Directory UMM :Data Elmu:jurnal:A:Advances in Physiology Education:Vol277.Issue6.Dec1999:

A SIMPLE, INEX PENSIVE METHOD FOR TEACHING

HOW MEMBRANE POTENTIALS ARE GENERATED

William M. Moran, Jerod Denton, Kelly Wilson, Matt Williams, and Steven W. Runge Depa rtm ent of Biology, University of Centra l Ark a nsa s, Conwa y, Ark a nsa s 72035-0001

W

e have developed a simple laboratory exercise that uses an inexpensive dialysis membrane (molecular weight cutoff5100) to illustrate the genera-tion of membrane potentials (Vm) across plasma membranes of animal cells. A

piece of membrane,2.0 cm2is mounted in an Ussing-like chamber. One chamber half is

designated cytosol and the other half external. Chamber sidedness helps students relate their findings to those of real cells. As in real cells, outward directed K1_{concentration}

gradients [high cytosolic K1_{concentration ([K}1_]

c) and low extracellular K1

concentra-tion] generate cytosol electrically negative Vm with a slope of approximately 245

mV/decade change in [K1_]

c. The polarity ofVmreflects the outward flow of potassium

ions because flow of the larger counterion, H2PO42, is restricted by the pores in the

membrane. A slope less than Nernstian (,59 mV/decade) suggests that the membrane is slightly permeable to H2PO42. Importantly, this facilitates teaching the use of the Nernst

equation to quantify the relationship between ion concentration ratios across mem-branes and magnitude of Vm. For example, students use their data and calculate a

permeability ratioPK/PH₂PO₄that corresponds to a slope,24% less than Nernstian. This

calculation shows that Nernstian slopes are achieved only when permeability to the counterion is zero. Finally, students use the concept of membrane capacitance to calculate the number of ions that cross the membrane. They learn where these ions are located and why the bulk solutions conform to the principle of electroneutrality.

AM. J. PHYSIOL. 277 (ADV. PHYSIOL. EDUC. 22): S51–S59, 1999.

Key words:diffusion potentials; Nernst equation; dialysis membranes

The membrane potential (Vm) in animal cells is

gener-ated largely by intracellular potassium ions (K1₎ flow-ing through plasma membrane K1_{-selective leak} chan-nels to the cell exterior (3, 7, 9). In animal cells, Vm

plays a vital role in a number of physiological pro-cesses, such as electrical signaling in excitable cells, Na-coupled transport of nutrients by small intestine, muscle contraction, brain function, sensory percep-tion, secretion of neurotransmitters, postsynaptic gen-eration of action potentials, cell signaling, secretion of insulin by pancreatic cells, secretion of Cl2_{by airway} epithelia, and ion transport across epithelial cells lining the nephron (7 ). Nearly one-third of the cell’s

supply of ATP powers the plasma membrane Na1_-K1 -ATPase (Na1_{pump) that maintains the high} intracellu-lar K1 _{concentration necessary to generate V}

m (1).

Because of the ubiquitous role ofVmin physiological

processes and the large fraction of the cell’s energy supply devoted to maintainingVm, it is essential that

beginning physiology students have a firm grasp on how membrane potentials are generated.

However, teaching the concept of a membrane poten-tial is very difficult (11). Students often become confused with the bewildering array of membrane pumps and channels and transmembrane ion

ents responsible for generatingVm. There are several

reasons for the confusion.1) In most cells, the plasma membrane electrogenic Na1 _{pump, whose activity} maintains the high intracellular K1 _{concentrations,} contributes little to Vm in part because the pump

behaves as a source of constant current; it has an infinitely high resistance (5).2) The high concentra-tion side of the membrane has an electrical polarity opposite that of the charge carried by the membrane permeant ion. 3) Even though ions cross the mem-brane to generateVm, bulk solution electroneutrality is

maintained (3, 4, 9). Therefore, before students can appreciate how the interaction between membrane pumps and leaks generate membrane potentials in real cells, they must understand how the flow of ions across membranes generates membrane potentials. For a current, detailed analysis of the origin of resting membrane potentials, see Sperelakis (13). A descrip-tion of the interacdescrip-tion between plasma membrane ion pumps and ion channels in generating membrane poten-tials in animal cells is found in Byrne and Schultz (3). Accordingly, we developed a simple, hands-on exer-cise that employs a mock cell to generate membrane potentials. The apparatus and approach presented here follow, in part, a laboratory exercise developed at California State University at Hayward (CSUH) that used cation- and anion-selective membranes to gener-ate membrane potentials (8). It is also similar to the experiment published by Manalis and Hastings in 1974 (11). In both of those laboratory exercises, K1 gradients were imposed across the cation-selective membrane andVmwere generated with nearly

Nern-stian slopes, 58 mV/decade. These exercises revealed the Nernst equilibrium potential for K1_{and showed} that sustained K1 _{gradients maintain V}

m constant.

Unfortunately, the ion-selective membranes used are no longer available, and the description of the CSUH experiment (8) is no longer in print. We searched for a suitable alternative membrane and found a new, low molecular weight cutoff (MWCO5 100) membrane, manufactured by Spectrum, that served our purpose. With the dialysis membrane, if one ion of a dissociable salt can pass through the membrane while the counte-rion is restricted because of its larger size, then a membrane potential will be generated when the two solutions bathing the membrane have different salt concentrations (see Fig. 1).

Compared with teaching the concept of membrane potentials with microelectrode techniques and the above-mentioned ion-selective membranes, the Spec-trum dialysis membrane offers several advantages. 1) It is relatively inexpensive; an 8 3 10-in. sheet costs about $62.00 (Spectrum part no. 133080), and,40 pieces of experimental membrane can be obtained from a single sheet. 2) The membrane is shipped ready to use and requires no time-consuming cleaning procedure.3) Because the relationship between trans-membrane ion concentration ratio andVmis less than

Nernstian, a permeability ratio can be calculated for the permeant ion and the less permeant counterion. Importantly, this helps students relate the magnitude of Vm to that predicted by the Nernst equation. 4)

Expensive equipment such as oscilloscopes and

elec-FIG. 1.

