physical principles and formalisms of electrical excitability
TRANSCRIPT
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C H A P T E R 6
A A * I I
A
A
A
Physical principles and formalisms
Departments of Physiology and Neurology, Albert Einstein
College of Medicine, New Yo rk , Ne w YoFk
Departmen t of Biophysics, T he Rockefeller Unive rsity,
New York , New York
of electrical excitability
C H A P T E R C O N T E N T S
Membrane Potentials and Ionic Fluxes
Fundamental concepts
Equilibrium state
Nonequilibrium state
Summary
The Donnan equilibrium
Phase-boundary potentials
Two important, examples of equilibr ium situ ations
Electrodes
-
he measurement of potential differences
Quasi-equilibrium systems
A membrane with a very large fixed-charge density
An oil membrane
Homogeneous uncharged membrane
Homogeneous membranes with special properties
Mosaic membranes
Formal Consequences of Voltage-dependent Conductances
Ion transport (the Nernst-Planck flux equations)
The nature of electrical excitability
Reasons for believing th at electrical excitability does not
Hodgkin-Huxley equivalent circuit
Current-voltage (I-V) characteristics
Negative-slope conductance
Changing the I - V characteristic without change of the g-V
result from the shifting of ionic profiles
characteristic
Voltage-dependent Conductance in Thin Lipid Membranes
The unmodified t hin lipid membrane
Formation
Permeability and electrical properties
Carriers
Channel formers
A mosaic membrane formed with two modifiers
Summary
Monazomycin
Alamethicin
Excitability-inducing material
Nonvoltage-dependent modifiers
Voltage-dependent modifiers
system, the modified thin lipid (or bilayer) mem-
brane, which illustrates most of the relevant phe-
nomena associated with nerve excitation. Before dis-
cussing this model, however, we develop more or less
from first principles the concepts of membrane poten-
tials and ionic fluxes. This forms a ra ther large pa rt
of the article and may be superfluous for the more
sophisticated reader. Nevertheless we have included
this material because, despite its importance for un-
derstanding nerve excitation, it is rather inaccessible
to students and investigators attempting to make
initial contact with the neurophysiological literature.
In the second section we discuss some of the formal
aspects of the behavior of systems containing ele-
ments whose conductances are profoundly affected by
the voltage across them. With thi s as background, we
then discuss the fascinating voltage-dependent phe-
nomena that can arise in suitably doped thin lipid
membranes.
MEMBRANE
POTENTIALS
A N D IONIC FLUXES
Fundamental Concepts
Here we consider two general situations: equilib-
rium and nonequilibrium states. We analyze the
equilibrium sta te from both th e thermodynamic and
the statistical mechanical viewpoint; the nonequili-
brium state is handled by the Nernst-Planck flux
equations.
EQUILIBRIUM STATE.
hermodynamic approach.
The
fundamental relation that we need from thermody-
Single channels
Summary and conclusion
_ _ _ _ ~
namics is that at thermal equilibrium the electro-
chemical potential, pi, f any species i is the same in
all phases to which the species has access. Thus, for
phases 1 and 2 we can write
pi (1) =
pi
(2) (1)
provided that species i can move between the two
phases. Equation
1
is the starting point of all our
IN
THIS CHAPTER
we intend to elucidate the presently
understood physicochemical principles and formal-
isms underlying the electrical excitability of biologi-
cal membranes. The primary analysis is of a model
161
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162
HA N D BOOK O F PHYSIOLOGY HE N ER V OU S SY STEM I
thermodynamic treatments. For a n ideal solution, p,
is given by
p , = p , + RT In
X ,
+ PV, + z , F
$
+ any other
relevant work terms (2)
where p , is the standard chemical potential (in the
particular phase being considered) of the ith species,
X ,
is mole fraction of ith species,
V ,
s partial molar
volume of ith species, z, is valence of ith species (0,
k l , +2,
.
. .), P
is
hydrostatic pressure of the phase, 4
is
electrostatic potential of the phase,
R is
the gas
constant, T is temperature in degrees Kelvin, and F
is the Faraday. (If the solution is nonideal, an activ-
ity coefficient, y , , is included with the mole fraction
term. In this article we deal only with ideal solu-
tions.) If species i is dilute, then Equation 2 can be
rewritten as
p, = p ,
+ RT
I n c, +
PV, + z,
F 4 +
. . . (2a)
where
c,
is the concentration of i.
( p l ( 0 )
n Eq. 2a is
different from p , n Eq. 2 because the standard state
must be newly defined in going from mole fraction
units to concentration units.)
A s
indicated in Equation 2 , other work terms must
be included if they contribute
to
the potential energy
of the species i. For example, if the system is in a
gravitational field (or ultracentrifuge), there will be a
term for the gravitational potential energy. For the
systems we consider, these terms do not arise. In fact,
even the
PV
term will generally be trivial and there-
fore not ente r into our treatments. Thus, for our
purposes,
p,
can be written as
p, =
p, (
+ RT
In c,
+
z , F
$
(2b)
Stat i s t ical mechanical approach. The basic rela-
tion tha t we need from statistical mechanics is tha t at
thermal equilibrium the particles satisfy the Boltz-
mann distribution a t all points in the system to which
they have access. That is, a t any point x (considering
only a one-dimensional situation)
c , x ) =
C,,e-%(x)/h7
(3)
where,
w ,
XI is the potential energy per particle of
the ith species a t
x ,
c,, is its concentration a t th e point
defined as zero potential energy, and
k
is th e Boltz-
mann constant. Multiplying the numerator and de-
nominator of the exponent by
N A ,
Avogadros num-
ber, we can write Equation 3 in the form
c , x ) = c , , e - , s ) lRT (3a)
where
W,
is the potential energy per mole of the ith
species. If W,
is
purely electrostatic energy, then
Equation 3a becomes
c , ( x ) =
~ ~ e - 2 ~W X ) l R T
(3a)
For example, consider particles in a gravitational
field. Then Equation 3 becomes
=
c e -l l l v s : k ? (4)
where m is the mass of the particle and g is the
acceleration due to gravity. If there were no thermal
energy, all the particles would sit a t x = 0, he point
of minimum potential energy. On the other hand if
there were no gravitational field and only thermal
energy, the particles would be homogeneously dis-
tributed throughout space. Equation 4 is the compro-
mise when both terms operate.
It is important to realize th at Equation 3a is equiv-
alen t to Equation
1.
(Take the logarithm of both sides
of Equation 3a and identify W, as all the terms on the
right in Equation
2a
except
RT
In
ci.)
Thus the ther-
modynamic statement that the electrochemical po-
tential of a species is the same in all phases is equiva-
lent to the statistical mechanical statemen t that the
molecules of the species satisfy the Boltzmann distri-
bution.
NONEQUILIBRIUM STATE.
The bases of our discussion of
ion transport a re the Nernst-Planck flux equations
where $,+and
$k-
represent the flux (in mol/s)
of
the
jth and kth ion, respectively, across unit area at any
point x in the system,
u ,
s th e molar mobility of the
jth cation,
uk
is the molar mobility of the kth
anion,
c,+
and
ck-
represent the concentration of the
jth and kth ions, respectively, a t any point I, nd
is the electrical potential
at
any point x . (Again we
are considering a one-dimensional situation with gra-
dients of concentration and electrical potential occur-
ring only in the x direction.)
Whereas our basic equilibrium equations (Eqs. 1
and 3) have
a
solid foundation in thermodynamics
and statistical mechanics, the flux equations are less
firmly grounded in theory, and in some instances
they are even grossly incorrect. We therefore wish to
make a few points concerning them tha t will give the
reader some feel for their meaning.
Equations 5 can be written i n the form
where
D = uRT
(7)
The Boltzmann distribution iS
a
quantitative state-
merit of the balanced but incessant competition be-
tween potential energy
( w )
nd thermal energy ( k T ) .
I some
heoretical problems associated with justifying the use
of the Boltzmann distribution in electrolyte theory do not concern
us
67).
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CHAPTER 6: PHYSICAL PRINCIPLES
A N D
FORMALISMS
OF ELECTRICAL EXCITABILITY 163
For
a
nonelectrolyte (z
=
O), Equation 6 simply be-
comes
dc
dx
4 = D-
which is Ficks
first
law of diffusion. [Equation
7
is
the famous Nernst-Einstein relation between the dif-
fusion constant an d th e frictional coefficient (13,
14);
the mobility (u)
s
simply the reciprocal of the fric-
tional coefficient.] Thus th e flux of a species is propor-
tional
to
the negative gradient of its concentration.
On the other hand for an ion
at
uniform concentra-
tion throughout the system dc/dx = 01, Equation
6
becomes
which is simply the equation for electrophoresis.
