB)

Lauryl Sulfate (Dodecyl sulfate)

Cholic Acid: X = OHDeoxycholic Acid: X = H

7

OH

CH3

CH3

H

X

CH3OH

H

H

H

-OOC

-O3S O C H2 (CH2)10 CH3

H3C N+

CH3

CH3

CH2 (CH2)14 CH3

Fig 2.6. Some ionic det ergen ts


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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 deter gent s

References

Stillinger, F.H. (1980) Water revisited, Science209, 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. USA71, 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. USA76, 4175-4176.C. Tanford (1980) The Hydrophobic Effect, Second Edition, John Wiley & Sons, New

York.


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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 , determined principally by the chain length of the fattyacids in the phospholipids. Each phospholipid molecule occupies a surface area of about70 2.

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.


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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. At

low 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 (Deamer

and 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.


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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 it

does 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).


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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 x

represents 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:yz' where x representsthe number of carbons, y represents the number of double bonds, and z' represents theposition of the first double bond counting from the carbon (methyl terminus). Forexample, the structure of palmitoleic acid (16:1, 9-cis or 17) is:

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

TABLE 3.3. Some Common Fatty acids

Saturated UnsaturatedNo. of Common Common NomenclatureCarbons Name Name Standard ___________________________________________________________________10 Capric Palmitoleic 16:1, 9-cis 1712 Lauric Oleic 18:1, 9-trans 1914 Myristic Vaccenic 18:1, 11-cis 1716 Palmitic Linoleic 18:2, 9-cis,12-cis 2618 Stearic -Linolenic 18:3, 9-cis,12-cis,15-cis- 3320 Arachidic -Linolenic 18:3, 6-cis,9-cis,12-cis- 3622 Behenic Arachidonic 20:4, 5,8,11,14 (all cis) 4624 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


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

Phosphat idic A cid (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


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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

6

5

4

3

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 proteins 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 aqueous

environments at either membrane surface. The -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 -helix consists of 3.61amino acids per turn spanning a distance of 1.5 per residue. Given a membranethickness of 30-40 , a transmembrane a-helix should contain 20-27 amino acid residues.


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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 hydropathy

is 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 -3.9

Arginine 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


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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 and

the 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, and

the 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. Acta443, 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, Biochemistry8, 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.


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CHAPTER 3: THE FLUID MOSAIC MEMBRANE

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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. USA81,6559-6561.

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. USA69,3561-3566.

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. Chem67, 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, Biochemistry22, 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. USA79, 7749-7753.

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


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CHAPTER 4: MEMBRANE ELECTROSTATICS

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

(4.8) = - qsx/o

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 ( = 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.


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Page 4.4

Outside Membrane Inside

0

surfacepotential (outside)

surfacepotential (inside)

dipolepotential(outside)

dipolepotential(inside)

membranepotential

potentialcreated bycapacitativecharge

Figure 4.2. Components of the membrane potential

When charge is moved from one side of a biological membrane to the other, we

assume 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/Ao

and an electrical potential difference across the membrane

(4.10) = qcx/Ao

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

(4.11) C = qc/ = Ao/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


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about 1 F/cm2. This corresponds to a membrane 35 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 charge

and 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-Hckel 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

qs q (x)v

o

(x)

() = 0

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


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(4.27) 2 = (F2/RTo) i Cio zi2

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

(4.28) 1/x =

One may think of 1/as a measure of the thickness of the diffuse double layer. That is, thesame surface potential o would result if all of the diffusable charge were placed at adistance 1/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 F/cm2. Typical surface charge densities are much larger than this (chromaffin granules

have 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 20C.


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Lm

Lmax

[L]

Lm

K

Lmax

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

(5.6) [L] = [L] exp (-ziFo/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 o is negligible. Since the surface potential dependsupon both the ionic strength of the solution and upon the surface charge density, it can be

considered 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).


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concentrations that range about Ksp, so [L] will be very small relative to the non-specificbinding constant [L] > [L] >> Ksp). Under these conditions, equation 5.9 reduces to

(5.11) Lm = Lmaxsp + Lmax

ns [L] / Kns

Because Lmaxsp


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(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 (d/dx):

(6.11) v = -(1/Nf) (d/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)(d/dx)

For a non-electrolyte, d/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.


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

(6.17) Cm = Cw exp [(w- m)/RT]

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

rearranging:(6.18) J = -(RT/Nfd){exp [(w - m)/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 [(w - m)/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[(w-m)/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 [(w - m)/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 (m-w) 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 m to 500 m (Remember that the membrane itself is only 4 x 10-3 m


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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.