The por ous [molecular weight cutoff (MWCO) 5100] dialysis membrane separates two compartments, called the cytosol and ex ter nal compartments. The brane allows the small cations (1) to cr oss the mem-brane down the concentration gradient, wher eas the lar ger anions (2) ar e unable to pass thr ough the membrane por es. Although ions actually cr oss the membrane to generateV_m, they do not the enter bulk solution; thus electr oneutrality is maintained in bulk solutions on both sides of the membrane. As soon as cations begin to flow acr oss the membrane, a mem-brane potential (V_m) develops, which pr events further movement of cations acr oss the membrane. [Modified fr om Becker et al. (2)]

trometers is not required; rather, a simple pH meter operated in voltage mode suffices. 5) The dialysis membrane chamber is relatively inexpensive and can be purchased from Jim’s Instrument Manufactur-ing (Iowa City, IA). 6) The entire experiment is designed around a mock cell with cytosol and extracel-lular fluid compartments to help students relate their results to conditions found in real cells. 7) Because membrane thickness is known, a value for membrane capacitance can be derived with a simple assumption. Hence, students employ the concept of membrane capacitance and calculate the number of potassium ions that c ross the membrane to generate Vm. 8) The approach used

here employs the scientific method such that stu-dents formulate hypotheses about polarity and magni-tude of Vm before they conduct experiments and

analyze data.9) Changes in ion activities in different concentrations of the salt solutions are of no concern because dilute (0.1–10 mM) solutions are employed. Change in ion activity is a particularly difficult concept for students to grasp and adds to the complexity of data analysis when salt concentrations approach 100 mM.

The experiments described here give students a hands-on experience with the generation of mem-brane potentials. In addition, students calculate the number of ions that cross the membrane to generate Vm, and they are introduced to the Nernst equation as

an equilibrium expression and learn how this impor-tant electrochemical equation is used in electrophysi-ological studies. The second experiment illustrates the role of the transmembrane Na1_{gradient in reversal of} Vmduring an action potential in excitable cells (7 ).

EXPERIMENTAL APPARATUS

The experimental apparatus is relatively simple (Fig. 2). An ,2-cm2 piece of membrane is cut from the sheet of dialysis membrane and placed in 0.1 mM KH2PO4 for ,10 min to allow antibacterial agents to diffuse from the membrane pores. We have found that placing the piece of membrane in distilled water gives anomalous results. After a light layer of silicone grease is applied around the chamber opening on each chamber half, the membrane is mounted by gently forcing the membrane down over the chamber’s membrane holding pins with stainless steel forceps. The two chamber halves are slipped together, guided

by two large stainless steel guide pins (Fig. 2). The two chamber halves are held together in a utility vise (jaw width 2.5 in., jaw opening 29_⁄

16in., jaw depth 1.5 in.,

and overall length 71_⁄ 4in.).

The two chamber halves are filled with appropriate salt solutions, and KCl-agar bridges are inserted into each chamber half to make an electrical connection to the calomel electrodes (catalog no. 13-620-258, Fisher Scientific). The agar bridges are prepared by boiling agar (3%, bacteriological; catalog no. A-5306, Sigma) in 3 M KCl dissolved in distilled water. Polyethylene (PE) tubing is then filled with the hot agar-KCl mixture from a plastic syringe fitted with a large stainless steel needle. To comply with electrophysiological conven-tion, the extracellular electrode is connected to ground on the pH meter (Digital Chemcadet model 5984-50, Cole-Parmer Instrument), and the other electrode is connected to the cytosol side of the chamber. In our studies, the output of the pH meter (i.e.,Vm) is fed to a

Vernier analog-to-digital converter. The digitized sig-nal is displayed on the screen of a Macintosh com-puter, employing Vernier’s Data Logger software. SOLUTIONS

We have experimented with several salts and found three that give reliable and reproducible results: K2HPO4, KH2PO4, and Na2HPO4, all dissolved in

dis-tilled water to give stock concentrations of 10 mM. Salts that failed to give reliable results were KCl, potassium tetrathionate, potassium pyrosulfate, potas-sium nitrate, potaspotas-sium sulfate, potaspotas-sium gluconate, sodium diatrizoate, and sodium citrate. The stock solutions of K2HPO4, KH2PO4, and Na2HPO4 are

di-luted to give concentrations of 1 and 0.1 mM for use in the experiments. To obtain additional data points, concentrations of 0.5 mM and 5 mM may be used. Although identical results are obtained with the two K1 _{phosphate salts, it is best to use KH}

2PO4 to

calculate the permeability ratioPK/PH₂PO₄, as described

inEXPERIMENTS.

Str eaming potentials do not contribute toV_m.Ion

concentration gradients are established across the dialysis membrane by placing salt solutions of differ-ent concdiffer-entrations in the cytosol and extracellular compartments. Thus osmotic gradients are also gener-ated that could cause streaming potentials due to water flow ‘‘dragging’’ ions across the membrane

(10). To test for streaming potentials, we added mannitol to the salt solutions on the low salt concen-tration side of the membrane to achieve osmolarities equal to those on the high salt concentration side. We then compared Vm to measurements made in the

absence of mannitol and found no difference between the two. Therefore, streaming potentials do not make measurable contributions toVmin these experiments.

EXPERIMENTS

Effect of Incr easing Cytosolic K1_{Concentration} onV_m

In the first experiment, students explore the relation-ship between transmembrane K1_{gradients andV}

m. To

mimic K1_{gradients present across plasma membranes}

of real cells, students consult Table 1 and determine in which compartment they increase K1_{concentration.} Importantly, this gives students insight into the nature of the K1_{concentration gradient. Next, they develop} a hypothesis, which includes statements about magni-tude and polarity ofVm, to predict what will happen to

Vmwhen they increase the cytosolic K1concentration

([K1_]

c) while holding the extracellular K1

concentra-tion ([K1_]

e) constant. In keeping with convention (see

Figs. 5–14B, Ref. 7), instructors may choose to begin the experiment with high [K1_]

c and increase [K1]e,

which depolarizesVm.

The experiment is started by placing 0.1 mM KH2PO4

in each chamber half. The potential displayed on the pH meter represents zero Vm. A 3-ml syringe with a

FIG. 2.

