Goldman-Hodgkin-Katz Equation Research Articles

Validity of the Goldman-Hodgkin-Katz equation in paracellular ionic pathways of gallbladder epithelium

The Goldman-Hodgkin-Katz equation has been extensively used to determine cationic/anionic permeability ratios in the paracellular pathways of the gallbladder epithelium. Nevertheless, new experimental evidence suggests that none of the theoretical assumptions of the equation hold for these pathways. In order to assess the experimental validity of the Goldman equation the permeability ratios were calculated from zero-current diffusion potentials by means of the Goldman equation and compared with the cationic/anionic permeability ratios measured by simultaneous determinations of cation and anion tracer fluxes in the same membranes. The results indicate that the Goldman equation is empirically valid for the tested salts (KCl and RbCl) within the experimental range of concentrations (25 to 200 mM) at an electrochemical-potential difference of zero.

The diffusion and electrogenic components of the membrane potential of rabbit ventricle after long exposure to different K o

In cardiac muscle the steady membrane potential ( V m) behaves as a K-electrode at extracellular K concentrations (K o) above 10 m m [ 3, 13]. At lower concentrations the Goldman equation describes V m reasonably well [ 13]. Such a curve fitting assumes: (a) concentration-independent permeability coefficients, (b) a constant intracellular potassium concentration (K i), and (c) a passive distribution of chloride. Considering that K i varies with K o in rabbit ventricle [ 11] we tried to fit the V m v. K o relation to the Goldman equation using assumptions (a) and (c) together with the measured variation of K i. We will present data suggesting that the NaK exchange ratio is not rigidly linked to the activity of the sodium pump.

Membrane permeabilities determining resting, action and mechanoreceptor potentials inStentor coeruleus

In response to mechanical stimuli the protozoan,Stentor coeruleus, contracts in an all-or-none fashion and simultaneously reverses the direction of its ciliary beat. These behaviors have previously been shown to be correlated with the presence of a mechanoreceptor potential and all-or-none action potential (Wood 1970, 1973a). In the studies reported below the ionic bases of the resting, receptor and action potentials ofStentor were determined by use of intracellular microelectrodes penetrating animals chilled to 8.5–10 °C. The resting potential is most dependent on the extracellular concentration of KCl but some dependence on CaCl2 concentration was also observed. If allowance is made for the large increases in membrane conductance observed in solutions containing 2–8 mM KCl it is found that the resting potential data are well described by a modified form of the Goldman equation wherePCa/PK = 0.068 andPCl/PK = 0.072. The intracellular ionic activities (K i + = 13.1 mM, Cl i − = 9.9 mM, Ca i + = 0 mM) which provide the best fit of this equation to the resting potential data are in close agreement with the intracellular concentration values measured by flame microspectrophotometry (Ki=12.4 mM, Cli = 9.4 mM) except in the case of Cai where most of the intracellular concentration is presumed to be bound. 65 to 75 mV action potentials are produced by suprathreshold depolarizations but contractions were not generally seen in these chilled animals, only ciliary reversals. The action potential peak varies with CaCl2 concentration with a slope of 12.6 mV/10 fold change but varies only slightly with KCl or Cl− concentration. These peak potentials are well described by assuming that thePCa/Pk = 7.9 andPCl/PK=1.0 at the time of the action potential peak. Depolarizing receptor potentials and brief inward receptor currents were observed for all forms of punctate and gross bodily mechanical stimulation employed. No evidence was found for any form of hyperpolarizing mechanoreceptor potentials as observed in some other ciliates. The reversal potential of the mechanoreceptor current varied with CaCl2 concentration in a manner similar to that of the action potential peak. As in the case of the action potential both thePCa/Pk andpcl/pk ratios appear to increase as a result of mechanical stimulation to 9.3–15 and 1.2–1.95 respectively. Mechanoreceptor currents are voltage dependent being increased when the membrane is depolarized above resting potential and decreased when the membrane is hyperpolarized. In general the electrophysiological characteristics ofStentor appear similar to those ofParamecium andStylonychia, but its resting membrane appears more selectively permeable to K+, it produces only depolarizing receptor potentials when mechanically stimulated and the initial action potential elicited by depolarizing current pulses can be all-or-none even in culture medium.