That is, the ionic flux
is
proportional to the electric
field (-d+/dx). Thus Equation 6 says that if there is
both a concentration gradient and an electric field,
the ionic flux is a l inear sum of the fluxes that would
arise from each effect alone.
Another way of looking at Equations
5
is to say
that
flux = concentration x velocity
and
velocity = mobility x driving force,2
so that combining we have
flux = mobility
x
concentration
x
driving
Z F9)
T dc
4 = u x c x
dx
forc
Substituting Equation 2b, the thermodynamic
expression for the electrochemical potential, into
Equation 8 we obtain Equation 5, the Nernst-Planck
flux equations (if p,( is constant). Note that Equa-
tion
8
reduces to the thermodynamic equilibrium
condition, Equation
1,
when
4i = 0
at all points; or
Equation
5
reduces to the BoItzmann distribution,
Equation 3ar, when Cbi
=
0.
[The relationship of the
Boltzmann distribution (equilibrium state ) to the
flux equations (nonequilibrium state) is clearly seen
upon differentiating Equation 3a
d%
T
dci
CI dx du
(which is the same relation a s obtained by se tting di
= 0 in Equation 5). Thus a t equilibrium the sum of
the electrical force
( -
z,F d$ldx) and the diffusional
force
( -
RT/ci dc,/dx) cting on an ion is zero at
every point in the system, and hence there is no flux
of matter. When these two forces do not balance,
there is a net driving force on the ion and hence a
flux.
For our purposes the Nernst-Planck flux equations
(Eqs. 5) are perfectly adequate. Nevertheless the
reader should be aware th at there exist situations for
ziF 0
The flux equation states that the force on an ion is the
sum of two terms: -RTlc
dcidx
and -zF d$ldx. The
second term is th e electrical force familiar from ele-
mentary electrostatic theory, but the first term is
more subtle.
It is
a phenomenological force resulting
from the random (Brownian) motion of each individ-
ual ion. Though the random movements of each ion
are equally likely to be in the positive or negative
direction, statistically it appears as if there is a force
operating in the direction of the concentration gra-
dient (4 1).
Equation
5
is
a
special case of the more general
expression
which st ates tha t the driving force on the ith species
is the negative gradient of its chemical potential.
In a condensed phase such as water, an ion subjected to a
force very quickly accelerates to its terminal velocity. In practice,
therefore (unless we are dealing with very high frequencies),
force gives rise to velocity rather than to acceleration.
which these equations are insufficient. In particular,
Equations
5
(or more explicitly Eq.
8)
say that the
only driving force on a species i is the negative gra-
dient of chemical potential for that species alone; it
neglects the coupling of the gradients of chemical
potentials for other species j to th e flux of species i. (A
familiar example of such coupling is solvent drag,
where the gradient of the chemical potential of water
gives rise not only to a flow of water, but also to a flow
of solute dissolved in the water.) Though these cou-
plings, expressed as cross coefficients to forces acting
:e
(5)
on other species, are extensively used by those who
deal with the formalism of irreversible thermody-
namics (36), they do not concern us here. For the
systems we consider (which are
of
neurophysiological
interest), they introduce corrections tha t are a t best
second order.
SUMMARY. We consider dilute, ideal solutions of ions
in which gradients of concentration, potential, or
solvent composition occur in only the x di re~ t ion .~
The sta rting point for the t reatment of these systems
is one of the following equations
4.
-
dc, d%
I uiRT ziuiFci 5) Transport
dx dx
For the general case, d/dz is replaced by
C
n all equations.
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164 HA N D BOOK OF PHYSIOLOGY THE NERVOUS SYSTEM I
m e m b r a n e
C I -
F I G .
1.
The Donnan system.
Two Impor tan t Examples of Equi l ibr ium S i tua tions
THE DONNAN
E Q U I L I B R I U M .
Aside from its intrinsic
physiological intere st, the Donnan equilibrium illus-
trates almost all the important concepts and difficul-
ties associated with membrane potentials; we there-
fore analyze it in some detail. The Donnan equilib-
rium arises when a membrane separates two solu-
tions containing both permeant and impermeant
ions. For simplicity we restrict our trea tment to the
case of an infinitesimally thin membrane separating
two infinite solutions; both solutions contain a single
permeant univalent positive (Na+) nd negative (Cl-)
ion species, and, in addition, one solution contains a
univalent impermeant ion, zN, where
z = + 1
(Fig.
1). (The mechanism of membrane semipermeability
is irrelevant to our treatm ent. In practice, the Don-
nan situation commonly arises when a porous mem-
brane, such as dialysis tubing, holds back a macro-
ion (e.g., protein) which cannot
fit
through the pores.
Usually the solvent, water, is also permeant.)
Thermodynamic ana lys i s . The conditions that
must be satisfied are, from Equation
1
/- .I
( l ) ~ N . I 2 ) (9a)
k l
(1 )
= P(I
( 2 ) (9b)
which become, upon substituting from Equation 2b
&,+ +
RT In a+], -t F ,
(10a)
= &;;+ + RT In [Na+In+
F I / J ~
Note that there is no such equation for N, since we
have specified that it cannot move between the two
aqueous phases. (We have excluded a third equation
equat ing the chemical potential
of
water on the two
sides of the membrane. Although the osmotic effect
accompanying the Dorinan equilibrium is of consider-
able general physiological interest, it is not signifi-
can t for electrophysiology.) Adding Equation 10a and
10b we obtain
or
This is known a s the Gibbs-Do nnan condi t ion, and
r
is called the Don nan ratio .
From Equations 10a or lob we can directly obtain
the potential difference across the membrane
RT a+],
v =
($, - J J ~ )=
~
In
~
F
a+],
RT
[Cl-ll RT
F
[Cl-]? F
-
- -
In ~=
-
~ In r (13)
To
calculate r, and hence the membrane potential,
we invoke the familiar electroneutrality condition
on both sides of the membrane
[Na+I1= [Cl-1, =
c I
a+],
+
z[N]
=
LCl-1, (14b)
(I t is to be understood tha t the electroneutrality con-
dition holds only for remote regions on eith er side
of
the membrane, that is, for macroscopic regions. As
will be seen presently, the al terna tive treatment by
means of the Poisson-Boltzmann equation establishes
the concentrations and potential as continuous func-
tions ofx, and indeed regions very close to the mem-
brane are not electrically neutral
.)
Combining Equa-
tions 14 and 12 we obtain
(14a)
(electroneutrality condition)
Figure 2 is a sketch of r as a function of the imper-
meant ion (N) concentration for
z =
+1. As [N+]
increases,
r
decreases from
1;
that is, permeant posi-
tive ion concentration decreases on side 2 and per-
meant anion concentration increases. In the limit of
large +I, there is virtually no Na+ on side 2 and
[Cl-1,
=
+I. The converse occurs for large -1.
Note also from Equation 13 th at t he membrane po-
tential, \Ir, will be positive or negative depending on
whether
z
is
+
1 or -1 and th at the absolute value of
\I
increases as [N] increases. Figure 3 is a diagram of
r
-
~
= r
Na+;l,
-
[Cl-1,
a+:[, [Cl-1,
FI G . 2 .
Qualitative plots of the Donnan ratio, r, as a function
of the impermeant ion concentration,
IN],
for an impermeant ion
of
valence
+1
or - 1 . (See Eq.
1 5 . )
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166
HANDBOOK OF PHYSIOLOGY THE NERVOUS SYSTEM I
and
[Cl-]+,
=
[Cl~]~,e+~YH
FIG.
5.
The two-phase system.
generated from a given initial condition. Thus, al-
though the impermeability of N is the ultimate cause
for the membrane potential and ion asymmetries,4
the $ and concentration functions are inexorably
coupled. That is, the
$
profile affects the concentra-
tion profiles which in turn affect the
J,
profile, and
which is the Donnan potential as given in Equation
13. Furthermore by equa ting the two expressions for
q
we see that
lCI-I+,lNal,
=
[Cl-lL,[Na+l-, so on.
We have dealt in some detail with the Donnan
equilibrium because the general nature of the
results
which we shal l discuss shortly. Basically the thermo-
which is the Gibbs-Donnan condition a s in Equation
we have the more general statement
2. (Note tha t bY
Equations 16a and 16b
applies to many other equilibrium systems, one of
lC1-l[Natl = [C-J-,[Na+]_,
=
constant
dynamic approach gives the-concentrations and po-
at every point x . )
Second, we now finally see the specific charge sepa-
ration that gives rise to the electrostatic potential.