The ex perimental apparatus consists of a dialysis membrane chamber, agar bridges, calomel electr odes to detectV_m, a pH meter that serves as an electr ometer, and an analog-to-digital (A/ D) converter [Serial Box Inter-face (SBIMac) Ver nier Softwar e, Portland, OR; www.ve r nie r .com] and a Macintosh computer for displaying and recording the signal from the pH me-ter. The ex ternal calomel electrode is connected to a ground on the pH meter in keeping with electrophysiological convention. BNC, coax ial cable connector.

piece of PE tubing attached is used to empty and refill the chamber halves when changing salt solutions. When the salt concentration is increased in a chamber half, the chamber should be rinsed at least once with the new salt solution beforeVmis recorded. AfterVmis

recorded with the higher KH2PO4concentrations, 0.1

mM phosphate salt should be returned to the cytosolic chamber to check for drift. Usually little drift occurs. However, in some cases drift of a few millivolts occurs, and either this must be added to or subtracted from Vm or the experiment must be repeated. With

the recording arrangement illustrated in Fig. 2, in-creases in [K1_]

cgenerate a cytosol negative Vm. After

Vm is recorded with 0.1, 1, and 10 mM KH2PO4

(students should also include concentrations of 0.5 and 5 mM) in the cytosol compartment, the students plot a graph ofVmversus [K1]c(Fig. 3A) and then plot

Vmversus the ratio of [K1]eto [K1]c, that is, ratios of 1,

0.1, and 0.01 (Fig. 3B). This graph shows that the ratio of potassium ion concentrations across the membrane is the important factor in generation ofVmbecause a

ratio of 1.0 gives a Vm of zero. With this plot, Vm

becomes increasingly more negative as a function of [K1_]

e/[K1]c, but not in a linear manner.

The next objective is to establish a linear relationship between Vm and [K1]e/[K1]c. When doing this,

stu-dents will be introduced to the Nernst equation and to the reason for taking the log [K1_]

e/[K1]cto establish a

linear relationship between Vm and the ratio of K1

concentrations on the two sides of the membrane. Ideally, measurements ofVmare made when diffusion

forces (concentration gradient) and electrical forces (Vm) are equal, that is, at equilibrium. Thus we can

equate these two forces. We can writeEfor electrical force and Ce/Cc for concentration ratio (extracellular

to c ytosolic ) or c onc entration forc e in the K1 transmembrane chemical gradient, where C repre-sents concentration (in mol/l). At equilibrium we then have

E5Ce/Cc (1)

The pr oblem with Eq. 1. At this point the students should be asked what the problem is with this equation. Clearly, the units of the two forces are not the same. To rectify this situation, the concentration ratio (Ce/Cc) is multiplied by a constant k that must

have units of volts. Now the units on the two sides of

FIG. 3.

A: plot ofV_m vs. cytosolic K1_{concentration ([K}1_]c) while ex ter nal K1_{concentration ([K}1_{]e) is held at 0.1} mM. B: plot of [K1_{]e/ [K}1_{]c vs.} V_m gives a nonlinear r elationship. As [K1_]

cincr eases (i.e., decr ease in [K1]e/ [K1_]

c),Vmbecomes mor e negative.Vmr epr esents the electrical potential of the cytosol compartment r efer-enced to the ex ter nal compartment (n₅3; standar d deviations ar e smaller than data point symbols). TABLE 1

Cytosol and ex ter nal Na1_{and K}1_{concentrations} and their ratios in squid nerve and fr og muscle

Tissue

Cytosolic Concentration,

mM

External Concentration,

mM

Ce/Cc

Na1 _K1 _Na1 _K1 _Na1 _K1

Squid nerve 49 410 440 22 8.8 0.05 Frog sartorius muscle 10 140 120 2.5 12 0.02 Data are from Eckert et al. (6). Ce/Cc, ratio of external to cytosolic

the equation are the same, and we can rewriteEq. 1as

E5k(Ce/Cc) (2)

Next, the constant k should be explored. In other words, where doeskcome from and why does it have units of volts?kis equal toRT/zF, where Ris the gas constant (2 cal/mol 3 Kelvin), T is the absolute temperature (Kelvin units), z is the valence of the membrane permeant ion, and F is the Faraday con-stant (23,062 cal/mol3 V). The Faraday constant is the conversion factor that converts energy in a chemi-cal concentration gradient (chemi-cal/mol) into equivalent energy in an electrical gradient (volts). At this point we have found it useful to have the students derive the units of k to convince themselves that it has units of volts. NowEq. 2 can be rewritten to include RT/zFas

E5(RT/zF)(Ce/Cc) (3)

However, a plot of E, orVm, against (RT/zF)(Ce/Cc)

still gives a nonlinear relationship. To establish a linear relationship between E and (RT/zF)(Ce/Cc), the

stu-dents should be asked how the numbers on thex- and y-axes differ. On the y-axis the numbers represent an arithmetic progression, such as 10, 20, 30, 40, etc., because the difference between any two successive numbers is constant. On the x-axis the numbers represent a geometric progression in which Ce/Cc

decreases as 1, 0.1, 0.01 (12). In this progression, the ratio between any two successive numbers is constant and the numbers decrease exponentially. If we take negative logarithms of the numbers on thex-axis, we obtain numbers that represent an arithmetic progres-sion of 0, 1, 2, etc. Now, if 2ln (Ce/Cc) is plotted

against Vm, one arithmetic progression is plotted

against another arithmetic progression and a linear relationship is established between the two variables (Fig. 4).

If we expandEq. 3to include ln (Ce/Cc) as follows

E5(RT/zF) ln (Ce/Cc) (4)

then we have the familiar Nernst equation. This is usually written as the log (Ce/Cc)

E5(2.3RT/zF) log (Ce/Cc) (5)

Eis now referred to as the equilibriumVm. The value

of 2.3RT/zFis 59 mV at 23°C. With a monovalent salt, a 10-fold ratio for Ce/Ccwill generate aVmof 59 mVif

the m em bra ne is perm ea ble to one ion only. Accord-ingly, the slope of the relationship between Vm and

2log (Ce/Cc) has a value of 59 mV/decade change in

Ce/Cc and is referred to as a Nernstian slope. The

significance of the Nernst equation is that it describes the mathematical relationship between an ion chemi-cal concentration difference (or gradient) across a membrane andVmunder equilibrium conditions, i.e.,

when there is no net flow of ion across the membrane andVmis constant (2, 3).

Why is the slope less than Ner nstian?We suggest that the following analysis be performed by the students as a homework assignment.

At this point, the students should be asked why the slope of their relationship between Vm and 2log

FIG. 4.