Measurements of Ion Concentrations and Fluxes inDunaliella parva

Measurements of K+, Na+, and CI were made on a halotolerant strain of Dunaliella growing at 500 mM NaCl, 25 °C, and a relatively low light intensity (6000 lx). Much effort was spent in searching for a means of measuring the extracellular volume of fluid trapped between the cells of centrifuged pellets. All of the sugars tried as markers were rejected because they were found to be digested in the cell suspension. The most suitable marker was found to be [14C]polyethylene glycol2 (mol. wt. 4000); although this substance was apparently adsorbed to the cell exterior, it was found possible to correct for adsorption and then obtain a reasonable figure for the trapped fluid. The final concentrations of cell K+ and Na+ were 128 ± 53 mM and 131 ±117 mM respectively. Cl~ balanced the sum of K+ + Na+. Influxes of 22Na+, 42K+, and 36C1 were measured in cells in which the ions were in the steady state. Averages of 610 and 6-6 nmol m~2 s_1 were obtained for Na+ and K+ respectively. CI influx was divided into 2 phases with values of 1540 and 178 mmol m2 s1. The faster influx was considered to be across the outer cell membrane. The membrane responsible for the slower influx has not been identified. By comparing values of the potential difference calculated from the Nernst and Goldman equations, it was concluded that Na+ and K+ are probably controlled by active mechanisms, whereas cell CI is likely to be at thermodynamic equilibrium with

Effects of amphotericin B on the electrical properties and electrolyte content of frog sartorius muscle.

We studied the effect of amphotericin B (52 microM) on the membrane potential, membrane resistance, and intracellular Na+ and K+ concentrations in isolated frog sartorius muscles to characterize further the nature of the ionic conductance induced by the antibiotic. After 5 h of exposure to amphotericin B, the membrane depolarized from -89.9 to -51.0 mV, the membrane resistance decreased from 4537 to 907 omega cm2, [K]i decreased from 122 to 31.2 mmol/L fiber H2O, and [Na]i increased from 30.9 to 88.7 mmol/L fiber H2O. The relative sodium permeability, PNa/PK, calculated with the Goldman equation remained apparently constant at a value of 0.01 in treated and untreated muscles. We hypothesize that amphotericin B creates either a nonselective cation channel or a completely nonselective ionic leak channel whose equilibrium potential is equal or close to the membrane potential.

Investigation of the mechanism of temperature sensitivity of the electroreceptors of ampullae of lorenzini

In experiments on Black Sea skates (Raja clavata), the potential of the receptor epithelium of the ampullae of Lorenzini and spike activity of single nerve fibers connected to them were investigated during electrical and temperature stimulation. Usually the potential within the canal was between 0 and −2 mV, and the input resistance of the ampulla 250–400 kΩ. Heating of the region of the receptor epithelium was accompanied by a negative wave of potential, an increase in input resistance, and inhibition of spike activity. With worsening of the animal's condition the transepithelial potential became positive (up to +10 mV) but the input resistance of the ampulla during stimulation with a positive current was nonlinear in some cases: a regenerative spike of positive polarity appeared in the channel. During heating, the spike response was sometimes reversed in sign. It is suggested that fluctuations of the transepithelial potential and spike responses to temperature stimulation reflect changes in the potential difference on the basal membrane of the receptor cells, which is described by a relationship of the Nernst's or Goldman's equation type.

