QUESTION
IEEE TRANSACTIONS ON BIOMEDICAL ENGINEERING, VOL. BME-19, NO. 5, SEPTEMBER 1972
[21 J. Ekstedt, “Human single muscle fiber action potential,” Acta
Physiol. Scand., vol. 61, suppl. 226, pp. 1-96, 1964.
(31 K. D. Wise, J. B. Angell, and A. Starr, “An integrated-circuit ap-
proach to extracellular microelectrodes,” IEEE Trans. Bio-Med.
Eng., vol. 17, pp. 238-247, July 1970.
[4] C. Gould, “A glass-covered platinum microelectrode,” Med.
Electron. Biol. Eng., vol. 2, pp. 317-327, 1964.
[51 H. A. Baldwin, S. Frank, and J. Y. Lettvin, “Glass-coated tungsten
microelectrodes,” Science, vol. 148, pp. 1462-1463, 1965.
(61 D. H. Hubel, “Tungsten microelectrodes for recording from single
units,” Science, vol. 125, pp. 549-550, 1957.
[71 G. Svaetichin, “Low resistance microelectrodes,” Acta Physiol.
Scand., vol. 24, suppl. 86, pp. 5-13, 1951.
[8] R. D. Adrain and D. W. Bronk, “Discharge of impulses in motor
nerve fibers,” J. Physiol., vol. 67, pp. 119-151, 1929.
[9] F. Buchthal, C. Gould, and P. Rosenfalk, “Volume conduction of
the spike of the motor unit potential investigated with a new
type of multielectrode,” Acta Physiol. Scand., vol. 38, pp. 331-
354, 1957.
[101 -, “Action potential parameters in normal human muscles and
their dependence on physical variables,” Acta Physiol. Scand.,
vol. 32, pp. 200-218, 1954.
[111] C. J. De Luca, “Myo-electric analysis of isometric contractions of
the human biceps brachii,” M.Sc. thesis, Univ. of New Brunswick,
Fredericton, N.B., Canada, 1968.
[121 D. A. Robinson, “The electrical properties of metal electrodes,”
Proc. IEEE, vol. 56, pp. 1065-1071, June 1968.
[131 R. C. Gesteland, B. Howland. J. Y. Lettvin, and W. H. Pitts,
“Comments on microelectrodes,” Proc. IRE, vol. 47, pp.
1856-1862, Nov. 1959.
Diflfusion Effects of Liquid-Filled
Micropipettes: A Pseudobinary Analysis of
Electrolyte Leakage
C. DANIEL GEISLER, MEMBER, IEEE, EDWIN N. LIGHTFOOT, FRANK P. SCHMIDT,
AND FRANCISCO SY
Abstract-The importance of diffusion phenomena in glass micro-
pipettes filled with concentrated electrolyte solutions is well known.
Nonequilibrium diffusion-rate equations for electrolyte-filled glass mi-
cropipettes are given for a model of situations of current interest:
diffusion into an infinite medium. The solutions of these equations
give the flux of material out of a micropipette as a function of time,
concentration, and geometry.
INTRODUCTION
A LTHOUGH electrolyte-filled micropipettes have proven
Aextremely useful in electrophysiological experimenta-
tion, interpretation of potential measurementsusing these
electrodes is subject to significant ambiguities. For one,
“tip potentials” exist that may change with time or upon
penetration of a cell [1]. The filtering effects of the electrical
circuit to which the micropipette is attached may also be im-
portant [2]. Moreover, a liquid-junction potential exists be-
tween the electrolyte in the micropipette and the physiological
fluid in which the tip is immersed [3]1. The use of 3M KCI to
minimize these effects is widespread. The use of this highly
Manuscript received April 12, 1971; revised November 8, 1971.
C. D. Geisler is with the Department of Electrical Engineering and the
Laboratory of Neurophysiology, University of Wisconsin, Madison,
Wis.
E. N. Lightfoot, F. P. Schmidt, and F. Sy are with the Department of
Chemical Engineering, University of Wisconsin, Madison, Wis.
concentrated KCI reduces the tip potential [4], reduces the
resistance of the micropipette [1], and fixes the liquid-
junction potential between the micropipette’s electrolyte and
axoplasm at no more than 3-4 mV [3].
The resulting difference in concentrations between the mi-
cropipette electrolyte and the less-concentrated ionic environ-
ment into which the micropipette is inserted will cause an
outward diffusion of electrolyte. The effects of such diffusion
may alter the external ionic environment and thus are of con-
cern [5]. Such diffusion effects have in fact been experi-
mentally demonstrated for molecules such as acetylcholine [6].
