Auditory Neuroscience, Vol. 3(4), pp. 351-361
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Equilibrium Behavior of an Isotropic Polyelectrolyte
Gel Model of the Tectorial Membrane:
The Role of Fixed Charges
THOMAS F. WEISS* and DENNIS M. FREEMAN
Department o f Electrical Engineering and Computer Science and Research Laboratory o f Electronics, Room 36-857,
Massachusetts Institute o f Technology, Cambridge, Massachusetts 02139; Eaton-Peabody Laboratory o f Auditory Physiology,
Massachusetts Eye and Ear Infirmary, Boston, Massachusetts 02114
(Received 8 March 1996; Accepted 2 August 1996)
We describe the equilibrium behavior of an isotropic polyelectrolyte gel model that is
intended to help us interpret chemical, electrical, mechanical and osmotic properties of
the tectorial membrane (TM). The gel is homogeneous, isotropic, and contains water,
fixed ionizable charge groups and mobile ions. The gel is in contact with an aqueous ionic
solution (bath). At equilibrium, the gel characteristics are derived from the following
physical principles: macroscopic electroneutrality, electrodiffusive equilibrium, osmotic
equilibrium and Hooke’s law. These physical principles lead to a pair of coupled alge­
braic equations that are solved analytically for certain special cases and numerically in
general to yield the concentrations of all mobile ions, the concentration of fixed charge,
the osmotic pressure difference between the gel and the bath, the volume of the gel, and
the electric potential difference between the gel and the bath. The gel model indicates
that fixed charges play a key role in such properties of the TM as the capacity to con­
centrate ions, the occurrence of a difference in electric potential between the TM and the
bath solution, and the capacity to imbibe water and to swell. Using the gel model to fit
measurements of the electric potential of the TM (Steel, 1983a), we estimate that the con­
centration of fixed charge in the TM at neutral pH is in the range -6.4 to -8.4 mmol/L.
Using the measured biochemical composition of the TM (Thalmann et al., 1993), we esti­
mate that the fixed charge concentration of the TM due to its glycosaminoglycan con­
stituent is -18 mmol/L.
Keywords: Cochlea, tectorial membrane, osmotic responses, polyelectrolyte gel
numerous investigations, there is very little information
about the TM which is generally agreed upon.” This sit­
uation has changed recently. New methodologies have
yielded appreciable knowledge of the biochemical
INTRODUCTION
A comprehensive 1983 review of the mammalian tecto­
rial m em brane (TM) stated (Steel, 1983b) “Despite
♦Corresponding author. Tel.: (617) 253-2594. Fax: (617) 258-7864. E-mail: tfweiss@mit.edu.
351
352
T. F. WEISS and D. M. FREEMAN
composition, molecular structure and physicochemical
properties o f the TM.
The TM is a Connective Tissue
Biochemical Composition
and Molecular Architecture
Connective tissues (Hay, 1991) include such diverse
m aterials as blood vessel w alls, bone, cartilage,
cornea, ligament, skin, synovial fluid, tendon, the vit­
reous body and, apparently, the tectorial membrane.
In general, connective tissues contain a m atrix con­
sisting of m acrom olecules o f insoluble proteins (both
collagen and elastin fibers), soluble polysaccharides
(glycosam inoglycans or GAGs), small solutes includ­
ing ions, and water. The m acrom olecules contain ionizable charge groups that are fixed in the matrix. At
neutral pH, the charge groups o f the proteins are ion­
ized, but the net charge of the protein is near zero.
GA G s are polym ers o f disaccharides (e.g., chondroitin sulfate, keratan sulfate, derm atan sulfate and
hyaluronic acid) attached to polypeptide cores to form
proteoglycans. GAGs contain a high density of ionizable carboxyl and sulfate groups. A t a pH near 7 these
groups are ionized and are responsible for the pres­
ence o f fixed (nondiffusible) negative charges in the
tissue. These fixed charges electrostatically attract
m obile (diffusible) counterions to achieve m acro­
scopic electroneutrality. T hese m obile counterions
contribute to the osm otic pressure in the tissue which
induces osm otic w ater influx. T he resu ltan t tissue
swelling is opposed by the m echanical rigidity o f its
matrix. The GAGs, because o f their effect on im bibi­
tion o f w ater, give the tissue the capacity to resist
com pressive forces. In contrast, the protein fibers
give connective tissues their tensile strength and their
capacity to withstand swelling.