Ozone Alteration of Membrane Permeability in Chlorella

The addition of ozone to a suspension of Chlorella sorokiniana causes a rapid loss of K(+), as measured by efflux of (86)Rb from prelabeled cells. The efflux of the tracer is stimulated some 15 to 20 times over that of the control. For about 100 microliters per liter ozone, about 25 minutes (6 x 10(-8) moles O(3) delivered per minute) of exposure are required for a 50% depletion of the intracellular K(+). The stimulation of K(+) efflux is nearly linearly dependent upon the amount of ozone delivered into the solution. Following short pulses of ozone (lasting 1 to 5 minutes), efflux rates return to the control level but only after about 15 minutes.While influx of K(+) is ultimately inhibited by ozone, at low concentrations or for short exposure times the tracer influex is stimulated 100 to 200%. Ozone stimulation of an active pump mechanism is unlikely in view of a concomitant decrease in respiration. Thus, this influx may represent movement of K(+) along its electrochemical gradient. Assuming that influx and efflux are in steady-state according to the Goldman equation, it was calculated that the membrane potential for K(+) of -80 to -90 millivolts in control cells drops to -40 millivolts with ozone exposure and is accompanied by a calculated increased permeability to K(+) of 2- to 3-fold.

Dynamic Oscillations in the Membrane Potential of Pancreatic Islet Cells

Glucose and other metabolizable sugars which elicit insulin release from the beta-cell of the pancreatic islet induce repetitive oscillations in the beta-cell transmembrane potential. Upon each phasic depolarization are superimposed rapid fluctuations in potentials, i.e. 'action potentials' or 'spikes' which occur as bursts of electrical activity; the duration and frequency of each burst is a function of glucose concentration. These established electrophysiological features of glucose-islet cell interaction are described in detail together with a consideration of their possible molecular and ionic basis. Based on these observations, a dynamic mathematical computer model of the beta-cell membrane electrical behaviour is presented which utilizes the Goldman equation extended to include divalent ions. The model illustrates how the ionic mechanisms deduced from experimental observations can account for the electrical patterns produced by the beta-cells in the presence of D-glucose; it also allows systematic changes to be made in a number of state variables in order to assess their relative importance and possible contribution to the integrated processes actually observed. Finally, distinction is made between aspects of the model which are well supported by experimental results and those areas which require further analysis.

Transmembrane electrical and pH gradients across human erythrocytes and human peripheral lymphocytes

Transmembrane electrical and pH gradients have been measured across human erythrocytes and peripheral blood lymphocytes using equilibrium distributions of radioactively labelled lipophilic ions, and of weak acids and weak bases, respectively. The distributions of methylamine, trimethylamine, acetic acid and trimethylacetic acid give calculated transmembrane pH gradients (pHe-pHi) for erythrocytes of between 0.14-0.21 for extracellular pH values of 7.28-7.16. The distributions of trimethylacetic acid. DMO and trimethylamine were determined for lymphocytes, establishing upper and lower limits of the calculated pH gradient over the external pH range of 6.7 to 7.7. Tritiated triphenylmethyl phosphonium ion (TPMP) and 14C-thiocyanate ion (SCN) equilibrium distributions were measured in order to calculate transmembrane electrical potentials, using tetraphenylboron as a catalyst to facilitate TPMP equilibrium. Transmembrane potentials of -7 to -10 mV were calculated from SCN and TPMP, respectively for red cells, and -35 to -52 mV respectively, in the case of lymphocytes. Distributions of TPMP and potassium ions were determined in the presence of valinomycin over a wide range of extracellular potassium concentrations for red cells and the calculated Nernst potentials for TPMP compared to the calculated potential using the Goldman equation for chloride and potassium ions. Distributions of TPMP, SCN and potassium ions were also determined for lymphocyte suspensions as a function of extracellular potassium and the calculated Nernst potentials for TPMP and SCN compared to the calculated potassium diffusion potential.