Electrolyte diffusion has even been used as an intracellular
marking technique: while recording intracellularly, Harris et
al. [7] used diffusion of Niagara Sky Blue dye from the record-
ing electrode to mark the penetrated cells.
Quantitative descriptions of diffusion rates from micro-
pipettes are not common in the literature, but a steady-state
solution has been obtained by Nastuk and Hodgkin [8] and
by Krnjevic et al. [6]. It is the purpose of this paper to present
a more general, time-dependent description of electrolyte
diffusion.
ANALYSIS
Since the geometry of experimental systems is quite variable,
and frequently not known in detail, it is desirable to consider
simplified models for possible limiting cases. We consider here
GEISLER et al.: DIFFUSION EFFECTS OF MICROPIPETTES
Ci(t,ri) r-
Fig. 1.
re
Ce(t, re)
one such model: diffusion from a long capillary tip into a very
large unstirred pool of electrolyte, representing the interior of
a large cell; a cell with high permeabilities for the ions of the
micropipette’s electrolyte; or a large volume of extracellular
fluid. We could have considered other situations as well, but
analysis of this model suggests that capillary diffusion can be
treated by a pseudosteady model for situations of present in-
terest. For this reason, it appears unnecessary to model ex-
ternal behavior in detail.
Diffusion in microelectrodes is actually multicomponent in
nature. Diffusional interactions can, however, be safely ne-
glected for our present purposes, and we shall use a pseudo-
binary approach here. In other words, we shall treat the dif-
fusion of a single electrolyte from a micropipette as if the
solution consisted only of that electrolyte and water.
For purposes of this analysis we shall also omit the surface
charges in the glass and the corresponding electrical “double
layer” [9]. Strictly speaking, such an analysis is accurate only
for a micropipette having a wide tip opening and filled with a
highly concentrated electrolyte. Nevertheless, it may also pro-
vide a reasonable approximation to micropipettes with smaller
tips and lower concentrations under some conditions. For ex-
ample, if the micropipette is made out of the commonly used
borosilicate glass, it will have a relatively low surface charge
[4]. Furthermore, it is known that the Debye length, the
effective width of the double layer, is extremely small at salt
concentrations of physiological magnitude or above (e.g.,
g3 m at 100 moles/l in a 1: 1 electrolyte [9]). Hence, for
electrolyte concentrations greater than 0. IM in a borosilicate-
glass electrode with a 0.5-l,m diameter tip, double-layer effects
should be negligible over most of the cross section of the
tip. This contention is supported by several different experi-
mental fmdings. First, there is good agreement between ob-
served electrode resistances and those calculated for several
types of micropipettes assuming free field diffusion (10].
Secondly, pH, which is known to greatly affect Debye length,
has relatively little effect on the resistance of Pyrex micro-
pipettes [4]. Thirdly, “tip potentials,” thought by some to
be due to the double layer, are relatively low for Pyrex micro-
pipettes filled with 0.5M KCI [4], [ 1].
We consider here the idealized system of Fig. 1 in which a
capillary is in contact with a limitless unstirred volume. The
capillary is assumed to be conical, with an angle of taper Oi,
and is truncated near the tip. Initially, the electrolyte within
the tip and that in the external solution are at different uni-
form concentrations. The surface of contact between the
capillary and external solution is assumed to be a sphere with
a radius R equal to that of the capillary bore. This is a con-
siderable sinplification of the actual situation, but it will be
shown that a more detailed geometric model is not needed.
It is convenient to use the spherically symmetrical form of the
diffusion equation in the external solution [12]. It is now
possible to write the differential equations and boundary conditions
for the problem as follows.
A. External Solution
aCe = De a aCe\
_= _ r -1
at r2 are e are
ce(O, re) = O, re > R
Ce(t, re) -* 0, re -+ o0, t finite (3)
where Ce is the solute concentration in the external solution
and De is the effective binary diffusivity of the solute in the
external solution.
B. Intracapillary Solution
aci Di, a 2 aci\
=r.
at ri2 ari ‘ ari
ci(O, ri) = 1, ri > a
ci(t,ri)+ 1, ri 0°° t finite (6)
where ci and Di are the concentration and effective binary dif-
fusivity of the same solute in the intracapillary solution.
Notice that ce and ci are dimensionless salt concentrations
defined by
Csc(Cse5)O
(Csi)o- (Cce)o
where c, is salt concentration (moles/liter) in the specified
solution, (cs,)o is initial salt concentration (moles/liter) in the
external solution, and (cg)o is initial salt concentration (moles/
liter) in the internal solution.