There exist narrow space-charge regions on the two
sides of th e membrane; in solution 1 he space charge
is negative and in solution 2 it is positive, with of
course
.r:& = -
J1:pbr
(24)
The potential, $, and the ion concentrations, a+]
and [Cl-I, vary continuously in these regions to their
final values in the remote regions.
A
comparison of
Figure 3 with Figure 4 clearly illustrates the differ-
ences between the thermodynamic and th e statistical
mechanical point of view. In the former case, the
$
and concentration functions are constant a t thei r re-
mote values on the two sides of the membrane, with
a discontinuity in these values occurring at the plane
of the membrane; in the latter case these functions
are seen to vary continuously from one remote region
to the other.
The ex ten t of the space-charge regions is deter-
mined by the Debye length, L,) Eq. 23). Roughly
speaking, p falls about e-fold for every Debye length.
For 0.1
M
salt solution in water
( E =
801,
L,, =
10
A;
thus the region where electroneutrality is signifi-
cantly violated extends about 40-50
A,
too small for
direct sampling with a pipette. Note from Equation
23 th at L,, is directly proportional to 4 2 nd inversely
proportional to f l a a Thus in low dielectric con-
stant media, L,, will tend to be smaller, whereas in
media of low ionic strength,
L,,
will be larger. We
shall refer back to this point later when discussing
ion distributions in a lipid, or hydrocarbon, phase.
Finally it should be realized that the system is a
typically nonlinear one, which makes it difficult to
describe how the J, and concentration functions are
tentials far from the membrane, or interface,
whereas the Poisson-Boltzmann treatment explicitly
describes how the membrane potential arises from
space-charge regions near the membrane; electroneu-
trality holds only a t remote (many Debye lengths)
regions.
PHASE-BOUNDARY POTENTIALS. In th e Donnan equilib-
rium, the asymmetry of permeable-ion distributions
(the Donnan condition) and the membrane poten-
tia l ar ise because a macro-ion in one of the aqueous
solutions
is
impermeant. Here we consider the case in
which no impermeant ion is present, but the two
phases are different (for simplicity we consider them
immiscible); for example, one phase
is
water and the
other is oil (Fig. 5).
Thermodynam i c anal y s i s . From Equations
1
and 2b
we have
pi;;,+,,
+ RT In a+], + F$,
= p l&t )2
RT In [Nail2
+
F$2
(25a)
+ RT In [Cl-1,
-
F $ ,
= p;: )l-)s
RT In [ClV], - F$2
(25b)
These are the same equations employed for the Don-
nan equilibrium (Eqs. 10a and lob); this time, how-
ever, the p s are not the same on sides l and 2,
because the solvents are different. The conditions of
electroneutrality for remote regions also give
a+], = [Cl-I, = [NaCIl, (264
[Na+12
=
[Cl-I,
=
[NaCll, (26b)
Substituting Equation 26 into Equation 25 and add-
ing we obtain
For the t rue Donnan case, where N
is
not fixed
at
a constant
value to th e right of the membrane, we would h ave
an
Equation
16c, stating th at N satisfies the Boltzmann dis tributio n for x
>
0.
The final result is not much different from th at shown in Fig. 4 ,
except that
N
is perturbed upward near the membrane.
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CHAPTER
6:
PHYSICAL
PRINCIPLES
AND
FORMALISMS
OF
ELECTRICAL
EXCITABILITY 167
Upon substituting Equation 26 into Equation 25 and
subtracting we also obtain
*O/\V =_ ( 2 - 1 )
(28)
(dk+,] P$L)J - (PL:%+)y PII.l-),)
-
2F
Equation 27 is an expression for the oil-water (o/w)
partition coefficient
of
NaCl in terms of the standard
chemical potentials of Na+and C1- in the two phases.
Equation 28 gives the phase-boundary potential in
term s of these same quantities.
It is instructive to transform these expressions into
ones containing the intrinsic partit ion coefficient of
each ion. At the boundary between the two phases
there is a discontinuity in standard chemical poten-
tial and hence a discontinuity in ion concentrations.
Of course, in (classical) reality there are never actual
discontinuities; all functions are continuous. Never-
theless, because th e boundary between two phases is
established through short-range van der Waals inter-
actions, the phase transition occurs over molecular
distances (-2 A) and hence can be practically treated
as a discontinuity, compared to the space-charge
re-
gions that can extend over tens or even hundreds of
angstroms. Thus the p*.))s,nd hence t he ion concen-
trations, can change precipitously over a distance
where the electrostatic potential,
$,
remains un-
changed.s With this in mind, and assuming that
there is not a layer
of
dipoles at th e interface, we can
write for th e conditions at the boundary
Equations 30a and 30b are expressions for the intrin-
sic partition coefficients
G.1
of Na+ and C1-, respec-
a Image forces, which make a major contribution to the intrin-
sic partition coefficient, extend over distances longer than the
phase-transition region, but these are still in general much
shorter than the space-charge regions.
tively. That is, they a re th e partition coefficients tha t
would be observed in remote regions
i f
somehow elec-
trostatic interaction among ions did not occur. Sub-
stituting Equation 30 into Equation 27, we have
p o / w = d(PNa+)(PCI-) (31)
and substituting Equation
30
into Equation 28 gives
,
(32)
Equation 31 states that the macroscopically ob-
served partition coefficient for NaCl is the geometric
mean of the individual partition coefficients
@ )
for
Na+ and C1-. Note th at if one of these approaches
zero, the parti tion coefficient for the sal t approaches
zero. Equation 32 states that the phase-boundary
potential is determined by the rat io of the individual
partition coefficients. Note that if these are equal,
the phase-boundary potential is zero. Also note that
in contrast to the Donnan potential, the phase-bound-
ary potential is not a function of the NaCl concentra-
tion (provided, of course, the partition coefficients
are
not concentration dependent).
Stat is t ical mechanical analys is .
Had we stopped
our thermodynamic analysis with Equations 27 and
28, we would have been in much the same position we
were in with our thermodynamic treatment of the
Donnan equilibrium; the concentrations of NaCl are
different in the two phases, and there is a potential
difference between th e two phases (Fig. 6). Again th e
formal expressions are perfectly correct, and again
the origin of the boundary potential
is
obscure. By
extending our thermodynamic analysis to the parti-
tioning occurring a t the boundary, we have given a
fairly strong hint as to the source
of
the boundary
potential. Clearly, applying the Poisson-Boltzmann
analysis with these boundary conditions will show
that space-charge regions exist in both the aqueous
and oil phases. Figure 6 will then be transformed into
the more complete Figure 7. We shall not go through
the formal treatment since, aside from the mathe-
matical details, there are no new physical principles
that have not already been considered for the Donnan
case. The extent of the space-charge regions in the
two phases will depend on their respective Debye
I
ql=O
I
FIG. 6 . Concentrations and potentials in the water and oil
phases. The potential, 2, in the oil phase i s positive, because we
have assumed that the partition coefficient for sodium, pNa+,
between oil and water
is
greater than that for chloride,
p c , - .
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168
HANDB OOK OF PHYSIOLOGY -
THE
NERVOUS
SYSTEM
I
lengths. Since
L D =
Jmhere tends to be a
compensating effect between dielectric constant and
salt concentration. Thus a low dielectric constant by
itself would lead to a smaller Debye length, but in
general a low dielectric constant
is
accompanied by a
small partition coefficient of the salt between water
and the low dielectric constant phase, which leads
to
a larger Debye length. In the cases we consider later
of membranes with an essentially hydrocarbon inte-
rior, the concentration term strongly predominates,
so that the space-charge region in the membrane
interior is much more extended than the one in the
aqueous phase. Of course, regardless of the extent of
the space-charge regions, overall charge conservation
must always obtain
pdx = - 1: p d x
(24)
We might also note that a truly continuous treat-
ment would not show the discontinuities in ion con-
centrations at = 0, as is depicted in Figure
7,
but
rather would show
a
continuous transition over a
distance of a few angstroms. Such a t reatment would
require a "van der Waals, image force,
. . .
-Boltz-
mann" analysis of this region, in analogy to the Pois-
son-Boltzmann treatment. Needless to say, the much
more complex nature
of
these forces makes such an
analysis extraordinarily difficult (if not impossible).
il ,
Electrodes - he Measurement
of Potential Difference
Up to this point in our discussion of potentials
associated with ionic systems, we have not described
how one goes about measuring these potentials. Be-
fore considering other examples
of
membrane poten-
tials, we must discuss this problem. We are con-
I
*O/
w
N a t , q
*--- - -
N o t l $/'
I /
A
r
I
I S o l u t i o n
1
I S o l u t i o n 2 I
FIG.