Plot of negative logarithm of concentration ratio (2log [Ce/ Cc]) vs.Vmgives a linear r elationship. The slopes obtained with K2HPO4(k;n57) and KH2PO4 (s;n₅3) wer e nearly identical, 45 vs. 43 mV per 10-fold change in [K1_]

e/ [K1]c, r espectively. Ther e-for e, a line was fit only to the K2HPO4data. For Na2HPO4 (q), n 5 4. Standar d deviations range fr om63–6 mV.V_mr epr esents the potential of the cytosol compartment r efer enced to that of the ex ter nal compartment. The Ner nstian r elation was calculated with the Ner nst equation for a monova-lent cation in which the salt concentration was incr eased in the ex ter nal solution. We assume that less-than-Ner nstian slopes r esult fr om a slight per-meability of the dialysis membrane to the counte-rion H2PO42(PH2PO4).

(Ce/Cc) for KH2PO4 is less than Nernstian. As noted

above, a Nernstian slope is obtained only if the membrane is permeable to one ion. This suggests that the dialysis membrane is somewhat permeable to the counterion, H2PO42. A small permeability to H2PO42is

not unreasonable, given that H2PO4

2_{has a molecular} weight of 96 and the dialysis membrane has an MWCO of 100. Hence, we assume that the membrane is slightly permeable to H2PO42. These data allow the

students to calculate a permeability ratio PH₂PO₄/PK

and to demonstrate how permeability to the counte-rion generates less-than-Nernstian slopes. Further-more, this calculation reveals that Nernstian slopes are achieved only when the membrane is permeable to one ion.

For a membrane permeable to both ions of a disso-ciable salt, such as KH2PO4,Vmis given by (3)

Vm5(PK2PH₂PO₄)/(PK1PH₂PO₄)

359 mV.log ([KH2PO4]e/[KH2PO4]c) (6)

This expression is derived from the equation that gives the diffusion potential arising from the diffusion of a salt that dissociates into monovalent cations and anions (3). Because permeability is proportional to the diffusion coefficient, we can substitute perme-abilities for diffusion coefficients (3). Furthermore, the students should be aware that this equation reduces to the Nernst equation for K1 _{if membrane} permeability to H2PO42approaches zero. From Fig. 4,

the slope is,45 mV/decade change in [K1]c(we use

the absolute value to simplify the calculation). We have consistently found slopes ranging from 40 to 50 mV/decade change in [K1_]

c. However, in the

stu-dents’ hands, slopes are sometimes less, 30 –45 mV/decade. For a 10-fold concentration gradient we have

45 mV5(PK2PH₂PO₄)/(PK1PH₂PO₄)359 mV

To facilitate calculation of a permeability ratio, we arbitrarily set PKto 1.0. We will solve for the

mem-brane permeability ratioPH₂PO₄/PK. Dividing each side

of the above equation by 59 mV gives

45 mV/59 mV50.765(PK2PH₂PO₄)/(PK1PH₂PO₄)

When we solve forPH₂PO₄/PK, we have

20.23/21.7650.135PH₂PO₄/PK

With PK5 1.0, then PK/PH₂PO₄ 5 1/0.13 5 7.7. The

dialysis membrane is 7.7 times more permeable to K1 than to H2PO42, and the permeability to H2PO42drives

the slope of the relationship between Vm and2log

([K1_]

c/[K1]e) to a value ,59 mV/decade change in

[K1_]

c.

How Many Ions Cr oss the Membrane to GenerateV_m?

One of the most confusing aspects for students in understanding membrane potentials is that relatively few ions cross the membrane to generateVm.

Further-more, the ions that cross the membrane do not enter the pool of ions present in bulk solution because the solution is a conductor (4). These ions are present in a layer adjacent to, or just beneath, the membrane (Fig. 5) (4, 7). The following calculation reveals the number of ions needed to generate Vm across the dialysis

FIG. 5.

Flow of cations fr om cytosol to ex ter nal solution causes char ge separation acr oss the dialysis mem-brane because the memmem-brane behaves as an electrical capacitor. Cations that cr oss the membrane leave behind an equal number of anions. These cations and anions ar e pr esent in thin layers adjacent to the membrane, and they interact electr ostatically acr oss the thickness of the membrane (7). This interaction holds these ions in the thin layers adjacent to the membrane. Thus the number of cations and anions in bulk cytosol and ex ter nal solution ar e the same, and both solutions confor m to the principle of electrical neutrality. [Modified fr om Eckert et al. (7)]

membrane, and Fig. 5 shows where these ions accumu-late. To calculate the number of ions that cross the dialysis membrane to generate Vm, students employ

the concept of membrane capacitance (Cm) (4).

Biologi-cal membranes behave as electriBiologi-cal capacitors be-cause they separate and store charge (4, 7, 10). The relationship betweenCmandVmis given by

Cm5Q/Vm (7)

where Q is the amount of charge stored by the membrane. Assuming that the cellulose ester dialysis membrane has a dielectric constant similar to that of biological membranes, we can estimate the capaci-tance of the dialysis membrane from the difference in thickness of the two membranes (4). Plasma mem-branes are ,5 nm thick, whereas the dialysis mem-brane is,20 µm thick, a difference of 4,000. Plasma membranes have a capacitance of 1 µF/cm2 _{(4, 9).}

Thus the dialysis membrane has a capacitance of ,2.5 3 1024 µF/cm2. For the KH2PO4 and Na2HPO4

salts, the slope of the relationship between Vm and

Ce/Ccis,45 mV/decade change in ion concentration. Therefore,

Q/Cm545 mV

and

Q5(2.5310210_F/cm2_{)(0.045 V)}

51.1310211_C/cm2

where C is coulomb (unit of electrical charge). To calculate the number of ions (N) that cross the membrane, we divideQby the charge per ion (e) (4)

N5Q(membrane area)/e (8) so that

N5(1.1310211_C/cm2₎₍

,1.0 cm2)/1.6310219C

5,6.93107

Approximately 7 million ions cross the,1.0-cm2area of dialysis membrane to generate aVmof 45 mV. This

seems like a large number of ions. However, these ions are stored near the membrane (see Fig. 5) and interact electrostatically with the layer of anions left

behind. As noted above, they do not change the number of ions in the bulk solution; hence, bulk electroneutrality is maintained (4). Furthermore, this number of ions, compared with the number of ions in bulk solution, represents a tiny fraction, as the follow-ing calculation shows. There are 6 3 1023 _{ions per}

mole, and the 3 ml of 0.1 mM phosphate (0.2 mM potassium and sodium ions) salt solutions have 3.63

1017_{ions. The ratio of ions that crossed the membrane}

to those present in bulk solution is

6.93107_/3.631017_51.9310210

This calculation shows that, relative to the number of ions in bulk solution, very few ions cross the mem-brane to chargeCmand generateVm.