C. Matching Conditions at the Interface Between the Capillary
and the External Solution
It remains to obtain a match of the two salt solutions in
the neighborhood of the capillary tip. Minute vibrations of
the tip are to be expected, and one can envision an area at the
tip where the solution is well stirred. Therefore, it seems
reasonable to assume that the concentrations just inside and
just outside the tip are equal. That is,
ci(t, a) = ce(t, R).
(7)
Furthermore, since no accumulation of material at the inter-
face can be expected, the flux of material out of the micro-
pipette will be equal to the flux of material into the cell. The
equality of material that flows at the micropipette-cell inter-
face is given in (8). The negative sign accounts for the dif-
ference in direction of the two flows:
AiDi = -AeDe –
lari ri =a are re=R
373
(1)
(2)
(4)
(5)
(8)
374
where Ai is the cross-sectional area [2ira2 (1 – cos Oi) * 2tR ]
of the part of the surface ri = a that is enclosed by the micro-
pipette and Ae is the area (47rR2) of the imaginary sphere at
the end of the micropipette from which diffusion occurs into
the infinite medium.
The partial differential equations (1) and (4) can be sim-
plified by taking their Laplace transforms with respect to time.
The resulting ordinary differential equations can be solved
using the Laplace-transform expressions of the boundary con-
ditions and matching conditions. If dimensionless variables are
used, the Laplace transform CQ(s) of cT(r) is found to be
cQ(S) = 1- 1 -es (m-1) [9a/+1+a)
s 1ijS +4 OI-
where rn = r1/a, r = tDl/a2 a = (Ae/Ai) (De/DI)A 2 nd 13
(RIa) (DlDe)112 Equation (9) along with a corresponding
equation for Ce(s), which we shall not need, completes the
formal solution.
A general inversion of (9) appears difficult to accomplish, so
we have concerned ourselves only with conditions at 7m = I
(r, = a). The concentration profile at r1 = a as a function of
time is found to be
ci(a,r)= + (a+g)(l+a)e erfc(bV1T) (10)
where b = (a + ,B)/,B(1 + a). Thus tip concentration is, in gen-
eral, time dependent. However, since the limit of the time-
dependent term for long times is zero, it is clear that
ci(a, oo) = (3/(ao + 13). Furthermore, since tip angles are usually
small (a > B), we find that ci(a, o) _ (01/a) < 1. Hence the
concentration in the neighborhood of the capillary tip ap-
proaches that of the external solution. The time-dependent
term in (10) can be accurately represented by a finite asymptotic
series for large values of b2’r [13]. Specifically, for
b2 r > 5, (10) can be rewritten as
ci(,-) + a(1) I
t+3+ (a +1)(I+o) V>rb (11)
with an error in the time-dependent term of less than approxi-
mately ten percent. Substituting the definitions of b and r in
(1 1) and assuming that Di = De and R < a, we find that
ci(a, t) -RAi I+ .1
Ae a -s/7TtDi )
5R2(I +a)2
a2Di
Equation (12) consists of two terms, each of which makes a
contribution to the value of the tip concentration at a given
time. The first term depends on the taper and approaches zero
as the micropipette approaches a cylindrical shape (a -+o)
The second term depends only on the diffusivities of the ma-
terial in the micropipette and the surrounding solution. If we
evaluate (12) numerically, assuming Di = De = 10-5 cm2/S
([14], [151),Ae/Ai =4, a = Ogm, andR 0.25 gm(cor-
responding to an angle Gi of approximately 1.5°), we find that
(12) is approximately correct by the time 0.5 ms has elapsed,
and the tip concentration has dropped to within a factor of 2
of its final value within 30 ms. Thus the interfacial concentra-
tion very rapidly approaches a constant value.
IEEE TRANSACTIONS ON BIOMEDICAL ENGINEERING, SEPTEMBER 1972
For any realistic estimate of Ai/Ae it may be seen that
ci(a, 00) < 1. Thus most of the diffusional resistance is con-
centrated within the pipette, and that in the external solution
is small. This situation, which results from the very small
taper of typical micropipette tips, will permit considerable
simplification when we consider diffusion into cells of limited
volume.
The dimensionless instantaneous flux at i= 1 is
3cj(T)/a7?jjn,=j. Upon Laplace inversion, it is found that
aci, a a/(1+a)
al?i 7jj. l a+: r
+ ( +a) (1 eb2T erfc(bv”T)
(13)
where b is defined in (10). Making the same approximations as
for (12), we find that
aci(t) = 1
ari ri=a a
SR2 (1 + a)2
t > o?D-
Once again, the time-dependent tenn will rapidly become in-
significant. The constant term on the right is the long-term
limit and agrees exactly with the steady-state results given by
Krnjevic et al. [6]. The second term is independent of taper
and would be the sole term if the micropipette were cylindri-
cal. Notice that R, the effective radius of the imaginary sphere
at the end of the micropipette, enters the approximate solution
only in the time that must elapse before (14) becomes
reasonably accurate. Hence, the particular R chosen for analy-
sis purposes is not of critical importance.