Step-like shifts of membrane potential in the stretch receptor neuron of the crayfish (Astacus fluviatilis) at high temperatures

1. The effects of temperatures between 10–32 °C on the slowly adapting stretch receptor neuron of the crayfish were studied with intracellular recording and potential clamp techniques. 2. The membrane potential was found to decrease about 0.3 mV/ °C with increasing temperature in the range of 10° to 30 °C. 3. At temperatures in the range of 25° to 32 °C stretch produced a brief intense impulse discharge followed by a maintained depolarization. A similar response could be elicited by intracellular cathodal pulses or appeared sometimes spontaneously. 4. Tetrodotoxin (TTX) at a concentration of 10−7–10−8 mg/ml completely abolished the plateau, the receptor potential remaining unaffected. 5. Lowering of Ca+ + concentration facilitated the development of the plateau while increase of Ca+ + concentration had the opposite effect. 6. At a resting potential of −68 mV and at +30 °C the pNa/pK was 0.03 using the Goldman equation, the PNa/PK at a plateau level of −35 mV being 0.18. —The time constant for the inactivation of the plateau response was in the order of 1 s. 7. The passive membrane conductance as measured with the potential clamp technique increased with increasing temperature. 8. The stretch induced current (SIC) decreased and the reversal potential (Erev) was shifted towards more positive values when temperature was raised. 9. It is suggested that the steplike shift of membrane potential at high temperatures may be attributed to an increase in PNa/PK of the regenerative portion of the membrane and that this increase as well as the increase in passive membrane conductance are accounted for by changes in the physical properties of the lipid phase of the membrane at high temperatures.

Ion fluxes and diffusion potentials in the Dungeness crab,Cancer magister

Transepithelial potentials (TEP's) were measured in Dungeness crabs exposed to a variety of experimental media. The TEP's can be accounted for as diffusion potentials. In sea water (SW) thePNa/PCl ratio (calculated by substitution in the Goldman equation) was 0.68, but in dilutions of SW the value increased, reaching a maximum of 3.33 in freshwater (FW). When the calcium and magnesium concentrations in the diluted media were maintained at SW levels thePNa/PCl remained close to that in SW.

The dependence of membrane potential on extracellular chloride concentration in mammalian skeletal muscle fibres.

1. The steady-state intracellular membrane potential of fibres in thin bundles dissected from mouse extensor digitorum longus or soleus muscles or rat sternomastoid muscles was measured with 3 M-KCl glass micro-electrodes. The steady-state membrane potential was found to depend on the extracellular concentrations of Na, K and Cl ions. 2. The resting membrane potential (3.5 mM-[K]o, 160 mM-[Cl]o) was -74 +/- 1 mV (mean +/- S.E.) and a reduction in [Cl]o to 3.5 mM caused a reversible steady-state hyperpolarization to -94 +/- 1 mV (mean +/- S.E.). 3. The steady-state membrane potentials recorded in fibres exposed to different [K]o and zero [Cl]o were consistent with potentials predicted by the Goldman, Hodgkin & Katz (GHK) equation for Na and K. The results of similar experiments done with Cl as the major external anion could not be fitted by the same equation. 4. The GHK equation for Na, K and Cl did fit data obtained from fibres in solutions containing different [K]o with Cl as the major external anion if the intracellular Cl concentration was allowed to be out of equilibrium with the steady-state membrane potential. 5. It is suggested that an active influx of Cl ions controls the intracellular Cl concentrations in these fibres and hence maintains the Cl equilibrium potential at a depolarized value with respect to the resting membrane potential. 6. The steady-state membrane potential of rat diaphragm fibres was independent of [Cl]o and it seems likely that the intracellular Cl concentration of these fibres is not controlled by active Cl transport.

Seasonal Change of Membrane Potential across the Proximal Tubular Epithelium in Bullfrog Kidneys

