Langmuir 1992,8, 1149-1153
1149
Studies on Concentration Dependence of Higher Order Phenomenological Coefficients Using Electrokinetic Studies across an Animal Membrane P. C. Shukla* and Gyanendra Misra Bio Physical Laboratory, Chemistry Department, St. Andrew's Post Graduate College, Gorakhpur, U.P., India Received September 5, 1990. I n Final Form: November 22,1991 Electrokinetic studies of aqueous solutions of urea-glucose mixtures with increasing concentrations of urea solution have been carried out across urinary bladder membranes. Higher order phenomenological Coefficients have been evaluated using extrapolation techniques. It has been predicted that the electrical double layer formed at the membrane/permeant interface undergoes structural changes due to specific adsorption of some of the ions of bulk phase and application of forces (electrical potential and pressure differences)in the same and in opposite directions. It has also been observed that straight phenomenological coefficientswhich are dependenton one force only behave differentlythan coupled or cross phenomenological coefficientswith increase in concentration of one of the permeants. Effects of concentration on the kinetic energy term (al),which is equivalent to velocity head which decreases the effective pressure across the membrane, and polarization term (a*)have also been analyzed.
Introduction Biological processessuch as generation of nerve impulse, muscle contraction, elasticity of the lung, intermittent ejection of blood from the left ventricle of the heart, filling and ejecting urine from the bladder, emergence of new structures, periodic oscillatory phenomena at all levels of organization, etc. are basically nonlinear The central role, in all physiological processes is played by membranes of diverse nature. Driving forces responsible for all changesare generally gradients of pressure, electrical potential, concentration, and temperature. These forces may act individually or collectively. Urinary transport is the one in which coupling5 of electrical potential and pressure across urinary bladder membranes gives rise to nonlinear behavior. Nonlinearity arises when the magnitude of forces operating in the system become too high or the membrane undergoes changes in the s t r ~ c t u r e . ~ J Phenomenological coefficients are the measure of interaction among forces, permeants, and the membrane used.s The coefficients, namely coupling, acquire much greater significance when the permeant becomes a mixture of diverse nature. The permeants chosen here are aqueous solutions of urea and glucose and the membrane used is the urinary bladder of goat. The permeants used have some similarity and many diversities"12 as regards their structural behavior, occurrence in the urine, mode of transport, etc. A close analysis of various phenomeno(1) Skalak, R.; Chien, S. Hand Book of Bio-engineering; McGraw-Hill Book Co.: New York, 1987. (2) Volkenstein, M. V. Bio Physics; MIR Publications: Moscow, 1983; n c 503. --(3) Guyton, A. C. Text Book of Medical Physiology; W. B. Saunders Co.: Philadelphia, PA, 1981. (4) Larter, R. Chem. Reu. 1990, 90,355-381. (5) Shukla, P. C. Membranes 1991, 16 (4) 192-198. (6) Shukla, P. C.; Misra, G.; Misra, J. P. Biophys. Chem. 1989, 33, 31-37. _(7) Raatogi, R. P.; Shabd, Ram Indian J. Chem. 1982, 21, 859. (8)Lorimer, J. W. J. Membr. Sci. 1985, 25, 211, 180. (9) Stain, W. D. Mouement of molecules across cell membranes: Academic Press: New York, 1967. (10) Jakee, J. G.; van Hook, W. A. J. Phys. Chem. 1981,85, 3480. (11) James, D. W.; Frost, R. L. J. Phys. Chem. 1974, 78, 1754. (12) Huppert, H. E.;Hallworth, M. A. J. Phys. Chem. 1984,88,12902.
logical coefficients is expected to be of very great use in understanding causesof nonlinearity in biological systems. Since the urinary process may be correlated with electrokinetic phen~mena,'~J~ evaluation of phenomenological coefficients and their concentration dependence may be of immense use in predicting the behavior of the membrane in varying situations.