It is also interesting to know the total amount of salt that
passes out of the micropipette and an expression for the
normalized efflux can be obtained by integrating (13). We
thus find that
O al?i 77i=l
2(l- 1)21 [eb2erfc(bVI 1]. (15)
(a + 13)3
The amount of salt M that passes out of the micropipette can
be found by multiplying (15) byA1a[(csj)o – (cse)0 . For the
same assumptions as for (12) we find that
M=Ai[(Csi)o – (Cse)o] L i + cD t >>.5R%r
a in Di
As before, the first term, which dominates at large time, is due
to micropipette taper, while the second tenn is the “cylin-
drical” contribution.
a a u(i nN2
dr = ‘r + 1 + P) 8/4.-Tji
(X + a I + a ((j + g)2
In the above approximation to the physical situation, diffu-
sion was considered to take place into an infinite medium.
Also considered was the case of diffusion out of a micropipette
into a cell of limited volume of well-mixed fluid, representing
the interior of a cell with low permneability for the ions of the
micropipette’s electrolyte. For short times it was found that
the initial salt flux for the two cases was identical. For ex-
(14)
(16)
GEISLER et al.: DIFFUSION EFFECTS OF MICROPIPETTES
ample, there is an initial salt flux of approximately 6 X 10-14
moles/s from a 3M KCI-filled micropipette with a 1/2-tim diameter
tip when inserted into a 20-,um diameter cell. After a
few minutes, this flux, which initially is exactly equal to that
calculated for the infinite medium case (14), decreased due to
the buildup of salt concentration in the limited volume.
DISCUSsION
It is clear from the above analysis that system behavior is not
sensitive to the geometry external to the pipette or to bound-
ary conditions, and that transients are of minor importance
under the usual conditions of micropipette operation. There-
fore, one can normally obtain satisfactory predictions by as-
suming pseudosteady behavior and neglecting diffusional re-
sistance outside the pipette.
The above two assumptions were used by Krnjevic et al. [6]
to calculate steady-state diffusion rates. There was good agree-
ment between their theoretical results, which are identical to
our long-term solution, and their experimental data (6].
Lanthier and Schanne also used such a pseudo steady-state
model in calculating micropipette impedances that agreed
quite well with their experimental measurements [101. The
ability of the pseudo steady-state model to accurately match
the experimental data of these workers [6], [101 therefore
provides support for the work presented in this paper.
The ion fluxes claculated from our models can be apprecia-
ble. For example, the flux of electrolyte from a 3M KCl-filled
micropipette with a 1/2-,um diameter tip is approximately
6 X 10-14 moles/s (14). This is approximately one quarter of
the flux produced by electrophoretic ion injections of KCI
using 50 nA of current. This latter magnitude of ion injection
is quite significant and has been shown [16] to have a con-
siderable effect on the resting and synaptic membrane poten-
tials of motoneurons. Even in individual muscle fibers, a salt
flux of 6 X 1014 moles/s is thought to have an appreciable
effect if the electrode is left in place for more than a few
minutes [8].
To minimize diffusion, it is suggested that the concentration
of the micropipette electrolyte be reduced whenever consistent
with other considerations. A reduction to 0.5M KCI, for ex-
ample, would reduce the diffusion flux by approximately a
factor of six (14). The resistance of the micropipette would
rise, but by considerably less than a factor of six [10]. Junc-
tion potentials would rise substantially [1], but time-varying
potentials should not be affected. “Tip potentials,” which do
not seem to be directly related to electrode resistance [1],
[17], should remain low at this concentration [4]. In short,
considerably reduced diffusion could be purchased at the cost
of somewhat larger resistance, and somewhat larger tip po-
tentials and liquid junction potentials.
REFERENCES
(1] R. H. Adrian, “The effect of internal and external potassium con-
centration on the membrane potential of frog muscle,” J. Physiol.,
vol. 133, pp. 631-658, 1956.
[21 T. F. Weiss, W. T. Peake, and H. S. Sohmer, “Intracochlear potential
recorded with micropipets. Part II. Responses in the
cochlear scalae to tones,” J. Acoust. Soc. Amer., vol. 50, pp. 587-
601, 1971.