8. Method for measuring the potential difference across a
membrane.
cerned not with the technical questions of which am-
plifiers to use
o r
what brand of oscilloscope is best,
but rather with an important theoretical question
(which also happens to have important practical im-
plications). The basic problem is the following: in
order to measure the potential difference across a
membrane, we must insert a pair of electrodes into
the system- ne electrode on each side of the mem-
brane (Fig. 8). By necessity there will exist
at
each
solution-electrode (soln/elec) interface a potential,
generally called an
electrode pot entia l .
Thus the po-
tential th at we measure,
Y,,, , ,~i5u,. , ,c, ,
s in principle the
algebraic sum of three potentials: the membrane po-
tential (the quantity
of
interest) plus two electrode
potentials
-
q'measured
-
Vrrnernbrane
4-
qeleclsoln I +
q s o i n
2leier
(33)
How then do we make contact with the solutions so
that the sum of the last two terms in Equation 33 is
negligible? This
is
crucial, for when we go to measure
a membrane potential, we want to measure a quan-
tity tha t is a unique property of the ionic system and
not a quantity that is dependent on the particular
pair
of
electrodes we happen to choose.
To illustrate more concretely the problem
of
meas-
urement, consider again the Donnan system of Fig-
ure 1. With two theoretical formulations we have
shown that
RT [Cl-1,
In~
[Cl-1,
- --
* ,,.ml,r;,nC.
-
( 1 3 )
and the question now is, can we measure it? Suppose
we use reversible Ag-AgC1 electrodes. [The electrode
reaction i s AgCl + e + Ag")'+ C1- (soln).] What then
is
~ , , , ( . ; l S U l . ( . t , ?
The answer turns out t o be zero. Let us
see why. The "electrode potential" of a Ag-AgC1 elec-
trode in contact with a solution containing C1- is
where qo s the so-called standard potential. (In thi s
case it is the potential
of
the half cell when the
solution is one molar in chloride.9 If we keep our sign
FIG.
7. Sketch of concentration and potential profiles for a
phase-boundary equi librium a s determined from the Poisson-
Boltzmann analysis (cf. Fig. 6, which is the result
of
the thermo-
dynamic analysis; as in
Fig. 6, P , ~ . >
p c i - ) . The space-charge
density is shown in th e lower par t
of
the figure.
It is interesting to note that Eq. 34 can be derived by exactly
the Same methods we have employed in treating our pure ionic
systems. Thus, since the system is in equilibrium and
C 1 ~
an
move between the two phases (solid and solution), we have from
our thermodynamic relation
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convention straight we then have
RT
+ (*(,
- p
n [Cl-I,) = 0
r
* i * Iv r / w1 n
2
The electrode potentials exactly cancel the mem-
brane potential, and operationally we measure no
potential difference
at
all. We certainly made a poor
choice of electrodes
We could have predicted th is result purely from the
second law of thermodynamics without going
through the above algebra. Since each electrode
is
in
equilibrium with its solution and the solutions are in
equilibrium (Donnan) with each other, the entire
system must be in equilibrium. If there indeed were a
potential difference between the electrodes, we could
construct a perpetual motion machine of the second
kind. That is, we could connect a load between the
two electrodes and do work. At one electrode, C1-
would go into th e solution, and at the other electrode,
C1- would come out; the overall chemical composition
of the solutions would not change. If the Donnan
condition became perturbed by the transport of NaCl
from one solution to the other, we could merely pause
for a while and let the Donnan condition reestablish
itself (utilizing, of course, only thermal energy).
When one electrode becomes almost depleted of AgC1,
we merely switch the electrodes from one solution to
the other, a process tha t, in principle, requires negli-
gible work. Thus we could indefinitely convert ther -
mal energy into work without any other change in
the universe -a clear violation of the second law of
thermodynamics.
Since a reversible pair of electrodes will always
give =
0,
we must try something different.
Why not use a pair of stainless-steel wires? We might
indeed measure the correct Donnan potential, but
then aga in we might not. The problem is that there is
not a well-defined process dominating the potential of
k I (solid) = k 1 soh)
&? solid) - FJlplrc
&','
(sold
+
R T In
ICI-I
- F&,
or
and rearranging, we obtain
Eq.
34 where
and
~ ' , 4 , . , h < , l ( elPC - l W d
1
Po = j
(d'i) solid) -
i' (soh))
Similarly a Poisson-Boltzmann analysis would show an extended
space-charge region in the solution near the electrode and a very
narrow one in the electrode itself.
the steel wires. What is the dependence of thi s poten-
tial on Na+ and C1- (or in th e more general case of the
Donnan equilibrium, on any other permeant ions
present in the system) concentration? What effect
does the macro-ion N have on this potential? It is
possible that none of these have significant effect on
the electrode potential, and therefore the two elec-
trode potentials will be equal and cancel each other
out, leaving the Donnan potential a s the only quan-
tity measured. But we cannot be sure , since we have
no theory to work from.
It turns out that we
can
measure the membrane
potential by introducing appropriate sa lt bridges. In-
stead of putt ing t he Ag-AgC1 electrodes directly into
the solutions, we place them into 3
M
(or saturated)
KC1 and make contact to the solutions through t he 3
M
KCl (Fig.
9) .
Now at
first
glance it might appear
that things are even worse, because the measured
potential is now the sum of five potentials instead of
three:
In addition to the two electrode potentials, there are
now two liquid junction potentials -one between 3 M
KCl and solution 1 and one between 3 M KCl and
solution 2. On reflection, however, it is clear that
things are not worse than before, for since the Ag-
AgCl electrodes are in identical solutions (i.e., 3 M
KCl), the electrode potentials must be equal and
hence cancel. This then leaves the two liquid junction
potentials with 3
M
KC1, and it t urn s out that these
are small, both because K+ and C1- have the same
mobility and because their concentration is large
(40).
Given a sal t bridge with a large concentration of
a salt whose ions have equal mobilities, it can be
shown tha t the liquid junction potential between this
bridge and any "reasonable" solution is small
(40).
(KCl happens to be the most convenient salt to use.)
Thus the only term left in Equation 35 hat is either
not negligible or does not cancel out is 9m ml ri nhe
quantity we wish to measure.
Note t hat , although solutions 1 and 2 in our Don-
nan example contain C1-, the presence of thi s partic-
ular ion is not relevant to our reason for using 3 M
KCl; these bridges would be equally effective with
chloride-free solutions. Also note th at since the metal
Ag/AgCI, ,Ag/AgCI
rne m i
one
FIG. 9. Method of measuring membrane potential by making
contact with the solutions through
3
M
KCI
junctions.
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HANDBOOK
O F
PHYSIOLOGY HE NERVOUS SYSTEM I
electrodes are in identical solutions, any identical
pair of electrodes in theory could be used (even stain-
less-steel wires). For practical reasons of stability and
convenience, the most commonly used electrodes ar e
either Ag-AgC1 or calomel (Hg-Hg,Cl,).
It should also be clear that the measurement, by
use of the arrangement shown in Figure
9,
of a finite
potential difference for a Donnan equilibrium is not a
violation of the second law, since with the salt
bridges present, the complete system of membrane
and
electrodes is not in equilibrium.
Finally, we must warn the reader who wishes to
pursue th is subject
of
the measurement of membrane
potentials fur ther tha t he will come across the view of
certain purists who claim that, since one can only
measure ~ l , l , , l , , , ~ , . , l l , , . through some such artifice as the
introduction of liquid junctions, it is not sensible to
even talk about membrane potentials. That is, it
is
impossible to assign values to the individual te rms in
Equation
35,
and hence one can only talk about
*,)
),,,,,
In fact, there are some ingenious argu-
ments that prove that most of the measured Don-
nan potential occurs not a t the membrane but a t the
contact between salt bridges and solution. It is not
our purpose to contribute to these polemics. We hope
that the Poisson-Boltzmann analysis of the Donnan
equilibrium will convince anyone th at indeed there is
a membrane potential intrinsic to that system and
that our treatment of the phase-boundary potential
(and other systems we shal l discuss shortly) will also
convince the skeptic of the reality of membrane po-
tentials. It might also be germane to point out that if
electrophysiologists had taken the purists critique of
membrane potentials seriously, the subjects of elec-
trical excitability, receptor potentials, and postsyn-
aptic potentials would never have gotten
off
the
ground, and most neurophysiologists today would be
out of business.
Quasi-equilibrium S,ystems
Before we take up the problem of diffusion poten-
tials involving the flux equations, we consider two
nonequilibrium systems t ha t a re sufficiently close to
equilibrium that they can be treated with good accu-
racy by the methods already employed. The consider-
ations developed here a re particularly relevant to our
future discussions of bilayer membranes.