In order to examine the seasonal changes in the relationship between the membrane potential and potassium activity of proximal tubular epithelium, a micropuncture study was performed with potassium selective microelectrodes on the kidney of the bullfrog (Rana catesbeiana) in two different seasons: winter (7 degrees C) and summer (20 degrees C). The potasssium activity in winter animals (7 degrees C) was 2.92 +/- 0.33 (mean +/- S.D., n = 20), 63.2 +/- 12.7 (n = 26), and 2.68 +/- 0.19 (n = 26) mM for the tubular fluid, cell, and plasma, whereas that in summer animals (20 degrees C) was 2.84 +/- 0.05 (n = 22), 61.8 +/- 11.2 (n = 24), 2.63 +/- 0.24 (n = 24) mM, respectively, indicating no seasonal difference. On the other hand, the mean values of the membrane PD in winter animals were 59.4 +/- 1.8 (n = 26) and 71.7 "/- 7.2 (n = 26) mV for the luminal and peritubular borders, whereas those in summer animals were 55.1 +/- 1.7 (n = 24) and 63.9 +/- 6.9 (n = 24) mV, respectivley, indicating that there was a significant seasonal difference (p less than 0.05 and p less than 0.001). Hence, compared to winter animals, the changes in the electrochemical profile for potassium in summer animals were: 1) the peritubular membrane PD is lower and 2) the transtubular electrochemical gradient is less steep. The sodium permeability calculated as the best fitting for a modified Goldman equation was 0.01 and 0.03 for winter and summer animals, respectively. In view of the fact the potassium in the cell and luminal fluid of the proximal tubule is kept at similar levels, potassium homeostasis is maintained in both groups of animals. The seasonal changes in electrical potentials are probably be due to an increase of cellular membrane permeability to ions other than potassium and to increased paracellular shunt conductance through the epithelium.

Mn2+ Ionic Permeability and Conductance Properties of Maize Primary Root Epidermis

Summary The results of in vitro experiments carried out on the maize primary root epidermis with a view to studying the permeability and conductance properties of this tissue are detailed. The transepidermal potential ET was measured as it varied with external concentrations of MnCl2 in the bathing solution. Relative and absolute permeability coefficients, as well as the relative conductance for the measured ET and the corresponding concentration gradients, were determined. The membrane potential Em was calculated according to a modified formulation of the Goldman equation and the Nernst potential was determined for the appropriate concentration gradients used in measuring ET.

A simple chamber for measuring lens asymmetry potentials

A simple method is outlined for the measurement of lens asymmetry potentials. The two surfaces of the lens were isolated by immersing in liquid paraffin at 35°C. Electrical contact at either surface was made via agar-Ringer electrodes to give a measure of the translens potential (A-P). Insertion of a glass microelectrode allowed the measurement of the potential differences across the posterior (I-P) and anterior (I-A) surfaces. All three potentials were sensitive to changes in temperature, but whereas the potentials I-A and I-P increased with temperature throughout the range, the translens p.d. (A-P) showed a maximum of approximately +6.5 mV between 30 and 35°C. All three potentials decreased with increasing external potassium and the values could be fitted by applying the Goldman equation to the data with α = 0·33 and β = 0·5 for the anterior membrane potential difference and α = 0·5 and β = 0·3 for the posterior membrane potential difference. Incubation for 1 1 2 hr in the presence of 10 −4 m-ouabain reduced the translens potential to near zero and this arose because of a more rapid fall of I-A relative to I-P. Preliminary experiments indicated that while there was no asymmetry potential (A-P) in human cataractous lenses (with high sodium content) in the oil chamber, a significant transmembrane potential difference of some −15 mV was present across both anterior and posterior surfaces.

Anomalous permeabilities of the egg cell membrane of a starfish in K+-Tl+ mixtures.

The electrical properties of "inward" rectifying egg cell membranes of the starfish mediastera aequalis have been studied in the presence of K(+)-Tl(+) mixtures. When the ratio of the external concentrations of these ions is changed while their sum is kept constant, both the conductance and the zero-current membrane potential go through a minimum, showing clear discrepancies from theoretical results based on conventional electrodiffusion models (E.g., Goldman's equation). By contrast, when the ration of the two concentrations is fixed and their sum varied, the potential follows an ideal Nernst slope, consistent with Goldman's equation. The membrane conductance which, according to previous studies on similar membranes, is to be viewed as a function of the displacement of the membrane potential from its resting value deltaV, shows marked differences between the cases in which K(+) or Tl(+) are the predominant ions: when K(+) is the predominant permeant ion in solution, the addition of small amounts of Tl(+) inhibits the current, while corresponding blocking effects of K(+) on the current are not observed when Tl(+) is the predominant permeant ion. Also, the time course of the conductance during voltage clamp is different in the two cases, being much faster in Tl(+) than in K(+) solution for comparable values of deltaV. Most of the above features are accounted for by a model in which it is assumed that the ionic channels have external binding sites for cations and that their permeability properties depend on the species of the cation bound (K(+)or Tl(+) in the present experiments).