[31 K. S. Cole and J. W. Moore, “Liquid junction and membrane po-
tentials of the squid giant axon,” J. Gen. Physiol., vol. 43,
pp. 971-980, 1960.
[41 0. F. Schanne, M. Lavell6e, R. Laprade, and S. Gagne, “Electrical
properties of glass microelectrodes,” Proc. IEEE (Special Issue on
Studies of Neural Elements and Systems), vol. 56, pp. 10721082,
June 1968.
[51 D. R. Curtis, “Microelectrophoresis,” in Physical Techniques in
Biological Research, vol. V, W. L. Nastuk, Ed. New York:
Academic, 1964.
[61 K. Krnjevic, J. F. Mitchell, and J. C. Szerb, “Determination of
iontophoretic release of acetylcholine from micropipettes,”
J. Physiol., vol. 165, pp. 421-436, 1963.
[71 G. G. Harris, L. S. Frishkopf, and A. Flock, “Receptor potentials
from hair cells of the lateral line,” Science, vol. 167, pp. 76-79,
1970.
[81 W. L. Nastuk and A. L. Hodgkin, “The electrical activity of single
muscle fibers,” J. Cell. Comp. Physiol., vol. 35, pp. 39-73, 1950.
[91 D. P. Agin, “Electrochemical properties of glass microelec-
trodes,” in Glass Microelectrodes, M. LavallUe, 0. F. Schanne, and
N. C. Hebert, Eds. New York: Wiley, 1969.
[10] R. Lanthier and 0. F. Schanne, “Change of microelectrode re-
sistance in solutions of different resistivities,” Naturwissenschaf-
ten, vol. 53, p. 430, 1966.
[111 M. Lavallee and G. Szabo, “The effect of glass surface conduc-
tivity phenomena on the tip potential of micropipette elec-
trodes,” in Glass Microelectrodes, M. Lavallee, 0. F. Schanne,
and N. C. Hebert, Eds. New York: Wiley, 1969.
[121 R. B. Bird, W. E. Stewart, and E. N. Lightfoot, Transport
Phenomena. New York: Wiley, 1960.
[13] M. Abramowitz and I. A. Stegun, Handbook of Mathenatical
Functions with Formulas, Graphs, and Mathematical Tables
(Applied Mathematics Series 55). Washington, D.C.: NBS, 1964,
p. 298.
[141 A. L. Hodgkin and R. D. Keynes, “The mobility and diffusion
coefficient of potassium in giant axons from sepia,” J. Physiol.,
vol. 119, pp. 513-528, 1953.
[15] R. A. Robinson and R. H. Stokes, Electrolyte Solutions. London,
England: Butterworths, 1959, p. 513.
[16] J. C. Eccles, R. M. Eccles, and M. Ito, “Effects produced on in-
hibitory postsynaptic potentials by the coupled injections of
cations and anions into motoneurons,” Proc. Roy. Soc., Ser. B,
vol. 160, pp. 197-210, 1964.
[171 P. G. Kostyuk, “Intrinsic potentials of glass microelectrodes,”
Fed. Proc., vol. 24, pp. T329-T332, 1965.
375
SOLUTION
Diffusion Effects of Liquid-Filled Micropipettes: A Pseudobinary Analysis of Electrolyte Leakage
C. DANIEL GEISLER, MEMBER, IEEE, EDWIN N. LIGHTFOOT, FRANK P. SCHMIDT, AND FRANCISCO SY
Abstract-The importance of diffusion phenomena in glass micropipettes filled with concentrated electrolyte solutions is well known. Nonequilibrium diffusion-rate equations for electrolyte-filled glass micropipettes are given for a model of situations of current interest: diffusion into an infinite medium. The solutions of these equations give the flux of material out of a micropipette as a function of time, concentration, and geometry.
INTRODUCTION
ALTHOUGH electrolyte-filled micropipettes have proven extremely useful in lectrophysiological experimentation, interpretation of potential measurements using these electrodes is subject to significant ambiguities. For one, “tip potentials” exist that may change with time or upon penetration of a cell [1]. The filtering effects of the electrical circuit to which the micropipette is attached may also be important [2]. Moreover, a liquid-junction potential exists between the electrolyte in the micropipette and the physiological fluid in which the tip is immersed [3]. The use of 3M KCI to minimize these effects is widespread. The use of this highly concentrated KCl reduces the tip potential [4], reduces the resistance of the
Manuscript received April 12, 1971; revised November 8, 1971.
C. D. Geisler is with the Department of Electrical Engineering and the Laboratory of Neurophysiology, University of Wisconsin, Madison, Wis.