SITY. uppose tha t we modify the Donnan system in
Figure
1 so
that the solution with the macro-ion
separates
two
solutions of NaCl (Fig. 10). Further-
more, let the macro-ion, N, be uniformly distributed
and immobile (as in the ion-exchange resin we dis-
cussed previously (see subsection
Statistical mechan-
ical analysis (Poisson-Boltzmann equation).
Then
the middle compartment (m) of Figure
10
is a mem-
brane (in point
of
fact, an ion-exchange membrane)
separating solutions 1and 2 . Let us also assume that
A
MEMBRANE
WITH A
V E R Y LARGE FIXED-CHARGE DEN-
the concentration of fixed charge is large compared to
the concentration of NaCl in the two compartments;
that
is,
[Nl
+
[NaCI],, [NaCll,. If the membrane
thickness is large compared to the Debye length in m,
then with solutions of equal concentration on the two
sides, there exist two Donnan potentials of the same
magnitude,
as
shown in Figure
1 l A
(where for the
sake of concreteness we take z = + l ) . igure 1 l A is
simply a symmetrical duplication of Figure 3 . At each
interface, there is a large Donnan potential jump
between solution and membrane, but there is no
potential difference across the membrane, because
these two jumps are equal.
Now consider the case in which th e concentrations
of NaCl in solutions
1
and 2 are not equal (e.g.,
[NaCll, < [NaCl],). This system is no longer in equi-
librium, and NaCl will diffuse slowly from solution
1
to 2; within the membrane there is a concentration
1 ) m ( 2 )
FIG. 10. A n ion-exchange membrane separating two solu-
tions.
Concentra t ion
I /
/i
P o t e n t i a l
p r o f i l e
63
\ C o n c e n t r a t i o n
I I p r o f i l e s
w
Pot en i a I
o r o f i l e
= O i
1)
( m )
2
FIG. 11.
A:
concentration and potential profiles for an ion-
exchange membrane of large positive fixed-charge density sepa-
rati ng two solutions of equal concentration of NaCI. B : same as
A , except tha t NaCl concentrations in compartments 1 and
2
a r e
unequal. (T he Na and C1- profiles within the m embrane have a
small negative slope tha t is not clearly seen in the figure.)
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EXCITABILITY
171
gradient of Na+ an d C1- -the concentration profiles
within the membrane have a finite slope. However,
the high concentration of fixed charge, N +, permits
only a very small concentration of Na+ in th e mem-
brane; thus the salt concentration gradient will be
quite shallow, and diffusion of NaCl across the mem-
brane will be very slow indeed. To a
first
approxima-
tion we can therefore neglect th e finite slopes within
the membrane and the slow flux of NaCl across the
membrane, and tr eat t he system by our equilibrium
methods. Thus, as in the case when the concentra-
tions of NaCl in solutions
1
and
2
were equal, we
again have two Donnan equilibria. This time, how-
ever, they are not identical, and the re exists across
the membrane a potential difference (Fig. 11B).
For highly charged membranes, the counter ion is
virtually the only mobile ion present within the
membrane. Thus the membrane is "permselective"
for C1- (if z = -1, it is permselective for Na+). Note
th at this permselectivity arises purely from the Don-
nan effect and is not dependent on an y steric factors.
On the basis of th e permselectivity for C1-, we can
immediately calculate the membrane potential from
our thermodynamic equilibrium criterion
k T ( 1 )
= k T ( 2 )
pi ;' +
RT
In [Cl-I, - FICII
= & ;)-
+
R T In [Cl-I, - FI, I~
If instead the membrane contained a negative fixed
charge,
it
would
be
permselective for Na+,and by the
same argument as above we would obtain
R T "af],
Y = + - l n -
F "a+],
In general, for a membrane that is permselective for
an ion, i, of valence z (regardless
of
the mechanism
for the permselectivity), we obtain, by equating the
electrochemical potential of the ion on the two sides
of the membrane
(37)
Equation
37
is often called the
Nerns t
equation, and
v'
is called the Nerns t potent ia l .
It is instructive to see how the membrane potential
can also be derived from the algebraic sum of the two
Donnan potentials. Since N+
s
very large, within the
membrane [Cl-1,
--
"+I. The Donnan potential be-
tween solution
1
and th e membrane
(m)
is
and between solution 2 and the membrane
Combining we have
which is Equation 36. Thus the transmembrane po-
tential is made up of the difference between two large
Donnan potentials.
We have been assuming that the membrane thick-
ness is large compared to the Debye length within it ,
and we have therefore been able to draw the concen-
tration and potential profiles as in Figure
11,
without
worrying about the very th in space-charge regions. If
the membrane thickness and the Debye length were
comparable, the continuous profiles in the space-
charge regions would have to be explicitly calculated.
Also it would now be meaningless to speak of the
transmembrane potential as the sum of two Donnan
potentials
at
each interface, since the distinction be-
tween interface and electroneutral membrane inte-
rior no longer exists; the space-charge region extends
throughout the enti re membrane. To explicitly calcu-
late th e membrane potential would require t he solu-
tion of the Poisson-Boltzmann equation. It turns out,
however, that even in such a thin membrane, vir-
tually the only mobile ion present is C1-, for which
the thin membrane is still permselective. Thus our
equilibrium results are still applicable, and the mem-
brane potential will still be the Nernst potential for
c1-.
AN
OIL MEMBRANE.
Let us now, in the same way th at
we extended the Donnan system of Figure
1
o make
an ion-exchange membrane in Figure 10, extend the
water-oil system of Figure 5 to make an oil mem-
brane bounded by water phases
as
in Figure 12.
Suppose, to start with, that the membrane
is
thick
compared to
the
Debye length within it and that
the
NaCl concentrations on the two sides differ. If we
assume tha t the partition coefficients of one or both of
the ions is very small, then again we can neglect the
small gradients of NaCl within the membrane and
the slow flux of NaCl across the membrane and
treat
the system
as
being
at
equilibrium. The concentra-
tion and potential profiles are shown in Figure 13,
where for the sake of concreteness we have made the
Na+ partition coefficient considerably larger tha n
th at of C1-. We see that there are two large positive
(H,O)
; (o i l ) ;
H,O)
1) m ) ( 2 )
FIG. 12.
An oil membrane separating two NaCl aqueous solu-
tions,
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172 H ANDBO O K O F
PHYSIOLOGY
- THE NERVOUS SYSTEM I
Not
1 - 1 -
I
2
I
I
Concentration
:
( o i l )
I
Na+.CI- prof I
les
I +m I
I I p r o f i l e
-
Potential
I
1
I I
I I
I
F IG . 13 .
Concentration and potential profiles for a thick oil
membrane separating two NaCl aqueous solutions. The potential
within the membrane
($,)
is positive, because we have assumed
that N a+ partitions better into the membrane than C1-.
phase-boundary potentials, but they are equal and
given by Equation
32
As we pointed out earli er, the phase-boundary poten-
tial is not
a
function of the NaCl concentration in the
aqueous phase. Even though N a+ is much more fa-
vored in the oil phase than C1- and consequently
there are large phase-boundary potentials, the total
membrane potential is zero. This result contrasts
sharply with that for the high-density fixed-charge
membrane we discussed earlier,
across which the
Nernst potential appears.
The above analysis was predicated on the mem-
brane being thick compared to the Debye length
within it. Consider now a membrane of thickness
comparable with t he Debye length. Instead of Figure
13,
we must now draw the complete ionic profiles
including the space-charge regions, since there is no
electroneutral region within the membrane. The pro-
files in Figure 14 ar e an extension of those in Figure
7 . For comparison we have redrawn Figure 13 in
Figure
14A,
exaggerating the space-charge regions
for
a
thick membrane; in Figure 14B we have ex-
panded the scale, since the space-charge regions ex-
tend through the enti re membrane.
To calculate the membrane potential we cannot use
Equation
32,
but must solve the Poisson-Boltzmann
equation. By inspection of the profiles, however, we
see
that Na+
is
virtually the only ion in the mem-
brane; that is, the membrane is permselective for
Na+. Consequently the membrane potential mus t be
given by the Nernst potential for Na+
This is true provided the mobilities of Na+ and C1- in the
membrane are equal. If they are not, then,
as
we shall see in a
later section, there will be a diffusion potential due to the asym-
metry in mobility. Nevertheless our main conclusion continues to
hold; namely, the phase-boundary potentials do not contribute to
the membrane potential.
Thus for the thick membrane (membrane thickness
* LJ
the membrane potential is zero, whereas for
the th in membrane (membrane thickness c , ;
if
c ,
> c p ,V,, the potential a t
I =
0 )
would change sign. T hat is, th e plots for
u
>
u
and
u < u
would
be interchanged.
current-voltage characteristic of thi s circuit for u
=
u,
u
> u, and u oes
change during
the
tran-
FIG.