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CHAPTER 7: PERMEABILITY AND CONDUCTANCE OF ELECTROLYTES

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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:

CNf

(7.1) J = - RT dC zFC dNf dx RT dx+[= -]( )[ + RT ln C + zFo ]( ) ddx

As before, we must integrate this from one side of the membrane to the other. Inthis case, however, we have two parameters, C and , 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)C edxd

(7.2) [ ]dC zFC d

+dx RT dx

(zF /RT)e

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

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

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

zF/RT zFmi/RT zFmo/RT

(7.4) J 0

de 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(i- mi)/RT]

(7.6) Cmo = Co Kp exp[zF(o- mo)/RT]

Therefore,zF /RTi

C e

dx = - RTKNfp

izF /RTo

- C eo(7.7) J 0d zF /RT

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


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Page 7.2

[ ]( )RTKNfd p oC - C eizF /RTm(7.8) J =

0d

d

exp (zF /RT) dx

where m 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 (zFm/RT)]

where Q is defined as

(7.10) Q = d/0

dexp (zF/RT) dx

MembraneOutside Inside

0 d

mo

mi

C i

o

CmoC o

Cmi

i

Figure 7.1. Concentration profile for steady-state electrolyte flow

The problem now is to evaluate Q. To integrate exp (zF/RT), we need to define(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 = -d/dx) is constant

throughout the membrane. Therefore,

(7.11) (x) = - Ex + (0)

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


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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 =

w) and Ei will differ from this by the membrane potential (Ei = w + zFm). Uponintroducing these values for the energies, equation 7.19 reduces to

(7.20) J = exp [-(Em- w)/RT] [Co - Ci exp (zFm/RT)] x constant

If we let PQ = exp [-(Em- w)/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) short

range 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.


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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 is the dielectric constant of the medium, the Borncharging energy is

(7.21) W = q2/8oa

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'/4ox2

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

(7.23) dW = -

aF dx = -

a(q' dq'/4ox

2) dx

= q' dq'/4oa

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

(7.24) W = W = 0

qq' dq'/4oa = q

2/8oa

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

(7.25) W = (q2/8oa) (1/hc - 1/w)

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):


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Page 7.7

( )( ) ]( )( )[ xd1 - - 1.2(7.26) W =

2q8ao

2

hc

1 a2x

ad

Section 7.3. The Goldman-Hodgkin-Katz EquationIf ions are in equilibrium across a membrane, then the membrane potential will begiven 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 = cations zjFJj + anions 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 = cationsFPj{(Fm/RT)/[exp(Fm/RT)-1]}{Coj - Cijexp(Fm/RT)}- anionsFPj{(-Fm/RT)/[exp(-Fm/RT)-1]}{Coj - Cijexp(-Fm/RT)}

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

(7.29) 0 = cationsFPj{1/[exp (Fm/RT) - 1] }{ Coj - Cij exp (Fm/RT) }- anionsFPj{ (-1/[1 - exp (Fm/RT)] }{ Coj exp (Fm/RT) - Cij }

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

(7.30) 0 = cations Pj{Coj - Cij exp (Fm/RT)}- anions Pj{Coj exp (Fm/RT) - Cij}

Then solving for the exponential term gives

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

cations anionsj ijP C + j ojP Cm

Therefore, the membrane potential is defined by the ion concentrations and permeabilities

as follows:

(7.32) = (RT/F) lnm

cations anionsj ojP C + j ijP C

cations anionsj ijP C + j ojP C

This is the Goldman-Hodgkin-Katz equation.


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CHAPTER 8: CHANNELS AND EXCITABLE MEMBRANES

David Njus


D. Njus, 2000


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Pa ge 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 structural

simplicity 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-NHCH 2CH2OH

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 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 a

barrel. 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 subunit. Inthe eel electroplax, that is the only subunit. The sodium channel from mammalian skeletal


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Pa ge 8.2

muscle also contains a 1 subunit (38 kDa), while the sodium channel in mammalian braincontains both a 1 (36 kDa) and a 2 subunit (33 kDa) along with the 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 CaK

channel 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 -conotoxin GVIA while P channels are blocked by -agatoxin IVA.Structurally, these channels are thought to be similar. The L-type Ca2+ channel fromskeletal muscle has five subunits: 1 (170 kDa), 2 (175 kDa), (52 kDa) and (32kDa). The 1 subunit contains binding sites for Ca

2+ channel antagonists and is thought tobe the subunit forming the functional Ca2+ channel. The 2 piece consists of a large 2

subunit linked to the 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: 2, 1, 1and 1 . Each subunit has at least four transmembrane

segments. The acetylcholine binding sites are located on the subunits.The second group of extracellularly activated ligand-gated ion channels, the

glutamate-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


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Pa ge 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 by

releasing more Ca

2+

from the SR. Thus, the ryanodine receptor mediates Ca

2+

-inducedCa2+ release.A related protein, the inositol-1,4,5-trisphosphate receptor releases Ca2+ from the

endoplasmic 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 8-10transmemb

fundamental principles of membrane biophysics

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