E. N. Lightfoot, F. P. Schmidt, and F. Sy are with the Department of
Chemical Engineering, University of Wisconsin, Madison, Wis.
micropipette [1], and fixes the liquid-junction potential between the micropipette’s electrolyte and
axoplasm at no more than 3-4 mV [3].
The resulting difference in concentrations between the micropipette electrolyte and the less-concentrated ionic environment into which the micropipette is inserted will cause an outward diffusion of electrolyte. The effects of such diffusion may alter the external ionic environment and thus are of concern [5]. Such diffusion effects have in fact been experimentally demonstrated for molecules such as acetylcholine [6]. Electrolyte diffusion has even been used as an intracellular marking technique: while recording intracellularly, Harris et al. [7] used diffusion of Niagara Sky Blue dye from the recording electrode to mark the penetrated cells.
Quantitative descriptions of diffusion rates from micropipettes are not common in the literature, but a steady-state solution has been obtained by Nastuk and Hodgkin [8] and by Krnjevic et al. [6]. It is the purpose of this paper to present a more general, time-dependent description of electrolyte diffusion.
ANALYSIS
Since the geometry of experimental systems is quite variable, and frequently not known in detail, it is desirable to consider simplified models for possible limiting cases. We consider here one such model: diffusion from a long capillary tip into a very large unstirred pool of electrolyte, representing the interior of a large cell; a cell with high permeabilities for the ions of the micropipette’s electrolyte; or a large volume of extracellular fluid.
We could have considered other situations as well, but analysis of this model suggests that capillary diffusion can be treated by a pseudosteady model for situations of present interest. For this reason, it appears unnecessary to model external behavior in detail.
Diffusion in microelectrodes is actually multicomponent in nature. Diffusional interactions can, however, be safely neglected for our present purposes, and we shall use a pseudo-binary approach here. In other words, we shall treat the diffusion of a single electrolyte from a micropipette as if the solution consisted only of that electrolyte and water.
For purposes of this analysis we shall also omit the surface charges in the glass and the corresponding electrical “double layer” [9]. Strictly speaking, such an analysis is accurate only for a micropipette having a wide tip opening and filled with a highly concentrated electrolyte. Nevertheless, it may also provide a reasonable approximation to micropipettes with smaller tips and lower concentrations under some conditions. For example, if the micropipette is made out of the commonly used borosilicate glass, it will have a relatively low surface charge [4]. Furthermore, it is known that the Debye length, the effective width of the double layer, is extremely small at salt concentrations of physiological magnitude or above (e.g., g3m at 100 moles/l in a 1: 1 electrolyte [9]). Hence, for electrolyte concentrations greater than 0.IM in a borosilicateglass electrode with a 0.5-l m diameter tip, double-layer effects should be negligible over most of the cross section of the tip. This contention is supported by several different experimental findings. First, there is good agreement between observed electrode resistances and those calculated for several types of micropipettes assuming free field diffusion (10]. Secondly, pH, which is known to greatly affect Debye length, has relatively little effect on the resistance of Pyrex micropipettes [4]. Thirdly, “tip potentials,” thought by some to be due to the double layer, are relatively low for Pyrex micropipettes filled with 0.5M KCl [4], [11].
We consider here the idealized system of Fig. 1 in which a capillary is in contact with a limitless unstirred volume. The capillary is assumed to be conical, with an angle of taper, and is truncated near the tip. Initially, the electrolyte within the tip and that in the external solution are at different uniform concentrations. The surface of contact between the capillary and external solution is assumed to be a sphere with a radius equal to that of the capillary bore. This is a considerable simplification of the actual situation, but it will be shown that a more detailed geometric model is not needed. It is convenient to use the spherically symmetrical form of the diffusion equation in the external solution [12]. It is now possible to write the differential equations and boundary conditions for the problem as follows.
A. External Solution
where is the solute concentration in the external solution and is the effective binary diffusivity of the solute in the external solution.
B. Intracapillary Solution
whereand are the concentration and effective binary diffusivity of the same solute in the intracapillary solution. Notice that ce and ci are dimensionless salt concentrations defined by
where cs is salt concentration (moles/liter) in the specified solution, (cse)o is initial salt concentration (moles/liter) in the external solution, and (csi)o is initial salt concentration (moles/liter) in the internal solution.