The m am m alian TM is a heterogeneous, acellular,
gelatinous structure that overlays the organ o f Corti
and is in close proxim ity to or in contact with the hair
bundles o f hair cells. The com position o f the TM is
becom ing clearer. The wet w eight o f the TM o f the
m ouse has been estim ated as 50 /ig (Richardson et al.,
1987; Thalm ann et al., 1987). Since the dry weight
has been estim ated to be 1.5 /ig, the TM is apparently
97% water. Approxim ately 50% o f the dry w eight of
the TM is protein, and about 40% o f the protein is col­
lagen, largely collagen type II although sm aller
amounts o f other collagens have also been reported
(T halm ann et al., 1986; T halm ann et al., 1987;
Richardson et al., 1987; Thalm ann, 1993). The TM
contains carbohydrates (K halkhali-Ellis et al., 1987;
Santi et al., 1990; Prieto et al., 1990; Sugiyam a et al.,
1991; Suzuki et al., 1992). It has been estim ated that
15-20% o f the dry w eight o f the TM is G A G s
(Thalm ann e ta l., 1993).
A prom inent feature o f the TM is the radial fibrillar
structure that can be seen in both fresh and fixed
preparations (Lim , 1972). U ltrastru ctu ral studies
reveal that the TM contains two types o f fibrils: type
A and type B protofibrils (K ronester-F rei, 1978).
Type A protofibrils are straight, unbranched, approxi­
m ately 10 nm in diam eter, collagenase sensitive and
trypsin resistant (H asko and R ichardso n , 1988).
B undles o f type A protofibrils apparently form the
radially oriented fibrillar structure o f the TM that can
be seen with light m icroscopy. Type B protofibrils are
coiled, branched, striated sheets o f fibrils 7 nm in
diam eter, collagenase resistant and trypsin sensitive
(Hasko and R ichardson, 1988). T ype B protofibrils
form a matrix in which the type A fibrils are enmeshed.
It seems likely that the type A protofibrils are com ­
posed o f collagen fibers and that the type B protofibrils
are proteoglycans w ith their high density o f GAG s
(Thalmann et al., 1987; Hasko and Richardson, 1988;
A rim a e fa /., 1990).
Material Properties
Connective tissues shrink and swell in response to a
variety o f changes in bathing solution com position
(Grodzinsky, 1983). Studies of osm otic responses of
the TM (Freem an et al., 1994; Shah et al., 1995;
Freeman et al., 1996) reveal that the TM swells when
sodium is substituted for potassium in the bathing
solution, when the calcium concentration o f the bath is
reduced, and at pH < 6 or pH > 9. T hese sw elling
responses have been interpreted as due to increases in
the concentration o f fixed charge in the TM.
GEL MODEL OF THE TECTORIAL MEMBRANE
353
Connective tissues exhibit an electric potential that
differs from the potential o f the bathing solution
(Grodzinsky, 1983). A negative electric potential is
m easured betw een the TM and the bathing solution
and the m agnitude o f this potential increases as the
osmolarity (and ionic strength) o f the bathing solution
is reduced (Steel, 1983a). This negative potential has
been interpreted as a Donnan potential that develops in
a gel that contains fixed negative charge.
Conclusion. Its biochemical composition, structure,
and material properties suggest that the TM resembles
a poly electrolyte gel (Tanaka, 1981) as do other con­
nective tissues (Steel, 1983a; Thalmann et al., 1993;
Shah et al., 1995).
Polyelectrolyte Gel Models
P olyelectrolyte gel m odels have been investigated
since the original w ork o f D onnan (D onnan and
Harris, 1911; Donnan, 1911; Donnan, 1924), and have
been reviewed extensively (Katchalsky, 1954). Such
m odels have been very effective in interpreting
diverse experim ents on both m an-m ade (Rieka and
Tanaka, 1984) and naturally occurring (Grodzinsky,
1983; M ow et al., 1984; Lai et al., 1991) gels such as
connective tissues. Interactions between fixed macrom olecular charge groups and m obile ions play a cen­
tral role in these gel models. These models indicate the
linkage between the m olecular architecture, biochem ­
ical composition and the material (mechanical, electri­
cal, chemical and osmotic) properties of gels.