Sodium permeability and conductance of maize primary root epidermis

With the aid of experimentally determined values of transepidermal potential (ET) the following values were derived: 1. the permeability coefficient and the relative permeability of the epidermis in relation to the external concentrations Na + , K + , Ca 2+ , Mg 2 , Fe 3+ , Mn 2 and Zn 2 ; 2. the relative conductance of the epidermis and the calculated membrane potential (E m ), from the modified Goldman equation. In addition, the interdependence of the Nernst potential, the diffusion potential of Na + -ions and the electrode potential in relation to the external Na + concentration were examined.

Characterization of the resting potential in Chironomus salivary gland cells : Evidence for an electrogenic sodium pump

The ionic basis of the resting potential ( E r ) across the plasma membrane of Chironomus thummi salivary gland cells has been investigated. E r was found to vary with developmental stage. The dependence of E r on [K] 0 was studied in the presence and absence of external chloride. The results could not be accounted for by the Goldman equation, and they suggested that an electrogenic sodium pump might be present. This was evaluated by repeating the experiments with: (1) external sodium replaced by Tris; (2) addition of 10 −5 M ouabain; (3) temperature reduction to 5 °C, and by a combination of these treatments. The results, along with earlier studies by Loewenstein and co-workers, provide strong evidence that 6–10 mV of the resting potential in standard saline is due to a metabolically dependent component. Most, but not all, of the results are consistent with the presence of an electrogenic sodium efflux produced by the Na,K-ATPase. This pump can produce changes in intracellular cation composition consistent with theories of the ionic variation of gene activity. It also explains the depolarization produced by most agents which destroy intercellular electrical communication.

Explanation for constancy of intracellular chloride concentration with respect to membrane potential in guinea pig mammary gland.

1. The relative permeabilities for sodium, potassium and chloride in guinea pig mammary gland slices are determined by means of ion flux studies with radioisotopes. 2. Assuming that there are no significant electrogenic potential components, we calculate permeability ratios PNa/PK = 0.97 and Pc1/PK = 1.25. 3. Substitution of these values in the Goldman equation yields membrane potentials of--15 mV before and--13 mV after ouabain treatment. 4. This small change in membrane potential explains the absence of a significant change in chloride content upon ouabain application, which leads to large changes in intracellular sodium and potassium concentrations.

Membrane potentials and ion permeabilities in flexor cells of the laminar pulvini of Phaseolus coccineus L.

The internal potential of flexor cells in slices of the laminar pulvini of Phaseolus coccineus has been recorded by standard microelectrode techniques in 100 eq m(-3) external salt solutions of various ionic compositions. The measured values are between-15 and-60 mV depending on the external medium. Treating the results with the Goldman equation yields the following relative permeabilities: K(+), 1.00; Na(+), 0.24; Cl(-), 0.19; NO 3 (-) , 1.6. The membrane potential was only slightly sensitive to external pH and Ca(2+). Metabolic inhibitors (azide, cyanide and salicylhydroxamic acid, carbonyl cyanid m-chlorphenyl hydrazone) caused only slight depolarizations (ca. 4 mV), which differed from the ion-induced changes by their slow time courses. The results are consistent with the hypothesis that the relatively impermeable Cl(-) is actively transported and osmotically efficient, whereas the well-permeable K(+) passively follows Cl(-) to maintain electroneutrality and is osmotically of only minor significance.