C. Matching Conditions at the Interface Between the Capillary and the External Solution
It remains to obtain a match of the two salt solutions in the neighborhood of the capillary tip. Minute vibrations of the tip are to be expected, and one can envision an area at the tip where the solution is well stirred. Therefore, it seems reasonable to assume that the concentrations just inside and just outside the tip are equal. That is,
Furthermore, since no accumulation of material at the interface can be expected, the flux of material out of the micro-pipette will be equal to the flux of material into the cell. The equality of material that flows at the micropipette-cell interface is given in (8). The negative sign accounts for the difference in direction of the two flows:
where Ai is the cross-sectional area ] of the part of the surface ri = a that is enclosed by the micropipette and is the area () of the imaginary sphere at the end of the micropipette from which diffusion occurs into the infinite medium.
The partial differential equations (1) and (4) can be simplified by taking their Laplace transforms with respect to time. The resulting ordinary differential equations can be solved using the Laplace-transform expressions of the boundary conditions and matching conditions. If dimensionless variables are used, the Laplace transform of is found to be
where , , , and . Equation (9) along with a corresponding equation for, which we shall not need, completes the formal solution.
A general inversion of (9) appears difficult to accomplish, so we have concerned ourselves only with conditions at (). The concentration profile at as a function of time is found to be
where. Thus tip concentration is, in general, time dependent. However, since the limit of the time dependent term for long times is zero, it is clear that . Furthermore, since tip angles are usually small (, we find that). Hence the concentration in the neighborhood of the capillary tip approaches that of the external solution. The time-dependent term in (10) can be accurately represented by a finite asymptotic series for large values of [13]. Specifically, for , (10) can be rewritten as
with an error in the time-dependent term of less than approximately ten percent. Substituting the definitions of and in (11) and assuming that = and , we find that
Equation (12) consists of two terms, each of which makes a contribution to the value of the tip concentration at a given time. The first term depends on the taper and approaches zero as the micropipette approaches a cylindrical shape. The second term depends only on the diffusivities of the material in the micropipette and the surrounding solution. If we evaluate (12) numerically, assuming = ([14], [15]), and (corresponding to an angle of approximately 1.5°), we find that (12) is approximately correct by the time 0.5 ms has elapsed, and the tip concentration has dropped to within a factor of 2 of its final value within 30 ms. Thus the interfacial concentration very rapidly approaches a constant value.
For any realistic estimate of it may be seen that . Thus most of the diffusional resistance is concentrated within the pipette, and that in the external solution is small. This situation, which results from the very small taper of typical micropipette tips, will permit considerable simplification when we consider diffusion into cells of limited volume.
The dimensionless instantaneous flux at is .[M1] Upon Laplace inversion, it is found that
where b is defined in (10). Making the same approximations as for (12), we find that
Once again, the time-dependent term will rapidly become insignificant. The constant term on the right is the long-term limit and agrees exactly with the steady-state results given by Krnjevic et al. [6]. The second term is independent of taper and would be the sole term if the micropipette were cylindrical. Notice that R, the effective radius of the imaginary sphere at the end of the micropipette, enters the approximate solution only in the time that must elapse before (14) becomes reasonably accurate. Hence, the particular R chosen for analysis purposes is not of critical mportance.
It is also interesting to know the total amount of salt that passes out of the micropipette and an expression for the normalized efflux can be obtained by integrating (13). We thus find that
The amount of salt M that passes out of the micropipette can be found by multiplying (15) by]. For the same assumptions as for (12) we find that
As before, the first term, which dominates at large time, is due to micropipette taper, while the second term is the “cylindrical” contribution.
In the above approximation to the physical situation, diffusion was considered to take place into an infinite medium. Also considered was the case of diffusion out of a micropipette into a cell of limited volume of well-mixed fluid, representing the interior of a cell with low permeability for the ions of the micropipette’s electrolyte. For short times it was found that the initial salt flux for the two cases was identical. For example, there is an initial salt flux of approximately 6 10-14 moles/s from a 3M KCl-filled micropipette with a 1/2-m diameter cell. After a few minutes, this flux, which initially is exactly equal to that calculated for the infinite medium case (14), decreased due to the buildup of salt concentration in the limited volume.
DISCUSSION
It is clear from the above analysis that system behavior is not sensitive to the geometry external to the pipette or to boundary conditions, and that transients are of minor importance under the usual conditions of micropipette operation. Therefore, one can normally obtain satisfactory predictions by assuming pseudosteady behavior and neglecting diffusional resistance outside the pipette.
The above two assumptions were used by Krnjevic et al. [6] to calculate steady-state diffusion rates. There was good agreement between their theoretical results, which are identical to our long-term solution, and their experimental data [6]. Lanthier and Schanne also used such a pseudo steady-state model in calculating micropipette impedances that agreed
quite well with their experimental measurements [101. The ability of the pseudo steady-state model to accurately match the experimental data of these workers [6], [101 therefore provides support for the work presented in this paper.