Effect of Incr eases in Ex tracellular Na1 Concentration onV_m

In the second experiment the students explore the relation between Vm and transmembrane Na1

gradients. This experiment will give students in-sight into the ionic basis of ac tion potentials in excitable cells. Again, students consult Table 1 and determine the compartment in which they must increase Na1 _{concentration to mimic the Na}1 gradient found in animal cells. Next, they develop a hypothesis about magnitude and polarity of Vm that

result from increases in external Na1 _{concentration} ([Na1_]

e). This experiment is conducted in the same

manner as with increases in [K1_]

c. After conducting

the experiment, students should answer the following questions.

1) Why is the absolute value of the slope (see Fig. 4) obtained with Na2HPO4 similar to that obtained with

KH2PO4?

2) Why is the polarity ofVmgenerated with Na2HPO4

opposite to that obtained with KH2PO4?

3) How do the results with Na2HPO4 relate to the

change inVmthat occurs during an action potential in

excitable cells?

Major Points of Lab Ex er cise

By the end of this laboratory exercise, the students should have grasped the following major points.

1) AVmdevelops when there is an ion gradient across

a membrane if one ion of a dissociable salt is restricted from crossing the membrane while the other ion is freely permeant.

2) The electrical polarity of Vm reflects flow of the

membrane permeant ion down its concentration gradi-ent.

3) In real cells and as demonstrated in the mock cell, Vm results from potassium ions from flowing down

their concentration gradient from the cytosol to the cell exterior.

4) The slope of the relation between Ce/Cc for the

membrane permeant ion andVmis Nernstian only if

the membrane is predominantly permeable to one ion of a salt that dissociates into monovalent cations and anions.

5) Relatively few ions cross the membrane to charge Cm and generate Vm; bulk electroneutrality is

main-tained.

6) Increases in external Na1_{concentration reveal the} concentration gradient responsible for reversal ofVm

during an action potential in excitable cells. SUMMARY

This laboratory exercise provides a simple set of experiments that can be reliably and reproducibly conducted by students in a three-hour laboratory period. The experiments reveal how membrane poten-tials are generated and show the ion gradient respon-sible for action potentials in excitable cells, and the students are introduced to the Nernst equation as an equilibrium expression.

We thank Dr. Paul Hamilton for critically reading the manuscript. This project was supported in part by a National Science Founda-tion InstrumentaFounda-tion Laboratory Improvement Grant (9352255) and

by funds supplied by the Biology Department at the University of Central Arkansas.

Present addresses: J. Denton, Dept. of Physiology, Dartmouth Medical School, Hanover, NH 03755; K. Wilson, Univ. of Arkansas for Medical Sciences, College of Medicine, 4301 W. Markham St. -Slot 551, Little Rock, AR 72205.

Address for reprint requests or other correspondence: W. M. Moran, Dept. of Biology, Univ. of Central Arkansas, Conway, AR 72035-0001 (E-mail: Mmoran@mail.UCA.EDU).

Received 12 March 1998; accepted in final form 20 July 1999.

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Physiol-ogy(3rd ed.). New York: Freeman, 1988.

7. Eckert, R., D. Randall, W. Bur ggr en, and K. Fr ench.

Anim a l Physiology(4th ed.). New York: Freeman, 1996. 8. Goldstein, N. N., and M. J. Fr ee.Founda tions of

Physiologi-ca l Instrum enta tion. Springfield, IL: Thomas, 1979.

9. Gutnecht, J.The origin of bioelectric potentials in plant and animal cells.Am . Zool. 10: 347–354, 1970.

10. Hille, B.Ionic Cha nnels of Excita ble Mem bra nes. Sunderland, MA: Sinauer Assoc., 1984.

11. Manalis, R. S., and L. Hastings.Electrical gradients across an ion-exchange membrane in student’s artificial cell. J. Appl. Physiol. 36: 769–770, 1974.

12. Morris, J. G.A Biologist’s Physica l Chem istry. Reading, MA: Addison-Wesley, 1968.

13. Sper elakis, N.Origin of resting membrane potentials. In:Cell Physiology Source Book (2nd ed.), edited by N. Sperelakis. New York: Academic, 1998, p. 178–195.

(10). To test for streaming potentials, we added

mannitol to the salt solutions on the low salt

concen-tration side of the membrane to achieve osmolarities

equal to those on the high salt concentration side. We

then compared

V

m

to measurements made in the

absence of mannitol and found no difference between

the two. Therefore, streaming potentials do not make

measurable contributions to

Vm

in these experiments.

EXPERIMENTS

Effect of Incr easing Cytosolic K

1

_{Concentration}

on

V

_m

In the first experiment, students explore the

relation-ship between transmembrane K

1

_{gradients and}

_V

m

. To

mimic K

1

_{gradients present across plasma membranes}

of real cells, students consult Table 1 and determine in

which compartment they increase K

1

_{concentration.}

Importantly, this gives students insight into the nature

of the K

1

_{concentration gradient. Next, they develop}

a hypothesis, which includes statements about

magni-tude and polarity of

V

m

, to predict what will happen to

V

m

when they increase the cytosolic K

1

concentration

([K

1

_]

c

) while holding the extracellular K

1

concentra-tion ([K

1

_]

e

) constant. In keeping with convention (see

Figs. 5–14

B

, Ref. 7), instructors may choose to begin

the experiment with high [K

1

_]

c

and increase [K

1

]

e

,

which depolarizes

Vm

.

The experiment is started by placing 0.1 mM KH

2

PO

4

in each chamber half. The potential displayed on the

pH meter represents zero

V

m.

A 3-ml syringe with a

FIG. 2.

The ex perimental apparatus consists of a dialysis membrane chamber, agar bridges, calomel electr odes to detectV_m, a pH meter that serves as an electr ometer, and an analog-to-digital (A/ D) converter [Serial Box Inter-face (SBIMac) Ver nier Softwar e, Portland, OR; www.ve r nie r .com] and a Macintosh computer for displaying and recording the signal from the pH me-ter. The ex ternal calomel electrode is connected to a ground on the pH meter in keeping with electrophysiological convention. BNC, coax ial cable connector.

piece of PE tubing attached is used to empty and refill

the chamber halves when changing salt solutions.

When the salt concentration is increased in a chamber

half, the chamber should be rinsed at least once with

the new salt solution before

V

m

is recorded. After

V

m

is

recorded with the higher KH

2

PO

4

concentrations, 0.1

mM phosphate salt should be returned to the cytosolic

chamber to check for drift. Usually little drift occurs.