21. Concentration profiles within the membrane for the
bi-ionic case of a membrane separating 0.1
M
NaCl from 0.1
M
KCI. A , I = 0; B , I is large and positive; C, I is large and negative.
10.1M NaCl
? I
FIG. 22. Sketch of the steady-state current-voltage character-
istic for the bionic case of Fig. 21. Note the characteristics(dashed
l ines) for a membrane separating either symmetrical 0.1 M NaCl
solutions or symmetrical 0.1 M KCl solutions, which the bi-ionic
characteristic approaches asymptotically for large positive and
negative potentials, respectively. Also shown are the chord and
slope resistance lines at point P .
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178 HANDBOOK OF
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- THE
NERVOUS
SYSTEM I
T
t I
F I G . 2 3 .
Nonlinear steady-state current-voltage characteristic
with chords drawn at
I , ,
9,)nd
I x ,
qJ. f current is stepped
fro ml, toZ,, the voltage originally attains the value a t P and then
increases with ti me along the vertical line from
P
until it reaches
the value q 2 . imilarly, if curre nt is stepped from I , to I,, the
voltage originally attains the value at Q and then moves along
the vertical line from Q until it reaches the value 9,.
sient from one steady state to another (81.1
The current-voltage characteristic in Figure 22 also
illustrates several concepts tha t will be important for
our later discussion of excitable systems: the system
showsrectification; tha t is, for voltages (i.e., values of
q
-
of the same magnitude but opposite sign, the
(absolute values of the) currents a re
not
equal. Also,
as for any circuit element with a nonlinear current-
voltage characteristic, the term
resistance
is ambigu-
ous and has a t least two reasonable meanings. One is
the chord resistance given by
(58)
In this case, since
\If,,
s not voltage dependent, the
chord resistance is identical to the integral resist-
ance. Another equally valid meaning for resistance is
the
slope resistance
given by
(59)
Both of these terms are illustrated in Figure
22
at a
given value
of
q.
n discussing membrane resistance
for a nonlinear system, one must always specify the
resistance referred to.
Of
course,
for
linear current-
voltage characteristics, such as in Figure 19, the two
resistances are identical, and no distinction need be
made.
It is very instructive to consider how the system
passes i n t im e from one steady-state point to another
(Fig. 23, which is a redrawing of the characteristic in
Fig. 22). Suppose we are sitting at the point ( I I ,PI)
and suddenly step the current up to I,. The ins tanta-
neous voltage will be given by I , times the chord
resistance at
I , ,
since the ionic profiles have not had
time
to
change. As they rearrange themselves, the
resistance rises (Na+
is
replacing K + in the mem-
brane) and the voltage increases until it attains the
steady-state value for I,. Similarly, if the current is
suddenly stepped from
I ,
to
14,
he instantaneous
voltage will be given by
I ,
times the chord resistance
at
I,,
and then the (absolute value of the ) voltage will
decrease
as the ionic profiles rearrange. The re-
sponses are schematized in Figure 24.
We see that the membrane resistance is not only
nonlinear, it is also
t ime uariant .
The time depend-
ence enters because of the time required for the ionic
profiles
to
shift from one steady-state configuration
to
another.
As
the profiles are shifting, the integral
resistance is continuously changing. (Y,, is also con-
tinuously changing, but we are neglecting that in
this discussion.) Note also th at the response in Figure
24A is phenomenologically similar to the response of
the RC network in Figure 25A, whereas the re-
sponse in Figure
24B
is phenomenologically similar
to
the response of the RL network in Figure 2%. It
can be shown formally tha t the AC-impedance char-
acteristics of time-variant resistances can display
phenomenological capacitances and inductances (42).
To
summarize the time-variant aspect of the system:
the instantaneous response to a change in current (or
voltage) is along the chord resistance; the voltage (or
1
A
I
F I G . 24. The change of voltage with time in response to steps
of cur ren t. These responses follow from the a nalysis of Fig. 23 as
given in the legend and text.
P ?
FIG. 25. RC
( A )
and RL ( B ) networks that would give re-
sponses similar to those shown in Fig. 24.
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CHAPTER 6: PHYSICAL PRINCIPLES AND FORMALISMS O F
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current) then relaxes to the point on the steady-state
current-voltage characteristic appropriate for the
new current (or voltage).
We have been discussing th e behavior of a membrane
whose only property is tha t
it
allows stir ring on both
sides for maintenance of boundary conditions; other-
wise the membrane imposes no new restrictions on
ion movement. Our real example has been coarse
filter paper. We now wish to consider two examples
in which the membrane phase is not merely a con-
fined salt solution of composition grading between
that of the two aqueous phases bathing the mem-
brane. After discussing the two examples, we shall
comment on the general situation of membranes with
special properties.
Fixed-charge membrane. We have already consid-
ered the quasi-equilibrium situation th at arises when
the fixed-charge density of
a
membrane is very large
and the membrane separates a single salt at two
different concentrations. We now extend the treat-
ment of fixed-charge membranes to the case of lower
charge densities. Since the co-ion is no longer ex-
cluded, transport (when
Z
= 0) of ions and sa lt cannot
be neglected. (Even in membranes of large fixed-
charge density, significant fluxes of counter ions oc-
cur; e.g., for the bi-ionic case of NaCl vs. KCl sepa-
rated by
a
high-density negatively charged mem-
brane, there is, a t
Z
= 0, a flux of Na+ from compart-
ment
1
o compartment 2, and an equal and opposite
flux of K+ from
2
to
1.
Our treatment will of course
also apply to these cases.) We do not go through the
details of the analysis, but merely point out the high-
lights of the principles involved.
The general analysis of fixed-charge membranes is
a straightforward combination of the Donnan equilib-
rium and the flux equations
(71). It
is assumed that
the interfacial processes are rapid compared to trans-
port through the membrane interior; hence one as-
sumes that equilibrium is maintained a t each inter-
face a t al l times (even in the face of curren t flow). For
the fixed-charge membrane, this means that the
Donnan equilibrium is satisfied at each interface.
Thus th e concentrations of ions just within the mem-
brane satisfy the Donnan condition with respect to
the ions just outside the membrane. Subject to these
new boundary concentrations, the ions then move
according to the flux equations. Essentially then the
fixed charge has established new boundary condi-
tions for the Nernst-Planck regime of ions. Current
flow will shift the concentration profiles within the
interior of the membrane, but
it
is assumed th at th e
concentrations Ijust inside the membrane are not
perturbed by the current. Figure 26A illustrates the
concentration profiles for the single-salt case in a
positive fixed-charge membrane. If +I = 0, the
system reduces to the single-salt case for an un-
charged membrane, and if +I % [NaCll,,,, the sys-
HOMOGENEOUS MEMBRANES WITH SPECIAL PROPERTIES.
I
I
( 1 ) ( m )
( 2 )
FIG.
26.
Concentration profiles (A ) and potential profile ( B ) or
a positive fixed-charge membrane separating NaCl solutions at
different concentrations. Note the Donnan
jumps
in concentra-
tions and potential at the interfaces.
tem approaches the quasi-equilibrium case of a mem-
brane with a very large fixed-charge density, de-
scribed earlier. Note tha t in th is lat ter case the mem-
brane is almost exclusively permeable to C1- not
because the mobility of chloride in the membrane is
so much larger tha n th at of sodium, but because the
concentration of chloride is so much greater than that
of sodium. This illustrates a n important general prin-
ciple applicable to both electrolytes and nonelectro-
lytes: the permeability of a membrane for
a
species is
generally dependent on two factors. One is the mobil-
ity of the species in the membrane phase, and the
other is the ability of the species to enter the mem-
brane in the first place. Unless one has other infor-
mation, i t is impossible to tell which is responsible for
a
given molecule being poorly permeable.
For a fixed-charge membrane, the total membrane
potential
is
the algebraic sum of three terms: two
Donnan potentials at the interfaces, generally called
rr, and
rr2,
and a Nernst-Planck diffusion potential
within the membrane, called (GI, - 2m). Thus
(
60)
(The potential profile is illustrated for the single-salt
case in Fig. 26B.) To find - 2m) one must solve
the flux equations subject to the new boundary condi-
tions established by the Donnan equilibria at the
interfaces
(71).