Theoretical Section By use of the methodology of nonequilibrium thermoand current (I)flow across dynamics,13J5J6volume flow (Jv) amembrane when it is subjected to pressure, and electrical potential gradients may be expressed as
where Lij, Lijk, and Lijkl (i, j , k, 1 = 1, 2) are the phenomenological coefficients. Li, and Ljj (i = j = 1, 2) are straight coefficients while Lij, Lijk, etc., are coupled phenomenological coefficients. Straight phenomenological coefficients are always positive: coupled phenomenological coefficients or cross phenomenological coefficients may have positive or negative values. Cross phenomenologicalcoefficients may also change their sign which depends upon nature of forces used. Again, when I = 0, eq 2 may be expressed as ~
(13) Shukla, P. C.; Misra, G. J. Membr. Sci. 1987, 31, 157-176. (14) Shukla, P. C.; Misra, G.; Misra, J. P. J. Colloid Interface Sci. 1989, 129,5342. (15) Lakshminarayaih, N. Equations of Membrane Bio Physics; Academic Press: New York, 1984; pp 277, 98. (16) Shukla, P. C.; Misra, G. J. Membr. Sci. 1986, 26, 277-286.
0743-7463/92/2408-1149$03.00/00 1992 American Chemical Society
1150 Langmuir, Vol. 8, No. 4, 1992
-(*) =-+--fl+-A++--L21 1L211 AP
1L212 (A$)2 2 L22 AP
2 L,, 1LZlll (AP)2 -1L2112 -L2122 (A$)2 + -AI' A$ 2 L22 6 L22 2 L22 l=O
L,,
L212 L22
Shukla and Misra
+
by using one force only, while cross phenomenological coefficients such as LI12,L1112, L1122, etc. are evaluated by using two forces together. Urea is the major constituent of urine while glucose is found in traces. The presence of urea in the urine depends upon diet taken while glucose appears in the urine when the system fails to absorb it.
+
+
The values of Lzl were determined by plotting the streaming potential against AP using eq 3. This gives L21/L22 as an intercept and higher order terms as slope. Equation 1 may also be expressed as J, = (J,)A+o
+ (J,).p=o + ~ 5 1 1 2@ A$ + 1/&1112(AP)2 A$ +
h p ( A J . ) 2 + ... (4)
Rearrangement of eq 4 yields
Ll12+ Lll12A P +
L1122A$
+ ...
(5)
Evaluation of phenomenological coefficients is carried out by using the extrapolation technique.13 Various phenomenological coefficients have been evaluated from the following measurements: (a) measuring J , at (A$) = 0 and at (AP)= 0; (b) measuring streaming potential at varying pressures; (c) measuring membranepermeant conductance by applying a certain voltage and noting the current flow in the system. Experimental Section Membrane Used and Experimental Techniques Employed. The membrane used was the urinary bladder of a goat. Its characteristics were preserved by using a formalin-alcohol system as described earlier.13J4s16The membrane is then repeatedly washed with double distilled water and fitted in the a p p a r a t ~ s . 1The ~ membrane is always maintained in a wet state by filling the apparatus with a dilute (20.01 M) urea solution. The apparatus is filled with liquid whose permeability is to be measured 8-10 h before observations are started. This is done to familiarize the membrane with the permeating material. The whole apparatus is then dipped in a liquid thermostat set at 35 "C which has an accuracy of hO.01 OC. Hydrodynamic permeability is measured by noting changes in the horizontal capillary tube. Variation of hydrostatic pressure is brought about by raising the level of the pressure head on one side of the membrane; the difference in height is noted by a cathetometer. The radius of the horizontal capillary tube is measured using a traveling microscope. The capillary tube is steamed for about an hour before use so as to get consistent flow. A potential difference could be applied across the membrane by an electronically operated power supply. Movement of fluid in the external capillary tube is noted so as to have an idea of electroosmotic flux. Streaming potential was measured by applying a pressure difference across one side of the membrane and noting the resultant potential using a digital multimeter with a comparator function with an impedence of the order of 10'3 R and current sensitivity of the order of 10-7A. Streaming current was calculated using streaming potential and resistance data. Conductance was determined by applying a certain voltage across the membrane and noting the resultant current.l8 Care was taken that no evolution of gas bubbles occurred at the electrodes. Applications of forces (1pand A$) were carried out both individually and collectively, depending on the coefficient to be evaluated. Straight phenomenological coefficients are obtained (17) Shukla, P. C.;Tripathi, B.N . Indian J . Biochem. Biophys. 1978, 15, 421.
(18) Katchalsky, A,; Curran, P. F. Non-equilibrium thermodynamics in Bio Phys; Harvard University Press: Cambridge, 1965; p 156.