In this paper w e develop a m acroscopic continuum
polyelectrolyte gel model o f the TM (Fig. 1) in which
we assum e that the m obile ions in the gel do not bind
to the gel binding sites .1First, we explore the effect of
fixed charge on the m aterial properties o f the TM.
Second, we estim ate the concentration of fixed charge
in the TM. The purpose of the model is to provide a
conceptual fram ework for measurements o f the m ater­
ial properties o f the TM and to guide further experi­
mental studies.
’A preliminary version of this work was presented earlier (Weiss
and Freeman, 1996).
Fixed ionizable groups
0 positive
0 negative
□ neutral
Mobile solutes
© cations
© anions
O uncharged
FIGURE 1 Polyelectrolyte gel model of the TM. Macromolecules
are represented in this figure by lines with fixed charge groups rep­
resented as square boxes that can be positive, negative or neutral.
Mobile solutes are represented as circles. The potential between the
gel and the bath is E.
THEORY
As shown in Figure 1, we assume the gel is immersed in
an aqueous ionic solution (bath). Let the concentration
of ion i in the bath be ct and in the gel be cf; its valence
is Zi- The gel is assumed to contain a macromolecule
which contains fixed charges with a charge density pf.
The volume of the gel is V and the hydraulic swelling
pressure of the gel is p. The potential of the gel minus
that in the bath is E. The gel is assumed homogeneous
and isotropic. The gel model consists of a set of funda­
mental relations which we describe next.
354
T. F. WEISS and D. M. FREEMAN
cK= ccl so that C1 = cK= c a . In this case, cK = cK/cK = 1
Electrodiffusive Equilibrium
Provided the m acrom olecular concentration is suffi­
ciently small so that the swelling pressure in the gel is
small (Overbeek, 1956), equilibrium of ion i between
the gel and the solution implies that the potential dif­
ference between the gel and the bath equals the N em st
equilibrium potential o f each ion, i.e.,
and ca - 1.
In term s o f the norm alized concentration, elec­
troneutrality in the gel is expressed as
Cf + ' £ lzicid Zi = 0.
(3)
Osmotic Equilibrium
which is called the Donnan potential. R, T and F are
the m olar gas constant, absolute tem perature and
F arad ay ’s constant, respectively. Therefore, ionic
equilibrium can be expressed as
At osmotic equilibrium between the gel and bath, the
flux o f water between the gel and bath is zero and the
hydraulic pressure difference betw een the gel and the
bath equals the osm otic pressure difference between
the gel and the bath. The condition for osm otic equi­
librium is
dZi = ( e -BFKmy i = ±
(2)
p - R T ^ c f = -R T '£ c i
... I- . ,
i
w here we call d the Donnan ratio. In term s o f the
D onnan ratio, the concentration o f ion i in the gel is
cf = C jdzl.
Electroneutrality in the Gel
W e assume that the total charge density in the gel is
zero, i.e., the gel obeys electroneutrality on a tim e
scale that is large com pared to the charge relaxation
time and over a distance scale that is large compared
to a Debye length (W eiss, 1996). Therefore,
where p is the hydraulic pressure o f the gel m inus
that o f the bath and is called the hydraulic sw elling
pressure o f the gel. If n is the osm otic pressure o f the
gel m inus that o f the bath then at osm otic equilibrium
p - n. W ith the use o f the D onnan ratio, osm otic
equilibrium can be expressed as follow s
£
= I< < p -
d c ,.
In terms o f norm alized variables, this relation yields
RTCX
^
P f + ^ Z iF c f = 0
i
which can be expressed in terms o f the Donnan ratio
(Eq. 2) as
y
+ E zicid Zi = 0 .