However, in some cases drift of a few millivolts

occurs, and either this must be added to or subtracted

from

Vm

or the experiment must be repeated. With

the recording arrangement illustrated in Fig. 2,

in-creases in [K

1

_]

c

generate a cytosol negative

V

m

. After

V

m

is recorded with 0.1, 1, and 10 mM KH

2

PO

4

(students should also include concentrations of 0.5

and 5 mM) in the cytosol compartment, the students

plot a graph of

V

m

versus [K

1

]

c

(Fig. 3

A

) and then plot

V

m

versus the ratio of [K

1

]

e

to [K

1

]

c

, that is, ratios of 1,

0.1, and 0.01 (Fig. 3

B

). This graph shows that the ratio

of potassium ion concentrations across the membrane

is the important factor in generation of

V

m

because a

ratio of 1.0 gives a

V

m

of zero. With this plot,

V

m

becomes increasingly more negative as a function of

[K

1

_]

e

/[K

1

]

c

, but not in a linear manner.

The next objective is to establish a linear relationship

between

V

m

and [K

1

]

e

/[K

1

]

c

. When doing this,

stu-dents will be introduced to the Nernst equation and to

the reason for taking the log [K

1

_]

e

/[K

1

]

c

to establish a

linear relationship between

V

m

and the ratio of K

1

concentrations on the two sides of the membrane.

Ideally, measurements of

Vm

are made when diffusion

forces (concentration gradient) and electrical forces

(

Vm

) are equal, that is, at equilibrium. Thus we can

equate these two forces. We can write

E

for electrical

force and C

e

/C

c

for concentration ratio (extracellular

to c ytosolic ) or c onc entration forc e in the K

1

transmembrane chemical gradient, where C

repre-sents concentration (in mol/l). At equilibrium we then

have

E

5

C

e

/C

c

(

1

)

The pr oblem with Eq. 1.

At this point the students

should be asked what the problem is with this

equation. Clearly, the units of the two forces are not

the same. To rectify this situation, the concentration

ratio (C

e

/C

c

) is multiplied by a constant

k

that must

have units of volts. Now the units on the two sides of

FIG. 3.

A: plot ofV_m vs. cytosolic K1_{concentration ([K}1_]c)

while ex ter nal K1_{concentration ([K}1_{]e) is held at 0.1}

mM. B: plot of [K1_{]e/ [K}1_{]c vs.} V_m gives a nonlinear r elationship. As [K1_]

cincr eases (i.e., decr ease in [K1]e/

[K1_]

c),Vmbecomes mor e negative.Vmr epr esents the

electrical potential of the cytosol compartment r efer-enced to the ex ter nal compartment (n₅3; standar d deviations ar e smaller than data point symbols). TABLE 1

Cytosol and ex ter nal Na1_{and K}1_{concentrations}

and their ratios in squid nerve and fr og muscle

Tissue

Cytosolic Concentration,

mM

External Concentration,

mM

Ce/Cc

Na1 _K1 _Na1 _K1 _Na1 _K1 Squid nerve 49 410 440 22 8.8 0.05 Frog sartorius muscle 10 140 120 2.5 12 0.02 Data are from Eckert et al. (6). Ce/Cc, ratio of external to cytosolic

the equation are the same, and we can rewrite

Eq. 1

as

E

5

k

(C

e

/C

c

)

(

2

)

Next, the constant

k

should be explored. In other

words, where does

k

come from and why does it have

units of volts?

k

is equal to

RT

/

zF

, where

R

is the gas

constant (2 cal/mol

3

Kelvin),

T

is the absolute

temperature (Kelvin units),

z

is the valence of the

membrane permeant ion, and

F

is the Faraday

con-stant (23,062 cal/mol

3

V). The Faraday constant is

the conversion factor that converts energy in a

chemi-cal concentration gradient (chemi-cal/mol) into equivalent

energy in an electrical gradient (volts). At this point

we have found it useful to have the students derive

the units of

k

to convince themselves that it has

units of volts. Now

Eq. 2

can be rewritten to include

RT

/

zF

as

E

5

(

RT

/

zF

)(C

e

/C

c

)

(

3

)

However, a plot of

E

, or

Vm

, against (

RT

/

zF

)(C

e

/C

c

)

still gives a nonlinear relationship. To establish a linear

relationship between

E

and (

RT

/

zF

)(C

e

/C

c

), the

stu-dents should be asked how the numbers on the

x

- and

y

-axes differ. On the

y

-axis the numbers represent an

arithmetic progression, such as 10, 20, 30, 40, etc.,

because the difference between any two successive

numbers is constant. On the

x

-axis the numbers

represent a geometric progression in which C

e

/C

c

decreases as 1, 0.1, 0.01 (12). In this progression, the

ratio between any two successive numbers is constant

and the numbers decrease exponentially. If we take

negative logarithms of the numbers on the

x

-axis, we

obtain numbers that represent an arithmetic

progres-sion of 0, 1, 2, etc. Now, if

2

ln (C

e

/C

c

) is plotted

against

V

m

, one arithmetic progression is plotted

against another arithmetic progression and a linear

relationship is established between the two variables

(Fig. 4).

If we expand

Eq. 3

to include ln (C

e

/C

c

) as follows

E

5

(

RT

/

zF

) ln (C

e

/C

c

)

(

4

)

then we have the familiar Nernst equation. This is

usually written as the log (C

e

/C

c

)

E

5

(2.3

RT

/

zF

) log (C

e

/C

c

)

(

5

)

E

is now referred to as the equilibrium

V

m

. The value

of 2.3

RT

/

zF

is 59 mV at 23°C. With a monovalent salt,

a 10-fold ratio for C

e

/C

c

will generate a

V

m

of 59 mV

if

the m em bra ne is perm ea ble to one ion only.

Accord-ingly, the slope of the relationship between

V

m

and

2

log (C

e

/C

c

) has a value of 59 mV/decade change in

C

e

/C

c

and is referred to as a Nernstian slope. The

significance of the Nernst equation is that it describes

the mathematical relationship between an ion

chemi-cal concentration difference (or gradient) across a

membrane and

Vm

under equilibrium conditions, i.e.,

when there is no net flow of ion across the membrane

and

V

m

is constant (2, 3).

Why is the slope less than Ner nstian?

We suggest

that the following analysis be performed by the

students as a homework assignment.

At this point, the students should be asked why the

slope of their relationship between

Vm

and

2

log

FIG. 4.