Results and Discussion A majority of biological processes proceed to completion under n ~ n l i n e a rsituations. ~,~ For example, in the present case, when no urine at all is in the bladder, intravesical pressure is approximately zero. Collectionof urine beyond certain values (say 400 to 500 mL) causes the pressure to rise very r a ~ i d l y . Pressure ~ gives rise to streaming potential, which in turn produces streaming current.'3 Thus electrokinetic measurements may be of immense use in understanding urinary transport, which may be explained as follows. Biological membranes are known to develop electrical potentials and hydrostatic pressure across them~e1ves.l~ Electroosmosis occurs due to application of electrical potential gradient across the membrane whereas streaming potential is the potential induced as a result of application of hydrostatic pressure. The streaming current is the current produced due to application of these forces. The results of these measurements may be summarized as follows: (i) Hydrodynamic permeability and streaming potential decreases with the increase in the concentration of urea. (ii) Electroosmotic permeability and membrane permeant conductance increase with increase in concentration of urea solution. Electroosmotic permeability, streaming potential, and streaming current are all related with electrokinetic potential or zeta potential.15~~~ Since an electrical double layer is formed as a result of preferential adsorption of ions by matrix of the membrane and subsequent balance of their charged ions dispersed in solution, structure of the membrane acquires prime importance. Biological membranes are built up in an infinite number of ways21as regards the nature of elements, the fraction of area covered by each element, and the geometrical array. Mechanical property of each tissue depends not only on chemical composition but also on its structural or ultrastructural details. Two tissues made of the same proportions of collagen, elastin, and ground substances may behave quite differently depending on how these basic elements are put together.' The urinary bladder is an example of transitional epithelial tissue which has elastic muscular walls and membraneous folds.z2 Studies across urinary bladder membranes suggest that cells are essentially coupled in series23 and the relation between cell length and bladder radius deviates from linearity a t a very small bladder volume. A series membrane may be composed of a number of solid or liquid layers. Some of the layers may be intermediate solutions. In such cases, continuity of potentials across cell boundaries appears to be of prime importance. The continuity of electrochemical potential does not imply continuity of concentration, pressure, or electrical potential. It is very likely that each term may undergo a drastic jump on passing from one phase to another. Urinary (19) Heinz, E.Electrical Potentials i n BiologicalMembrane Transport; Springer Verlag: New York, 1981. (20) Bergethon, P.R.;Simons, E. R. Biophysical Chemistry; Springer Verlag: New York, 1990. (21)Kedem, 0.;Katchalsky, A. Trans. Faraday SOC.1963,59, 1918. (22) Copenhaver, W. M.; Kelly, D. E.; Wood, R. L. Bailey's Text Book of Histology; Williams & Wilkins: Baltimore, MD, 1979. (23) Uvelius, B.; Gavele, G. Acta Physiol. Scand. 1980, 110, 257.
Langmuir, Vol. 8, No. 4, 1992 1151
Electrokinetic Studies of Membranes
Table I. Values of Different Coefficients for Urea-Glucose Mixture Systems 35 f 0.1 O C temperature of the system 1.95 X m2 membrane area 0.17 X m membrane thickness
+
phenomenological coefficients
+
urea (0.02M) glucose (0.01M)
urea (0.02M) + glucose (0.01M)
urea (0.04M) glucose (0.01M)
urea (0.05 M) + glucose (0.01M)
1.77 2.50 2.70 4.20 0.72 -0.43 -4.02 -1.40 1.02 4.60 -26.90 24.87
1.56 2.70 3.30 6.50 2.00 -0.66 -3.44 -1.01 1.42 1.60 -35.70 24.46
1.38 3.20 3.50 8.40 3.40 -0.86 -3.08 -0.52 1.12 0.10 -43.70 36.31
1.15 3.50 3.90 11.20 4.52 -2.24 -2.60 -0.24 3.67 -4.10 -63.70 60.36
0.91 4.20 4.05 13.50 5.62 -4.08 -1.40 -0.13 3.77 -3.80 -69.00 70.96
bladder membranes may be considered as composed of many separate membrane entities which may be isotropic or anisotropic. Most biological membranes have heterogeneous membrane surfaces where both cationic or anion groups are active at a given pH. Groups or sites on the membrane surface in general are of weak acid or weak base type and so ionization will depend on the pH of the medium in which they exist.15 Since urinary bladder membranes are sensitive to formation of urinary calculi, pH of the medium plays important role. Thus adsorption of the ions present in the bulk phase to the membrane surface describes the overall situation. Dissociation of water molecules takes place as follows: H20 + H 2 0+ H,O+