The quantity pf/F = Cf is the concentration o f fixed
charge, which can be either a positive or a negative
quantity. It is convenient to norm alize all concentra­
tions to half the osmolarity o f the bath so that c, = c /C ^
and Cf = C f/C i where
= (l/2 )£ , ct. W e illustrate the
normalization for a particularly simple bath— a binary
electrolyte. For example, if the bath were KC1 then
= (1/2){cK + cC[). Since the bath is electrically neutral
Mechanical constitutive relation
W e shall assum e that the gel is elastic and that the
hydraulic pressure and volum e changes are propor­
tional, i.e., they obey H ooke’s law, so that
w here Va is the volum e o f the gel when the fixed
charge density is zero, and /cis a stiffness bulk m odu­
lus. W e define the volum e ratio as v - V/V0 so that the
mechanical constitutive relation can be expressed as
p=
k (v
- 1).
(5)
GEL MODEL OF THE TECTORIAL MEMBRANE
METHODS
355
RESULTS
Equations 3, 4 and 5 can be combined to yield the fol­
lowing coupled simultaneous equations
C f + ^ ZiCtd Zi = 0 and
at|
£ (t> -
1
i
)
(
( 6)
7
)
w here k — k /(RTC^). E quation 6 expresses elec­
tro n eu trality in the gel and E quation 7 expresses
osm otic equilibrium betw een the gel and the bath.
E lectrodiffusive equilibrium and H ooke’s law are
incorporated in these equations to eliminate concentra­
tions in the gel and the sw elling pressure. Let us
assume that Cf is specified, the bath concentrations of
all solutes are known, and the modulus o f the gel is
known. Then the only unknowns in Equations 6 and 7
are the Donnan ratio d, and the volume ratio v. Hence,
in principle, these two equations can be solved for these
two variables. All other variables can be obtained from
d and V. However, the two equations are amenable to
analytic solution in special circum stances only. For
m ore general circum stances, we obtained numerical
solutions with M athem atica (W olfram , 1991) using
Newton’s method. The accuracy o f the numerical solu­
tions was assessed by computing the quantities
cf iB m m
and
K ( - 0 - 1 ) - X i d * — 1) Cj
for every sim ulation run. B oth eq and ep should be zero
(Eqs. 6 and 7); deviations from zero represent errors in
the num erical com putations. B oth errors represent
concentrations norm alized to half the osm olarity of
the bath: eq represents deviations from electroneutral­
ity and ep represents deviations from osmotic equilib­
rium. For all numerical results presented in this paper
eq < 10~10, i.e., the deviation of the charge concentra­
tion in the gel from zero is ten orders of magnitude less
than the osm olarity of the bath. Also ep < 10 15, i.e.,
the deviation of the osmolarity o f the gel from osmotic
equilibrium is fifteen orders o f m agnitude less than the
osm olarity of the bath.
W e exam ine the theoretical predictions o f the gel
model for two different assumptions about the fixed
charge concentration. W e start with a relatively simple
assumption to gain insight into gel osmotic properties
before considering the more general case.
The Role of the Fixed Charge Concentration
in a Stiff Gel
W e consider a gel model that yields analytic solutions
and considerable insights into many of the properties of
gels without undue algebraic complexity. We assume a
bath that is a binary electrolyte whose constituent ions
do not bind to the fixed charges in the membrane, and
a gel that has a large bulk modulus so that the change in
volume of the gel is negligible. Therefore, we have z+ —
+1, Z- = -1 , c+ = C, c_ —C, and
= C, so that c+ = 1
and c_ = 1, where these quantities are the normalized
concentrations of the cation and anion, respectively.
The normalized concentrations of the cation and the
anion in the gel are cf and cf, respectively. The normal­
ized fixed charge concentration is constant, i.e.,
Cf=C°f .
W ith these assumptions, all the key variables are func­
tions of one parameter, Cf.