Plot of negative logarithm of concentration ratio (2log [Ce/ Cc]) vs.Vmgives a linear r elationship. The

slopes obtained with K2HPO4(k;n57) and KH2PO4

(s;n₅3) wer e nearly identical, 45 vs. 43 mV per 10-fold change in [K1_]

e/ [K1]c, r espectively. Ther

e-for e, a line was fit only to the K2HPO4data. For

Na2HPO4 (q), n 5 4. Standar d deviations range

fr om63–6 mV.V_mr epr esents the potential of the cytosol compartment r efer enced to that of the ex ter nal compartment. The Ner nstian r elation was calculated with the Ner nst equation for a monova-lent cation in which the salt concentration was incr eased in the ex ter nal solution. We assume that less-than-Ner nstian slopes r esult fr om a slight per-meability of the dialysis membrane to the counte-rion H2PO42(PH2PO4).

(C

e

/C

c

) for KH

2

PO

4

is less than Nernstian. As noted

above, a Nernstian slope is obtained only if the

membrane is permeable to one ion. This suggests that

the dialysis membrane is somewhat permeable to the

counterion, H

2

PO

42

. A small permeability to H

2

PO

42

is

not unreasonable, given that H

2

PO

4

2

_{has a molecular}

weight of 96 and the dialysis membrane has an MWCO

of 100. Hence, we assume that the membrane is

slightly permeable to H

2

PO

42

. These data allow the

students to calculate a permeability ratio

PH

₂PO₄

/

PK

and to demonstrate how permeability to the

counte-rion generates less-than-Nernstian slopes.

Further-more, this calculation reveals that Nernstian slopes are

achieved only when the membrane is permeable to

one ion.

For a membrane permeable to both ions of a

disso-ciable salt, such as KH

2

PO

4

,

Vm

is given by (3)

V

m

5

(

PK

2

PH

₂PO₄

)/(

PK

1

PH

₂PO₄

)

3

59 mV.log ([KH

2

PO

4

]

e

/[KH

2

PO

4

]

c

)

(

6

)

This expression is derived from the equation that

gives the diffusion potential arising from the diffusion

of a salt that dissociates into monovalent cations

and anions (3). Because permeability is proportional

to the diffusion coefficient, we can substitute

perme-abilities for diffusion coefficients (3). Furthermore,

the students should be aware that this equation

reduces to the Nernst equation for K

1

_{if membrane}

permeability to H

2

PO

42

approaches zero. From Fig. 4,

the slope is

,

45 mV/decade change in [K

1

]

c

(we use

the absolute value to simplify the calculation). We

have consistently found slopes ranging from 40 to 50

mV/decade change in [K

1

_]

c

. However, in the

stu-dents’ hands, slopes are sometimes less, 30 –45

mV/decade. For a 10-fold concentration gradient we

have

45 mV

5

(

P

K

2

P

H₂PO₄

)/(

P

K

1

P

H₂PO₄

)

3

59 mV

To facilitate calculation of a permeability ratio, we

arbitrarily set

PK

to 1.0. We will solve for the

mem-brane permeability ratio

PH

₂PO₄

/

PK

. Dividing each side

of the above equation by 59 mV gives

45 mV/59 mV

5

0.76

5

(

P

K

2

P

H₂PO₄

)/(

P

K

1

P

H₂PO₄

)

When we solve for

PH

₂PO₄

/

PK

, we have

2

0.23/

2

1.76

5

0.13

5

PH

₂PO₄

/

PK

With

PK

5

1.0, then

PK

/

PH

₂PO₄

5

1/0.13

5

7.7. The

dialysis membrane is 7.7 times more permeable to K

1

than to H

2

PO

42

, and the permeability to H

2

PO

42

drives

the slope of the relationship between

Vm

and

2

log

([K

1

_]

c

/[K

1

]

e

) to a value

,

59 mV/decade change in

[K

1

_]

c

.

How Many Ions Cr oss the Membrane to

Generate

V

_m

_?

One of the most confusing aspects for students in

understanding membrane potentials is that relatively

few ions cross the membrane to generate

V

m

.

Further-more, the ions that cross the membrane do not enter

the pool of ions present in bulk solution because the

solution is a conductor (4). These ions are present in a

layer adjacent to, or just beneath, the membrane (Fig.

5) (4, 7). The following calculation reveals the number

of ions needed to generate

V

m

across the dialysis

FIG. 5.

Flow of cations fr om cytosol to ex ter nal solution causes char ge separation acr oss the dialysis mem-brane because the memmem-brane behaves as an electrical capacitor. Cations that cr oss the membrane leave behind an equal number of anions. These cations and anions ar e pr esent in thin layers adjacent to the membrane, and they interact electr ostatically acr oss the thickness of the membrane (7). This interaction holds these ions in the thin layers adjacent to the membrane. Thus the number of cations and anions in bulk cytosol and ex ter nal solution ar e the same, and both solutions confor m to the principle of electrical neutrality. [Modified fr om Eckert et al. (7)]

membrane, and Fig. 5 shows where these ions

accumu-late. To calculate the number of ions that cross the

dialysis membrane to generate

Vm

, students employ

the concept of membrane capacitance (

Cm

) (4).

Biologi-cal membranes behave as electriBiologi-cal capacitors

be-cause they separate and store charge (4, 7, 10). The

relationship between

Cm

and

Vm

is given by

C

m

5

Q

/

V

m

(

7

)

where

Q

is the amount of charge stored by the

membrane. Assuming that the cellulose ester dialysis

membrane has a dielectric constant similar to that of

biological membranes, we can estimate the

capaci-tance of the dialysis membrane from the difference in

thickness of the two membranes (4). Plasma

mem-branes are

,

5 nm thick, whereas the dialysis

mem-brane is

,

20 µm thick, a difference of 4,000. Plasma

membranes have a capacitance of 1 µF/cm

2

_{(4, 9).}

Thus the dialysis membrane has a capacitance of

,

2.5

3

10

24

µF/cm

2

. For the KH

2

PO

4

and Na

2

HPO

4

salts, the slope of the relationship between

Vm

and

C

e

/C

c

is

,

45 mV/decade change in ion concentration.