The ion fluxes calculated from our models can be appreciable. For example, the flux of electrolyte from a 3M KCl-filled micropipette with a 1/2-m diameter tip is approximately 6 10-14 moles/s (14). This is approximately one quarter of the flux produced by electrophoretic ion injections of KCl using 50 nA of current. This latter magnitude of ion injection is quite significant and has been shown [16]. to have a considerable effect on the resting and synaptic membrane potentials of motoneurons. Even in individual muscle fibers, a salt flux of 6 10-14 moles/s is thought to have an appreciable effect if the electrode is left in place for more than a few minutes [8].
To minimize diffusion, it is suggested that the concentration of the micropipette electrolyte be reduced whenever consistent with other considerations. A reduction to 0.5M KCI, for example, would reduce the diffusion flux by approximately a factor of six (14). The resistance of the micropipette would rise, but by considerably less than a factor of six [10]. Junction potentials would rise substantially [1], but time-varying potentials should not be affected. “Tip potentials,” which do not seem to be directly related to electrode resistance [1], [17], should remain low at this concentration [4]. In short, considerably reduced diffusion could be purchased at the cost of somewhat larger resistance, and somewhat larger tip potentials and liquid junction potentials.
REFERENCES
[1] R. H. Adrian, “The effect of internal and external potassium concentration on the membrane potential of frog muscle,” J. Physiol., vol. 133, pp. 631-658, 1956.
[2] T. F. Weiss, W. T. Peake, and H. S. Sohmer, “Intracochlear potential recorded with micropipets. Part II. Responses in the cochlear scalae to tones,” J. Acoust. Soc. Amer., vol. 50, pp. 587- 601, 1971.
[3] K. S. Cole and J. W. Moore, “Liquid junction and membrane potentials of the squid giant axon,” J. Gen. Physiol., vol. 43, pp. 971-980, 1960.
[4] O. F. Schanne, M. Lavell6e, R. Laprade, and S. Gagne, “Electrical properties of glass microelectrodes,” Proc. IEEE (Special Issue on Studies of Neural Elements and Systems), vol. 56, pp. 1072- 1082, June 1968.
[5] D. R. Curtis, “Microelectrophoresis,” in Physical Techniques in Biological Research, vol. V, W. L. Nastuk, Ed. New York: Academic, 1964.
[6] K. Krnjevic, J. F. Mitchell, and J. C. Szerb, “Determination of iontophoretic release of acetylcholine from micropipettes,” J. Physiol., vol. 165, pp. 421-436, 1963.
[7] G. G. Harris, L. S. Frishkopf, and A. Flock, “Receptor potentials from hair cells of the lateral line,” Science, vol. 167, pp. 76-79, 1970.
[8] W. L. Nastuk and A. L. Hodgkin, “The electrical activity of single muscle fibers,” J. Cell. Comp. Physiol., vol. 35, pp. 39-73, 1950.
[9] D. P. Agin, “Electrochemical properties of glass microelectrodes,” in Glass Microelectrodes, M. LavallUe, 0. F. Schanne, and N. C. Hebert, Eds. New York: Wiley, 1969.
[10] R. Lanthier and 0. F. Schanne, “Change of microelectrode resistance in solutions of different resistivities,” Naturwissenschaften, vol. 53, p. 430, 1966.
[11] Lavallee and G. Szabo, “The effect of glass surface conductivity phenomena on the tip potential of micropipette electrodes,” in Glass Microelectrodes, M. Lavallee, 0. F. Schanne, and N. C. Hebert, Eds. New York: Wiley, 1969.
[12] R. B. Bird, W. E. Stewart, and E. N. Lightfoot, Transport Phenomena. New York: Wiley, 1960.
[13] M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables (Applied Mathematics Series 55). Washington, D.C.: NBS, 1964, p. 298.
[14] L. Hodgkin and R. D. Keynes, “The mobility and diffusion coefficient of potassium in giant axons from sepia,” J. Physiol., vol. 119, pp. 513-528, 1953.
[15] R. A. Robinson and R. H. Stokes, Electrolyte Solutions. London, England: Butterworths, 1959, p. 513.
[16] J. C. Eccles, R. M. Eccles, and M. Ito, “Effects produced on inhibitory postsynaptic potentials by the coupled injections of cations and anions into motoneurons,” Proc. Roy. Soc., Ser. B, vol. 160, pp. 197-210, 1964.
[17] P. G. Kostyuk, “Intrinsic potentials of glass microelectrodes,” Fed. Proc., vol. 24, pp. T329-T332, 1965.
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