A Gel can Concentrate Ions. Substitution of the con­
centrations into Equation 3 yields the quadratic equation
d 2 + C°f d - 1 = 0 ,
whose only physically plausible solution is
m'f
w m m
2
Therefore, the concentrations of ions in the gel are
c i = d = l 1+
c°f
\2
a
and
K 2
m
d
i+
c o \2
r°
c/
( 8)
356
T. F. WEISS and D. M. FREEMAN
The norm alized concentrations o f cations and anions
in the gel and bath are shown as a function of the con­
centration of fixed charge in the gel in Figure 2. W hen
the fixed charge concentration is zero, the concentra­
tions o f cations and anions in the gel are equal to their
bath concentrations,
= c l = 1. The total concentra­
tion of cations and anions, or the osmolarity, is twice
that value. As the fixed charge concentration is m ade
more positive, the concentration o f m obile cations in
the gel decreases and the concentration o f m obile
anions in the gel increases in order to satisfy elec­
troneutrality. W ith a positive fixed charge in the gel,
the concentration of the anion in the gel exceeds that
in the bath. In addition, note that the osmolarity is a
minimum when the fixed charge concentration is zero
and increases if there is an increase in m agnitude of
either the positive or negative fixed charge. As long as
the fixed charge differs from zero the osmolarity o f the
gel exceeds that o f the bath. Equation 8 shows that as
Cf—» « , c i —> Cf. In other words, as the fixed charge
concentration is m ade large and positive, the anion
concentration approaches the concentration o f fixed
charge. Similarly, as C° —> -°°, c% —» -C f. As the fixed
charge concentration is made large and negative, the
cation concentration approaches the negative o f the
concentration o f fixed charge.
FIGURE 2 Concentrations of mobile cations c f and anions c i in
the gel (thick solid lines) and mobile cations c+and anions c_ in the
bath (thin horizontal solid line) as a function of the concentration C f
of fixed charge in the gel in normalized coordinates. Also shown is
the sum of the cation and anion concentration in the gel (thick
dashed line) and in the bath (thin horizontal dashed line). In addi­
tion, asymptotes for large magnitude fixed charge concentrations
(dotted lines) are also shown.
These results dem onstrate fundam ental properties
of polyelectrolyte gels— the concentration o f counteri­
ons in the gel exceeds that in the bath and the gel
osmolarity exceeds that of the bath.
Gels Exhibit an Electric Potential that Differs from
that o f the Bath. From Equations 1 and 8, w e obtain
1+
c/
+
c°<
(9)
The Donnan potential is plotted versus Cf in Figure 3.
W hen the fixed charge in the gel is zero, the potential
difference between the gel and bath is also zero. If the
fixed charge is positive then the Donnan potential is
positive and if the fixed charge is negative then the
Donnan potential is negative. This relation between the
sign o f the fixed charge and the polarity o f the potential
is a direct consequence o f the physical principles
em bodied in the gel model. For example, if the fixed
charge is negative, the gel attracts cations so that the
cation concentration in the gel exceeds that in the bath.
Therefore, at equilibrium the potential difference must
be negative (Eq. 1) so that diffusion o f cations ju st bal­
ances migration of the cation in the electric field that is
established between the gel and the bath. The results
also show that as the m agnitude o f the concentration of
fixed charge in the gel increases, the m agnitude of the
potential betw een the gel and the bath increases.
Figure 3 shows results for a range o f norm alized fixed
FIGURE 3 '' Donnan potential E of the gel as a function of the con­
centration C f of fixed charge in the gel at a temperature of 37°C.
GEL MODEL OF THE TECTORIAL MEMBRANE
charge concentration. Note that at the extremes of the
range, the fixed charge concentration is 4 times that of
the bath concentration o f cations. Such a condition can
occur experimentally if the bath osmolarity is made
sufficiently low. P hysiologically, such conditions
occur for lateral line organ cupulae o f freshw ater
aquatic animals. The potential, in a gel that has a fixed
charge concentration that is a small fraction of the bath
cation concentration, is a few millivolts at most.
These results dem onstrate a second fundam ental
property o f polyelectrolyte gels— a potential exists
between the gel and the bath whose sign equals that of
the fixed charge in the gel.
Gels Swell. W e can combine Equations 4 with 8 to
obtain the swelling pressure and then use Equation 5
to obtain the increm ent in volume as follows
p
_
i)
2 C R T ~ 2C RT
m m
c°f
The swelling pressure and volume increm ent due to
the fixed charge in the gel are shown as a function of
the fixed charge concentration in Figure 4. W hen the
fixed charge concentration is zero, both the swelling
pressure in the gel and the increment in gel volume are
m inimal. An increase in the m agnitude o f the fixed
charge concentration causes an increase in gel osm o­
larity (Fig. 2) which causes an increase in the swelling
pressure and an increase in the gel volume.
p
2C R T
k (v
— 1)
2C RT
These results demonstrate a third fundamental prop­
erty of poly electrolyte gels: the existence of a fixed
charge, with either a positive or a negative sign, causes
a hydraulic swelling pressure in the gel that increases
the gel volume.