The electroneutrality condition
within the membrane now takes the form
C
cj+ +
ZN -
X C ~ -
0 (61)
Note tha t for the bi-ionic case, the Donnan potentials
ml nd
rr2
are equal an d hence cancel, so that the
membrane potential in Equation 60 is given just by
the internal diffusion potential -
Oil membrane. The analysis of th is system is com-
pletely analogous to that of the fixed-charge mem-
brane, but the equilibria a t the boundaries are now
partition equilibria, not Donnan equilibria. Subject
to the new concentrations established just within th e
membrane, the ions once again diffuse according to
the Nernst-Planck flux equations. (The concentration
profile for the single-salt case is illustrated in Fig.
27.) Once again the membrane potential is the alge-
braic sum of three potentials: two phase-boundary
potentials and an inte rnal diffusion potential. Thus
9 = T(o/u)a +
T( u /o) z
+ ( I m - 2m)
(62)
= T +
~2
+ ( 1,
-
2m)
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HANDBOOK
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THE
NERVOUS SYSTEM I
As we pointed out earlier (see subsection
Quasi-egui-
Zibrium Systems.
AN OIL
MEMBRANE),
the phase-
boundary potentials are equal for the singe-salt case
and hence cancel. Thus, despite the relative solubili-
ties of the anion and cation in the oil phase, the
membrane potential will only be a function of their
relative mobilities in this phase. Therefore, for the
single-salt case, Equation 62 reduces to
u - u R T cZrn
In
J
=--
u + u F c,,,
On the other hand, the phase-boundary potentials
will in general be different for the bi-ionic case, and
all three terms in Equation 62 will contribute to
II/.
General considerations.
The two examples we have
jus t given illustrate the general approach taken with
membrane transport. One assumes that the mem-
brane can conceptually be divided into three regions:
two interfacial regions and an interior region. It is
assumed that interfacial processes are
so
fast that
equilibrium conditions prevail there. These processes
establish new boundary conditions Ijust within the
membrane, and transport then takes place through
the membrane interior according to the flux equa-
tions. (This analysis incidentally is equally applica-
ble to nonelectrolyte transport.) The membrane po-
tential is the algebraic sum of two equilibrium poten-
tia ls and one diffusion potential. Since the membrane
substance is discontinuous at the interface, there
exist discontinuities in concentrations and electrical
potential at each interface, but the assumption of
equilibrium is equivalent to assuming th at the elec-
trochemical potential of each species is nevertheless
continuous across the interface. [For example, in a
fixed-charge membrane there are jumps in concen-
tration and electrical potential at the interface (see
Fig. 261, but the electrochemical potential of each ion
on the two sides of the interface is the same. In fact,
these characteristic discontinuities of the Donnan
equilibrium were derived on the basis of equality of
electrochemical potential.] Of course, one could be
more sophisticated and expand these discontinuities
into the actual space-charge regions present at each
interface.
There are two points worth noting about this anal-
ysis of membrane transport and potentials. Fi rst , it is
possible that situations might be found where the
assumption of very fast interfacial processes relative
I C 1 n I
1 )
m )
( 2 )
FIG.
27. Concentration profile
for
an
oil
membrane separat-
ing a single salt at t w o different concentrations.
to diffusion within the membrane begins to break
down. It would th en no longer be possible to assume
equilibrium a t the boundaries, and a n explicit analy -
sis of the boundary kinetics would have to be included
in the overall transport. The previous type of analy-
sis, however, still gives a good qualitative picture
even in this instance. For example, suppose that
interfacial events were not fast enough to maintain
the Donnan conditions. Nevertheless the
sign
of the
change of ion concentrations across the interface can
be obtained from the Donnan analysis. That is, for a
positively fixed-charge membrane, t he anion concen-
tration will be elevated across the interface and the
cation concentration depressed. The interfacial po-
tentials will then be somewhat reduced from their
equilibrium values.
The second point is that the analysis presupposes
the possibility
of
dividing the membrane up into in-
terfacial regions and interior. Clearly this becomes a
very tenuous assumption when the membrane
is
of
the order of
50-p\
thickness, as with th e lipid bilayers.
In fact, the analysis breaks down.
It
is no longer
possible to break up the membrane potential as in
Equation 62, since the space-charge regions extend
throughout the membrane. Here, a n explicit analysis
combining the flux equations and Poissons equation
with the appropriate boundary conditions must be
used. It is interesting that for nonelectrolyte trans-
port across the bilayers (e.g., isotopic water flux), the
procedure of dividing the membrane into three re-
gions leads to predictions in good agreement with
experimental results
(18).
It appears that partition
equilibrium is attained rapidly with respect to diffu-
sion even through a very thin
(-50
A)
region.
MOSAIC MEMBRANES.e have been considering mem-
branes whose properties do not vary in a plane paral-
lel to the membrane surfaces; tha t
is,
the membranes
are homogeneous. We now wish to comment on mo-
saic membranes consisting of regions with different
permeability characteristics. For example, the mem-
brane may contain some regions permselective for
Na +, other regions permselective for
K + ,
and still
others relatively nonselective among univalent ions.
As we shall see later, such membranes are of direct
relevance to biological membranes, particularly ex-
citable membranes, and can be experimentally real-
ized with modified thin lipid membranes. At this
point we are not concerned with t he mechanisms for
the selectivities, but accept them as given. (If the
membrane consisted of cation and anion selective
regions, we could imagine that each region was a
patch of high fixed-charge density ion-exchange
resin; such membranes can be fabricated with ion-
exchange beads imbedded in an inert matrix .)
For a membrane with regions in parallel,
the
equivalent circuit has the form shown in Figure 28.
(For completeness we have included the membrane
capacitance, which we discuss in a later section.)
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CHAPT E R 6:
PHYSICAL PRINCIPLES
AND FORMALIS MS OF ELECTRICAL EXCITABILITY 181
I
- a
'.i
7
t
t 1
1
FIG
28.
Equivalent circuit for
a
mosaic membrane. The indi-
vidual elements are shown to
be
ideally selective
for
a given ion,
but this need not necessarily be the case.
Here we have indicated that in principle the conduct-
ances (g,)can be voltage dependent. (In treating par-
allel circuits, it is more convenient to use conduct-
ances than resistances.) The emfs , however, are in-
variant, being determined by the concentrations of
ions on the two sides of the membrane and the perme-
ability characteristics of the regions. If the regions
are permselective, as in Figure 28, each emf is the
Nernst potential for the permeant ion of tha t region.
This circuit should be contrasted to the one for the
homogeneous membrane given in Figure 16. In that
case there is a single conductance and one emf, both
of which can be voltage dependent. For the mosaic
membrane, we have from elementary circuit theory
z,
= gj 9
Ej)
(63)
for each element of the circuit. Summing over all
elements, we have from Kirchhoff
s
law
where
z = T Z I (65)
Equation 64 bears the same relation to the mosaic
membrane as Equation 38 bears to the homogeneous
membrane. Comparing the two, we see that each
equation has an Ohm's law term and also another
term that may be called the diffusion potential. In a
formal sense the two expressions are quite similar.
Although the flux equations describe the move-
ment of ions across the membrane, the equivalent
circuit of Figure 16 does not indicate that there is ion
flux when Z
=
0. In contrast, the equivalent circuit of
Figure 28 depicts the local currents th at flow through
the elements of the membrane even when no current,
I ,
is being passed across the membrane. Under these
circumstances, the membrane potential is given ex-
clusively by the second term in Equation 64, whereas
Equation 65 becomes
zzj = 0
j
It
should be understood that the dissipative pro-
cesses for the homogeneous and mosaic membranes
are quite different. In the former the ions flow
through a common region, and hence there are no
local current flows. In the mosaic membrane ion
movement is through local currents. Despite these
physical differences,
it
happens tha t the
steady-state
properties of a homogeneous membrane can also be
formally represented by the circuit in Figure
28 (20).
FORMAL CONSEQUENCES OF VOLTAGE-
DEPENDENT CONDUCTANCES
The N a ture
of
Electrical Excitability
In
the
subsection
Ion Transport (the Nern st-
Planck Flux Equations) we developed the properties
of a homogeneous membrane and showed that even
with such a simple membrane as coarse filter paper,
it
is
possible to observe rectification, nonlinearities,
and time transien ts in th e membrane potential. Since
these properties are also characteristic of electrically
excitable biological membranes, the question arises
whether the mechanisms operating in the simple
systems we discussed are also responsible for gener-
ating action potentials and bioelectric phenomena.
We remind the reader th at the nonohmic behavior we
have described is due to the shifting of the ionic
profiles of mobile ions within the membrane in the
face of an applied voltage or current; this change in
the ionic profiles leads to changes in the membrane
conductance and the membrane emf. The question
then is whether such a mechanism could account for
action potentials of nerve and muscle.