+ OH-
Urea, which is a weak base, will show a tendency to be protonated as follows:
HzNCNHz
I
-
+
urea (0.01M) glucose (0.01M)
HiN=CNHz
I
OH*
OH
-
H,NC=NH,*
I
OH
It is because of such autodissociation reactions that even highly purified amphoteric solvents display some electrical conductivity. Thus in the urea-water mixture, there exists the possibility of competition between water dipoles and protonated urea complexes for adsorption at the urinary bladder interface. But when the mixture becomes ternary, i.e. water, urea, and glucose, the possibility of competition among various dipoles becomes quite high. To allow the maximum interaction of hydroxyl groups with the surface, glucose is likely to be adsorbed with the plane of pyranose ring parallel to the surface.24 In the case of the ureaglucose mixture in water, the diffusivity in water of urea is greater than that of glucose.lO Interphase electrification may be due to charge established or imposed on one phase or charge generated by differential affinity between phases.20 This may be due to ionizable surface groups, physical entropment of charge, differential distribution of ions between phases, preferential attraction of ionic species to surfaces, etc. As interphase electrification leads to formation of a double layer, potential difference at an electrified interface is due to charges present and also due to dipole layers. Thus a conceptual separation of charge and dipole contribution to the total potential describes the overall situation. ~~~~~~~
(24)Dunstan, T.D. J.; Pincock, R. E. J . Phys. Chem. 1984,88, 5684.
A modified picture of the electrical double layer7J"27 uses an effective zeta potential which is made up of charge contribution potential (A+) and dipolar potential (AX)so that
The dipolar potential (AX) depends upon the total number of dipoles adsorbed, dipole momenta of permeants, and the number of such dipoles that interact with a particular dipole. Thus it is seen that the resultant field in the double layer will be due to charge in the double layer and the dipoles adsorbed at the interface. Phenomenological coefficients, namely ,512, L122, etc., depend upon influence of electric field, and L211 and L2111 are related with pressure difference on double layer characteristics. Cross phenomenologicalcoefficients such as L112, ~51112,L1122 and L212, L2112, L2m represent net interaction of forces toward electrical double layer. For example, L122 depends upon14 (a) the total number of adsorbed dipoles, (b) viscosity of the medium, and (c) dipole moment vector. It is possible to alter the double layer in such a way that L122 may change sign. Similarly Lzll can have both positive and negative sign. In processes where electrical potentials and pressure gradients change directions, higher order phenomenologicalcoefficients may reverse their signs. The value of these coefficients have been evaluated using extrapolation techniques, and they are given in Table I. From these values, it is evident that Onsager's reciprocal relationship holds in this case. Electrical potentials and pressure difference across an electrical double layer produce opposite effects28 in its characteristics. An electric field at the double layer causes ordering of molecular dipoles, thus increasing the field, which increases thickness of the interior layer. The driving pressure gradient, causing translational movement of the molecules, destroys the dipolar orientation. Increasing the pressure gradient will decrease the thickness of interior layer. Thus these factors act in opposite directions. The two forces acting in the same direction and in opposite directions are bound to make structural changes in the double layer characteristics. As a result of this, coupled phenomenological coefficients are expected to behave differently than straight coefficients when concentrations of permeants are changed. This is evident from Figures 1-7. The changes are due to the fact that concentration (25)Bockrish,J.0.M.;Reddy,AKN Modern electrochemistry; Plenum Publishing Corp.: New York, 1977. (26)Srivastava, M. L.;Ram, Bali. J.Non-Equilib. Thermodyn. 1985, 10, 57-82.
(27)Rastogi, R. P.; Shabd, Ram. J . Phys. Chem. 1977,81,1953-1955. (28)Gur, Y.;Ravina, I. J . Colloid Interface Sci. 1979, 72, 442.
Shukla and Misra
1152 Langmuir, Vol. 8, No. 4, 1992
I
001 UREA
002 003 004 005 CONCENTRATION MOLE
c'
Figure 1. Concentration dependence of L11 for urea-glucose mixture systems.
1
0.0I 5t
UREA
Figure 4. Concentration dependence of mixture systems.
0.01
I
0.02 0.03 0.04 C O N C E N T R A T I O N MOLE Lzll
i'
0.05
for urea-glucose
002 003 004 0.05 ,CONCENTRATION MOLE c'
UREA
Figure 2. Concentration dependence of L12 for urea-glucose mixture systems.