The Role of the Stiffness Bulk Modulus
The stiff gel model demonstrates that the fixed charge
concentration controls the concentration of mobile ions
in the gel, the gel potential, the gel hydraulic pressure,
and the gel volume. However, in this special case the
change in volume was assumed negligible so that the
fixed charge concentration was assumed constant. We
now remove this restriction to investigate the role of the
stiffness bulk modulus on gel properties. Now we
assume that the total quantity of fixed charge Nf is con­
stant but that the concentration of fixed charge changes
as the volume changes. Therefore, N f= CyV so that
N°f
C fi
where Nf = Nf(V„Cj_). Therefore, the gel properties
now depend on two parameters, Nf and K, which we
explore next.
There are some symmetries in the gel equations for
a binary electrolyte that are helpful in examining the
solution for a gel of arbitrary stiffness. For this case,
the gel equations (Eqs. 6 and 7) are
C f + d - d ~ 1= 0
k (v
FIGURE 4 Swelling pressure p and volume increment V - 1 in the
gel as a function of the concentration C f of fixed charge in the gel
in normalized coordinates.
357
and
- 1) - d - drx + 2 = 0 .
If the sign of the fixed charge quantity is reversed, i.e.,
if N f—> -Nf, then Cf —> -C f, d —» d~l, p —» p , v —> v,
and E —> -E . Hence, the hydraulic swelling pressure
and volum e ratio are even functions o f the fixed
charge quantity, whereas the fixed charge concentra­
tion and the Donnan potential are odd functions of the
fixed charge quantity.
Figure 5 shows the volume ratio as a function of the
fixed charge quantity for different values of the nor­
m alized bulk m odulus. Figure 6 shows the volume
ratio as a function of the normalized bulk modulus for
358
T. F. WEISS and D. M. FREEMAN
FIGURE 5 Volume ratio u o f the gel as a function of the quantity
N f of fixed charge in the gel in normalized coordinates. The lines
show the volume ratio for different values of the normalized bulk
modulus k under the assumption that the quantity of fixed charge is
constant.
d ifferent values o f the fixed charge quantity. As
expected from the results for a stiff gel (constant fixed
charge concentration), the volum e increases when the
m agnitude o f the fixed charge quantity is increased.
W ith a large bulk modulus ( ic~ 102), the volum e ratio
is small and approaches v = 1. As the bulk modulus
decreases, the gel volume increases.
Figure 7 shows the concentration o f m obile cations
and anions in the gel as a function o f fixed charge
quantity for a range of norm alized bulk moduli from
K
FIGURE 6 Volume ratio v of the gel as a function of the bulk
modulus k in normalized coordinates. The lines show the volume
ratio for different values of the quantity N f of fixed charge. Since
the volume is an even function of the quantity of fixed charge, the
results are the same for positive and negative charges that have the
same magnitude.
FIGURE 7 Concentrations of mobile cations c | and anions c£in
the gel as a function o f the quantity N f o f fixed charge in the gel
in normalized coordinates. The solid lines show results for differ­
ent values of the normalized bulk modulus icunder the assumption
that the quantity o f fixed charge is constant and therefore that the
concentration of fixed charge varies inversely with gel volume,
i.e., C f = Nf / v. For a stiff gel ( k = 102), v ~ 1 and C f ~ Nf.
Therefore, results for k = 102 almost completely overlie previous
results when the fixed charge concentration was assumed constant
(Fig. 2, shown here as a thick dashed line). Anion concentrations
are shown in black, cation concentrations in grey.
10-1 to 102. As the fixed charge quantity is increased,
the concentration o f the counterion increases and the
concentration o f the coion decreases exactly as was
the case for a stiff gel. In fact, for a large value o f the
bulk m odulus ( K ~ 102), the gel is relatively stiff and
shows little change in volum e (Fig. 5), and the cation
and anion concentrations approach those predicted
for a stiff gel. As the bulk m odulus is decreased, the
concentration o f the counterion for any value o f the
fixed charge quantity is decreased and the value o f the
coion is increased. T his is exactly w hat w ould be
expected if the increase in volum e reduces the fixed
charge concentration.