Therefore,

Q

/

C

m

5

45 mV

and

Q

5

(2.5

3

10

210

_F/cm

2

_{)(0.045 V)}

5

1.1

3

10

211

_C/cm

2

where C is coulomb (unit of electrical charge). To

calculate the number of ions (

N

) that cross the

membrane, we divide

Q

by the charge per ion (

e

) (4)

N

5

Q

(membrane area)/

e

(

8

)

so that

N

5

(1.1

3

10

211

_C/cm

2

₎₍

,

1.0 cm

2

)/1.6

3

10

219

C

5

,

6.9

3

10

7

Approximately 7 million ions cross the

,

1.0-cm

2

area

of dialysis membrane to generate a

V

m

of 45 mV. This

seems like a large number of ions. However, these

ions are stored near the membrane (see Fig. 5) and

interact electrostatically with the layer of anions left

behind. As noted above, they do not change the

number of ions in the bulk solution; hence, bulk

electroneutrality is maintained (4). Furthermore, this

number of ions, compared with the number of ions in

bulk solution, represents a tiny fraction, as the

follow-ing calculation shows. There are 6

3

10

23

_{ions per}

mole, and the 3 ml of 0.1 mM phosphate (0.2 mM

potassium and sodium ions) salt solutions have 3.6

3

10

17

_{ions. The ratio of ions that crossed the membrane}

to those present in bulk solution is

6.9

3

10

7

_/3.6

₃

₁₀

17

₅

_1.9

₃

₁₀

210

This calculation shows that, relative to the number of

ions in bulk solution, very few ions cross the

mem-brane to charge

Cm

and generate

V

m

.

Effect of Incr eases in Ex tracellular Na

1

Concentration on

V

_m

In the second experiment the students explore

the relation between

Vm

and transmembrane Na

1

gradients. This experiment will give students

in-sight into the ionic basis of ac tion potentials

in excitable cells. Again, students consult Table 1

and determine the compartment in which they

must increase Na

1

_{concentration to mimic the Na}

1

gradient found in animal cells. Next, they develop a

hypothesis about magnitude and polarity of

Vm

that

result from increases in external Na

1

_{concentration}

([Na

1

_]

e

). This experiment is conducted in the same

manner as with increases in [K

1

_]

c

. After conducting

the experiment, students should answer the following

questions.

1

) Why is the absolute value of the slope (see Fig. 4)

obtained with Na

2

HPO

4

similar to that obtained with

KH

2

PO

4

?

2

) Why is the polarity of

V

m

generated with Na

2

HPO

4

opposite to that obtained with KH

2

PO

4

?

3

) How do the results with Na

2

HPO

4

relate to the

change in

Vm

that occurs during an action potential in

excitable cells?

Major Points of Lab Ex er cise

By the end of this laboratory exercise, the students

should have grasped the following major points.

1

) A

Vm

develops when there is an ion gradient across

a membrane if one ion of a dissociable salt is restricted

from crossing the membrane while the other ion is

freely permeant.

2

) The electrical polarity of

V

m

reflects flow of the

membrane permeant ion down its concentration

gradi-ent.

3

) In real cells and as demonstrated in the mock cell,

V

m

results from potassium ions from flowing down

their concentration gradient from the cytosol to the

cell exterior.

4

) The slope of the relation between C

e

/C

c

for the

membrane permeant ion and

Vm

is Nernstian only if

the membrane is predominantly permeable to one ion

of a salt that dissociates into monovalent cations and

anions.

5

) Relatively few ions cross the membrane to charge

Cm

and generate

Vm

; bulk electroneutrality is

main-tained.

6

) Increases in external Na

1

_{concentration reveal the}

concentration gradient responsible for reversal of

V

m

during an action potential in excitable cells.

SUMMARY

This laboratory exercise provides a simple set of

experiments that can be reliably and reproducibly

conducted by students in a three-hour laboratory

period. The experiments reveal how membrane

poten-tials are generated and show the ion gradient

respon-sible for action potentials in excitable cells, and the

students are introduced to the Nernst equation as an

equilibrium expression.

We thank Dr. Paul Hamilton for critically reading the manuscript. This project was supported in part by a National Science Founda-tion InstrumentaFounda-tion Laboratory Improvement Grant (9352255) and

by funds supplied by the Biology Department at the University of Central Arkansas.

Present addresses: J. Denton, Dept. of Physiology, Dartmouth Medical School, Hanover, NH 03755; K. Wilson, Univ. of Arkansas for Medical Sciences, College of Medicine, 4301 W. Markham St. -Slot 551, Little Rock, AR 72205.

Address for reprint requests or other correspondence: W. M. Moran, Dept. of Biology, Univ. of Central Arkansas, Conway, AR 72035-0001 (E-mail: Mmoran@mail.UCA.EDU).

Received 12 March 1998; accepted in final form 20 July 1999.

Refer ences

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2. Becker, W. M., J. B. Reece, and M. F. Poenie.The World of the Cell (3rd ed.). Menlo Park, CA: Benjamin/Cummings, 1996.

3. Byr ne, J. H., and S. G. Schultz. An Introduction to Mem -bra ne Tra nsport a nd Bioelectricity. New York: Raven, 1988. 4. DeFelice, L. J. Electrica l Properties of Cells. New York:

Plenum, 1997.

5. DeWeer, P. Electrogenic pumps: theoretical and practical considerations. In:Electrogenic Tra nsport: Funda m enta l Prin-ciples a nd Physiologica l Im plica tions, edited by M. P. Blaus-tein and M. Lieberman. New York: Raven, 1984, p. 1–15. 6. Eckert, R., D. Randall, and G. Augustine.Anim a l

Physiol-ogy(3rd ed.). New York: Freeman, 1988.

7. Eckert, R., D. Randall, W. Bur ggr en, and K. Fr ench. Anim a l Physiology(4th ed.). New York: Freeman, 1996. 8. Goldstein, N. N., and M. J. Fr ee.Founda tions of

Physiologi-ca l Instrum enta tion. Springfield, IL: Thomas, 1979.

9. Gutnecht, J.The origin of bioelectric potentials in plant and animal cells.Am . Zool. 10: 347–354, 1970.

10. Hille, B.Ionic Cha nnels of Excita ble Mem bra nes. Sunderland, MA: Sinauer Assoc., 1984.

11. Manalis, R. S., and L. Hastings.Electrical gradients across an ion-exchange membrane in student’s artificial cell. J. Appl. Physiol. 36: 769–770, 1974.

12. Morris, J. G.A Biologist’s Physica l Chem istry. Reading, MA: Addison-Wesley, 1968.

13. Sper elakis, N.Origin of resting membrane potentials. In:Cell Physiology Source Book (2nd ed.), edited by N. Sperelakis. New York: Academic, 1998, p. 178–195.