I
0.01 UREA
0.02 0.03 CONCENTRATION
004 MOLE
i'
0.05
Figure 5. Concentration dependence of LIl2for urea-glucose mixture systems.
I 001 UPEA
0.02
0.03 0 04 CONCENTRATION MOLE
- , 0.05 L'
Figure 3. Concentration dependence of LlZ2for urea-glucose mixture systems.
of ions at the inner Helmholtz plane (IHP) and outer Helmholtz plane (OHP) of the double layer would change due to applied electrical field, typical structural behavior of urea and glucose and lateral interactions among water dipoles, urea dipoles, and glucose, leading to changes in net number of molecules in one direction and affecting values of dipolar potential. Thus there would be a change in the structure of interface giving rise to nonlinear behavior. The urinary bladder is an example of a membrane in which nonlinear behavior with pressure difference (hp)
and electrical potential difference (A+) have been observeda6 For a membrane of uniform capillary tubes, Poiseuille's law holds. But in situations where flow converges, diverges, or follows a curved path, this is no longer the case. Lorimers introduced a kinetic energy term ( q )and , the effect of this term is equivalent to velocity head which decreases the effective pressure across the membrane. For a well-developed (Poiseuille's) flow in a uniform capillary,al is zero. The kinetic energy term ( a d is related with phenomenological coefficients8 as (7)
where A is an effective cross-sectional area of the membrane and p is density. Values of a1 calculated for different solutions of the urea-glucose mixture show that it increases with increase in concentration of urea as shown in Figure 8.
Langmuir, Vol. 8, No. 4, 1992 1153
-
4,
'0 X
Y ;
5'
I 0.01 UREA
0.02
0.03
CONCENTRATION
0.04
-,
0.05
-20-
MOLE L
Figure 6. Concentration dependence of L l l l ~for urea-glucose mixture systems. 001
002 UREA
003 or4 CONCENTRAT!3N
oo? MOLE
L'
Figure 8. Concentration dependence of a1 for urea-glucose mixture systems. 70-
y>
60
-
8
-21
501
0.01 0.02 0.03 0.04 UREA CONCENTRATION MOLE
i'
0.05
Figure 7. Concentration dependence of L1122 for urea-glucose mixture systems.
Since electrical field effects produce changes in the interfacial property of surfaces, the polarization term (a21 is given as8
Ll12d2 3a1 PL122 -- L1222A2 = a1 "1PL113 A2 3pLl12L12a1 2AZ where L1122 and L1222 are higher order phenomenological coefficients. The term a2 is a constant and is a function of structure of membrane. Since urinary bladder membranes are elastic in nature, polarization and d e p o l a r i z a t i ~ nare ~ ~the ~ ~primary nature of the membrane: evaluation of this coefficient (a2) acquires prime importance. A plot of a2 vs concentration (Figure 9) shows that it increases with increase in concentration of urea. A close analysis of various plots reveals that electrokinetic behavior of the urea-glucose mixture shows almost identical behavior as those of urea s01utions.l~ Thus it may be inferred that increase of urea masks the effect of glucose which may be of immense use in physiological studies. a2= --
~
~~
(29) Wyker, A. W.; Gillinwater, J. Y. Method of Urology; Oxford &
IBH Publishing Co.: Oxford, 1977.
(30)Levin, R. M.; et al. J . Urol. 1986, 136, 517.
/
'I I
I
001 002 003 2.04 UREA C-NCENTRATION MOLE
0.05 I?
Figure 9. Concentration dependence of a2 for urea-glucose mixture systems. Nonlinear behavior due to changes in structure of membrane has been discussed. However, nature and magnitude of forces also decide nonlinearity. Most biological processes are regenerative;e.g. (urinarytransport which involves filling and ejection of urine due to generation of micturition waves) the forces reach their maximum value at some point and then begin to decrease.5 It is also very likely that all the forces do not act in the same direction. Thus membraneous structure bears all the constraints of forces whether in one direction or in opposite directions. That is why membrane geometry plays a vital role in nonlinearity. The functionalproperties of membranes often vary with their physiological conditions. Effect of age, temperature, radiation, season, environment (particularly ionic environment), etc., on permeability have been reported. Although goat to goat variation is possible due to several reasons, the trend of behavior will always be the same. Acknowledgment. G.M. thanks UGC (New Delhi) for financial support. Registry No. Glucose, 50-99-7; urea, 57-13-6.