The effect o f the change in volum e on the fixed
charge concentration is shown in Figure 8. As the bulk
modulus of the gel is decreased the resultant increase
in volum e (Fig. 6) leads to the decrease in the fixed
charge concentration shown in Figure 8.
The D onnan potential in the gel is shown as a func­
tion of the fixed charge quantity in Figure 9. The same
trends are seen as for the concentrations o f m obile
ions in the gel (Fig. 7). At a large value o f the bulk
m odulus, the potential approaches that for a stiff gel.
As the bulk m odulus is decreased and the gel volum e
GEL MODEL OF THE TECTORIAL MEMBRANE
10~2
KT1
10°
101
*
102
359
DISCUSSION
Estimation of the Fixed Charge Concentration
The concentration of fixed charge plays a pivotal role
in determining the properties o f a polyelectrolyte gel.
Hence, it is important to estimate the concentration of
fixed charge in the TM. W e have estimated this con­
centration at neutral pH in two independent ways.
-6 0
Cf
-8 0
-1 0 0 J
N
FIGURE 8 Fixed charge concentration Cf in the gel as a function of
the bulk modulus k in normalized coordinates. The lines show the
fixed charge concentration for different values of the quantity Nj of
fixed charge.
increases, the Donnan potential decreases as the fixed
charge concentration decreases.
Results for gels of arbitrary stiffness illustrate interre­
lations of mechanical, chemical, electrical and osmotic
properties o f polyelectrolyte gels. For example, changes
in fixed charge concentration result in changes in the ion
concentrations, the osmotic pressure, the hydraulic pres­
sure and the electric potential of the gel. As a second
example, changes in gel stiffness result in changes in the
fixed charge concentration, the osmotic pressure, the
hydraulic pressure and the electric potential in the gel.
Estimate Based on Measurements o f the TM Potential.
Measurements of the potential in the TM as a function
of bath concentration of KC1 (Steel, 1983a) are com­
pared with the Donnan potential predicted by the stiff
gel model in Figure 10. The stiff gel model predicts the
relation between the Donnan potential and bath concen­
tration (Eq. 9) provided the quantity Cf, which is the
fixed charge concentration, is known. W e computed the
value of this quantity that minimized the mean square
error between the measured potential and the predicted
Donnan potential. The value was C°f- -8 .4 mmol/L, but
the resultant fit of the model to the measurements is
crude (with an rms error of 2.3 mV). Measurements of
DC potentials with pipet electrodes always contain
some unknown junction potential. Therefore, we also
estimated the fixed charge concentration by assuming
that the m easurem ents contained an unknow n but
5
FIGURE 9 Donnan potential E of the gel as a function of the
quantity N f of fixed charge in the gel at a temperature of 37°C. The
solid lines show results for different values of the normalized bulk
modulus ic under the assumption that the quantity of fixed charge is
constant, i.e., Cf = N°f lv . The thick dashed lines, which almost com­
pletely overlie the solid lines for
102, show results for a stiff gel,
i.e., Cf = C°f (same computations as in Figure 3).
Bath ion concentration (mmol/L)
20
50
100
FIGURE 10 Comparison of measurements of the potential in iso­
lated mouse TMs as a function of bath concentration of KC1 (Steel,
1983a) with the Donnan potential predicted by the polyelectrolyte
gel model with constant fixed charge concentration, i.e., a stiff gel.
The data points are mean values of the measured potential. The solid
line is a plot of Equation 9 with Cj= -8 .4 mmol/L, and the dashed
line is a plot of the same equation with the addition of the junction
potential correction of -3.1 mV and with C°f— -6.4 mmol/L. The
temperature was 37°C.