It
is very difficult to give a
definitive
answer to this
question. It can probably be proved that
it
is impossi-
ble to reproduce electrical excitability with a filter
paper membrane, even with the most bizarre mixture
of ions on the two sides of the membrane. This does
not preclude the possibility that a membrane with a
fixed-charge density varying as arc coth x3", or one
with some continuously varying function of the die-
lectric constant of
its
''oil" interior, or a combination
of these st ruc tures could not generate action poten-
tials. Although no one has demonstrated, either theo-
retically or experimentally, that one can develop an
excitable membrane simply from electrophoretically
moving ionic profiles within the membrane, neither
has anyone presented proof that it is impossible to do
so. [In fact, if one allows the ionic profiles to be
shifted not only by the electric field (i.e., electropho-
resis) but also by solvent drag produced by electro-
osmosis and hydrostatic pressure gradients, then one
can indeed develop, with a membrane made merely of
sintered glass, oscillatory behavior and complex all-
or-none responses th at ar e in some ways phenomeno-
logically similar to excitable tissue (72, 73).] There
are nevertheless strong indications for believing th at
such
a
mechanism cannot be the basis for biological
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HANDBOOK
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-
THE NERVOUS
SYSTEM I
excitability, and indeed there is compelling evidence
that excitable membranes are not homogeneous, but
rather mosaic structures. We shall discuss the bases
for the above assertions and a t the same time present
the current view of the na ture of biological excitable
systems.
REASONS F O R
BELIEVING
THAT ELECTRICAL EXCITAB IG
ITY
DOES NOT RESULT
FROM THE SHIFTING
OF IONIC
PROFILES. Th e magn i tude of rect if ication. In excitable
cells such as the squid giant axon, the ratio of the
conductance a t large depolarizations to the conduct-
ance at large hyperpolarizations can be several
hundred to one. Since the major conducting ions are
Na+ (on the outside) and
K +
(in the axoplasm), with
approximately equal total sal t concentration on each
side of the membrane, the situat ion is approximately
bi-ionic. In such a case, rectification is achieved by
exchanging in the membrane a more mobile ion with
a less mobile ion, depending on the sign of the poten-
tial. Assuming tha t the contribution of all other ions
to the membrane conductance can be neglected (an
extreme case), the limiting rectification ratio is given
by the ratio of the sodium and potassium mobilities in
the membrane (i.e.,
u K / u N a ) ,
hich is approximately
1.5 in free aqueous solution.
If
this phenomenon ac-
counted for the observed rectification ratio of several
hundred, either the potassium mobility within the
membrane must be abnormally large, or the sodium
mobility must be abnormally small, or both.
Th e t ime scale of the a ct ion potent ia l . In our discus-
sion of trans ien t behavior during ionic profile shifts,
we omitted the time scale. Since the relaxation of the
profiles is basically a diffusion process, the times
involved will be roughly those required for the root
mean square displacement
(?)I/*
of an ion t o be equal
to the membrane thickness. This
is
given by the well-
known result from Brownian motion theory
(13,
14)
x2 = 2Dt (66)
In aqueous solution, the diffusion coefficient, D,
for
small ions such as Na+ and K+ is of the order
cm2/s. Taking the membrane thickness a s
100
A
(lo- ';
cm), we find from Equation
66
that the time involved
for the displacement of concentration profiles is of the
order of s, that
is,
0.1 p s . The events associated
with a n action potential, occurring
as
they do on the
millisecond time scale, are four orders of magnitude
slower. We would therefore have to postulate that t he
diffusion constants (i .e. , mobilities) of ions within the
membrane are at least a factor of
lo4
times smaller
th an t he mobilities of ions in free solution.
Dissociation
of
sod ium and po tass ium conduc t -
ances. One of the distinguishing features of the
permeability changes occurring during a n action po-
tential (or under voltrage-clamp conditions) is th e dif-
ferent time course of these changes for different ions.
In th e squid giant axon, for example, sodium permea-
bility rises and begins to fall before potassium perme-
-
ability changes significantly. In fact, Hodgkin and
Huxley's analysis of excitability in the giant axon
was possible only because they could clearly resolve
and separate the time course of the sodium and potas-
sium conductance changes (29). Since then, t he belief
that these components are separate has been rein-
forced with the discovery of pharmacological agents
that inhibit one, but not both, conductances. For
example, tetrodotoxin (TTX), the puffer fish poison,
can completely and reversibly block the sodium con-
ductance changes without affecting the magnitude
and the time course of the potassium conductance
transients (49). Conversely, tetraethylammonium
(TEA) reduces the potassium conductance without
significantly affecting the sodium transients (2) .
It is difficult t o imagine a homogeneous regime
of ions where th e permeability changes for one ion are
not intimately coupled with those of the other ions. In
the bi-ionic case tha t we considered, for example, th e
movement of Na+ into or ou t of th e membrane was
accompanied by the obligatory and opposite move-
ment of K + . It
is
even more difficult t o conceive of an
agent, acting in a homogeneous membrane, that
could inhibit transients associated with one ion and
yet not affect those associated with the other ions in
the membrane. For these reasons, virtually all elec-
trophysiologists believe that excitable membranes
are mosaic structures of the form described in the
subsection
MOSAIC
MEMBRANES, and although Hodg-
kin and Huxley did not explicitly stat e this in their
basic papers (29-331, they clearly had this picture in
mind. Certainly the simplest way in which the so-
dium and potassium conductances could be function-
ally independent is for the regions of sodium and
potassium permeability to be physically separated.
HODCKIN-HUXLEY EQUIVALENT CIRCUIT. nalysis of
electrical excitability of the squid giant axon mem-
brane, and any other excitable membrane, starts
with the equivalent circuit for a mosaic membrane of
the type shown in Figure 28. The ions involved in the
circuit can vary depending on the particular biologi-
cal membrane
o r
the ions in the solution that bathes
the membrane.
Because much is known about the
squid axon, we use this system as representative of
the others, with the understanding t hat the pr inc i -
ples operative there are generally believed to apply to
all excitable membranes. For the squid giant axon in
normal seawater , the equivalent circuit of Figure 28
takes the particular form shown in Figure 29 (32) .
The sodium and potassium conductances are voltage
dependent, whereas the leakage conductance (g,)-
probably a lumping together of several ion permea-
bilities- s constant. The sodium and potassium
emf's are,
of
course, given by the Nernst potentials
for these ions, and are assumed to be constant, be-
cause the concentrations of these ions in seawater
and axoplasm are constant (unde r the experimental
conditions). Since [Na+loutside [Na+linside nd
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CHAPTER 6: PHYSICAL PRINCIPLES
AND
FORMALISMS OF ELECTRICAL EXCITABILITY
183
O u t s i d e ( s e a w a t e r )
c
I n s id e ( o x o p l a s m )
FIG
29. The Hodgkin-Huxley equivalent circuit for the squid
giant axon membrane in normal seawater.
[K+loutside < [K+linside>
ENa and EK are of opposite
polarity. Ultimately this system would run down
(Na+coming in and K + going out), unless some coun-
terbalancing process exists. This energy-requiring
process
is
the role assigned to metabolism. The leak-
age emf
(E,)s,
like the leakage conductance, proba-
bly a lumped parameter. Although the leakage ele-
ment modifies the behavior of the system somewhat,
the dominating factors are the voltage-dependent so-
dium and potassium conductance elements; for the
most part, therefore, we confine o u r comments to
these and ignore the leakage element.
The Hodgkin-Huxley description of the action po-
tent ial is briefly summarized as follows: in the rest-
ing state,
g,
>>gNa o tha t the resting potential sits
near EK.At threshold depolarization,
g N a
has in-
creased enough tha t the membrane furthe r depolar-
izes. This leads to a further increase in g N a , which
leads to further depolarization, and so on. The net
result of thi s process is the rising phase of the action
potential, which tends toward EN, a t
its
peak. The
falling phase results from a turning off of g,, (called
sodium inactivation) and a turning on of
g K ;
both
processes act to move the membrane potential back to
EK.
Regenerative behavior develops because the
membrane potential is determined by the relative
conductances (permeabilities) of the membrane to
potassium and sodium [if we neglect the capacitance
current, th e membrane potential is given at all times
by the last ter m in Equation
64,
which in this case is
(gNaEXa
+
g,EK)/(gNa
+
gK)],
while a t the same time
these conductances are functions of the membrane
potential. The functional dependence ofg,, and
g,
on
membrane potential is determined in voltage-clamp
experiments. Tha t is, the membrane potential is held
fixed (via external electrodes) at various levels and
the transient changes in the sodium and potassium
conductances recorded (29-31, 33).
The physics underlying t he voltage-dependent con-
ductances is still not understood; it
is
a major un-
solved problem in electrophysiolo