360
T. F. WEISS and D. M. FREEMAN
constant junction potential as well as an unknown fixed
charge concentration. W e computed the values of the
fixed charge concentration and the junction potential
that m inimized the difference between the measured
potential and the Donnan potential predicted by the
model offset by the junction potential. The theoretical
curve with the junction potential fits the measurements
more accurately (with an rms error o f 0.9 mV) for a
value o f C j - -6 .4 mmol/L and a junction potential of
-3.1 mV. The value of the junction potential is in the
range expected for liquid junction potentials. The two
estimates o f the concentration of fixed charge in the TM
(C f= -8 .4 and -6 .4 mmol/L) underestimate the magni­
tude of the fixed charge concentration because they are
based on the stiff gel model. In the stiff gel model, the
change in volume of the gel is negligible. If the volume
increases then the gel potential behaves as if the magni­
tude o f the fixed charge concentration were reduced
(Fig. 9). The data are insufficient (only 4 points) to
allow fitting the measurements with the more elaborate
gel model that includes appreciable gel swelling.
Estimate B ased on GAG Composition. It has been
reported (Thalmann et al., 1993) that the mouse TM
composition (as a % of wet weight) is 97% water, 0.8%
collagen and 0.5% GAG with a ratio of chondroitin sul­
fate (CS) to keratan sulfate (KS) o f 1.7. W e can use
these quantities to estimate the concentration of fixed
charge in the TM. Since the TM is 97% water, the mass
density of the TM is close to that o f water or 103 gm/L.
Thus, the mass o f GAG per liter of TM is 0.005 x 103 =
5 gm, of which 3.15 gm is CS and 1.85 gm is KS. The
m olecular weight o f each disaccharide in CS is 457
gm/mol and in KS is 444 gm/mol. Hence, the molar con­
centration o f CS is 3.15/457 = 6.9 mmol/L and of KS is
1.85/444 = 4.2 mmol/L. Each disaccharide unit of CS
contains one COO- and one CH 2OSO _3 group each of
which is ionized at physiological pH. Hence, CS con­
tributes 13.8 mmol/L of negative charge. Similarly, each
disaccharide unit of KS contains one CH 2OSO ~3 group
that is ionized at physiological pH. Hence, KS con­
tributes 4.2 mmol/L of negative charge. Assuming that
the GAG is due only to CS and KS, the total charge con­
centration is C f= -1 8 mmol/L. At neutral pH, the con­
tribution o f collagen to the fixed charge is negligible.
Conclusion. These two independent estim ates o f the
fixed charge concentration in the TM one based on
m easurements of the TM potential and the other on the
GAG composition o f the TM — give estim ates o f the
concentration o f fixed charge in the TM that differ by
less than a factor o f 3. Given the paucity o f m easure­
ments of TM potential and the possibility o f inaccura­
cies in estimates of GAG concentration o f the TM , we
regard this agreem ent as remarkable.
Strengths and Weaknesses o f the Polyelectrolyte
Gel Model
Strengths. W e have found the sim ple polyelectrolyte
gel model o f the TM helpful as a conceptual fram e­
work for understanding the TM. The model shows the
pivotal role o f fixed charge in determ ining m aterial
properties o f the TM. The presence o f this fixed charge
allows the TM ion com position to differ from that of
the bath (Figs. 2 and 7), leads to the developm ent o f an
electric potential difference between the TM and the
bath (Figs. 3 and 9) and makes the TM swell (Figs. 4,
5, and 6) so that it consists prim arily o f water.
Weaknesses. The m odel is o f a hom ogeneous poly­
electrolyte gel. N either its structure nor its osm otic
responses suggest that the TM is hom ogeneous and
isotropic. The m odel is m acroscopic and does not take
the m icroscopic structure of the TM into account. In
the m odel, the fixed charge concentration causes vol­
ume changes by changing the osm otic pressure, but is
assum ed not to affect the m echanical constitutive rela­
tion (Eq. 5) directly. The m odel is described in terms
o f solute concentrations rather than activities; activity
factors have not been taken into account. The model
m acrom olecules are assum ed not to bind m obile ions;
the binding o f protons and m etal cations are consid­
ered elsew here (W eiss and Freem an, 1996; Freeman
e ta l., 1996).
Acknowledgments
This project was supported by grants from the NIH. T.
F. W eiss was supported in part by the Thom as and
Gerd Perkins professorship.
GEL MODEL OF THE TECTORIAL MEMBRANE
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