Gibbs energy (G) is the potential energy that can be absorbed or released during a chemical reaction in a closed system. G is a function of internal energy (U), pressure (P), volume (V), temperature (T), and entropy (S) as shown in Equation (1): (1) G = U + P V − T S .

Equation (1) can be differentiated to: (2) d G = d U + PdV + VdP − TdS − SdT .

Equation (2) can be simplified because dU = dQ − PdV (the first law of thermodynamics, in which Q is heat) and dQ = TdS (the second law of thermodynamics). Thus at fixed temperature (dT = 0), Equation (2) can be restated as: (3) d G = VdP .

Substituting the definition of V into Equation (3) according to the ideal gas law (V = nRT/P), where R is the ideal gas constant, describes the relationship between G and P: (4) d G = nRT P d P .

dG is defined as an infinitesimally small change in G in Equations (2), (3), and (4). But if we assume that our model system changes from an initial state “1” to a final state “2”, causing a measureable change in G (ΔG = G₂ − G₁), we can integrate Equation (4) between states 1 2, as shown below in Equation (5). (Note that ∫(dP/P) = ln P): (5) Δ G = ∫ 1 2 d G = G 2 − G 1 = ∫ 1 2 nRT P = d P = nRT ∫ 1 2 d P P = nRT ( ln P 2 − ln P 1 ) = nRT ln P 2 P 1 .

That is to say: (6) G 2 = G 1 + nRT ln P 2 P 1 .

Defining the initial state 1 as a standard state where G^⊖ is the standard free energy and P^⊖ is the standard state pressure (1 atm), provides us with Equation (7): (7) G = G Θ + nRT ln P P Θ .

In our closed system (at constant T, V) Equation (7) can be restated in terms of chemical potential, (μ) which can be described as Gibbs energy per mole (G/n): (8) μ = μ Θ + R T ln P P Θ .

In the case of a solution, it is more useful to consider the concentration of a molecule ([X]) rather than its pressure (_P). Henry's law relates pressure and concentration by a constant k_H that cancels out when we substitute the equation P = k_H [X] into Equation (8): (9) μ = μ Θ + R T ln [ X ] [ X ] Θ .

Because the standard state concentration of particle [X] ^⊖ = 1 M, this term does not need to be included in the equation. As we noted earlier, because of the influence of physical factors upon solubilization, the activity of a chemical species (a) is usually less than its molar concentration [14]. Thus we need to modify the relationship with an activity co-efficient (γ) which relates the activity and concentration of a species (γ [X] = a) such that: (10) μ = μ Θ + R T ln γ [ X ] = μ Θ + R T ln a .

In order to extend our consideration to charged particles (i.e., ions) we must further modify Equation (10) to describe the contribution of charge to the electrochemical potential (μ̃) of the system, where z is the valence of the ion, F is Faraday's constant, and Ψ is electrostatic potential of the system: (11) μ ˜ = μ Θ + R T ln a + z F Ψ .

Note that, if a particle is uncharged, z = 0 and μ̃ = μ, per Equation (10).

Now, let us imagine two compartments 1 and 2 that are separated by a selectively-permeable barrier such as a cell membrane or an ionophore cocktail that permits selective permeation of a certain ion (shown in Figure 1, panels A and B).

When the two compartments are at equilibrium (i.e., ΔG = 0), net ion movement across the membrane is zero and the electrochemical potentials are identical between the two compartments, such that μ̃₁ = μ̃₂. In this case, the electrical potential difference at equilibrium can be described as: (12) μ Θ + R T ln a 1 + z F Ψ 1 = μ Θ + R T ln a 2 + z F Ψ 2or: (13) z F ( Ψ 2 − Ψ 1 ) = R T ln a 1 a 2 (14) Δ Ψ 2 − 1 = R T z F ln a 1 a 2 .

Because a = γ [X^z+] (or [X^z-] in the case of an anion) and assuming that the physiochemical properties of the two compartments are similar such that γ₁ is approximately the same as γ₂, we derive the Nernst equation: (15) Δ Ψ 2 − 1 = R T z F ln [ X z + ] 1 [ X z + ] 2 .

At room temperature, RT/(F × loge) = 0.058 thus: (16) Δ Ψ 2 − 1 = 0.058 z log [ X z + ] 1 [ X z + ] 2 .

That is to say, when both compartments contain an equal concentration of [X^z+], ΔΨ₂₋₁ = 0 mV (Figure 1(A)) but a ten-fold difference in [X^z+] across an X^z+-permeable membrane will produce an electrical potential difference across that membrane of 0.058 V = 58 mV in the case of a monovalent ion (Figure 1(B)), or 29 mV in the case of a divalent ion. This predictable electrical response to changes in ion activity is the basis for the usefulness of ISMs.

In the case of an ISM, we can consider the ionophore cocktail in the tip of the electrode as the semi-permeable membrane. A ‘backfill’ solution within the ISM (Figure 1(C); equivalent to compartment 2 in Figures 1(A) and 1(B)) provides the electrical connection between a silver wire (Ag/AgCl half-cell) in the microelectrode holder (see Section 5.1) and the ionophore cocktail. The assay space into which the electrode tip is placed, of which we would like to know the ion concentration (Figure 1(C)), is equivalent to compartment 1 in Figures 1(A) and 1(B). Thus a ten-fold concentration difference between [X^z+] in the assay space and [X^z+] in the backfill solution is registered as a 58/z mV potential difference across the cocktail.

Crucially, given that the backfill solution in compartment 2 has a fixed composition ([X^z+] _backfull) and assuming that the activity co-efficient for [X^z+] is constant in the assay space, we can determine that the potential difference between two solutions A and B containing [X^z+]_solutionA and [X^z+]_solutionB is: (17) Δ Ψ solutionB − solutionA = 0.058 z ( log [ X z + ] solutionB [ X z + ] backfull − log [ X z + ] solutionA [ X z + ] backfull ) = 0.058 z log [ X z + ] solutionB [ X z + ] solutionA .

That is to say, a ten-fold concentration difference between solution A and solution B will register as a 58 mV potential difference between the two solutions for a monovalent ion [Figure 1(D)]: a Nernstian response. The exhibition of a Nernstian response by an ISM proves that it is selectively permeable to the ion of interest among all ions present in the solutions. This is the basis of calibration, an example of which is shown in Figure 2. Practical aspects of electrode calibration are considered in Section 5.3. Traditionally, K⁺- and H⁺-selective ISMs are more ideally ion-selective than Na⁺- or Cl⁻-selective ISMs. A consideration of the selectivity of each cocktail is presented in Sections 3.1 to 3.4 (guidelines for determining the selectivity coefficients for ISMs are provided in reference [15]). Note that the activity of Na⁺ decreases as [Na⁺] increases causing the calibration slope of Na⁺-selective ISM in Figure 2 to be slightly less than the Nernstian ideal of 58 mV/decade. Note that the estimated activity coefficients of the ions considered in this review (H⁺, Na⁺, K⁺, and Cl⁻), are 0.98 at 1 mM, 0.93 at 10 mM, and 0.82–0.86 at 100 mM [14].

There are three commercially-available cocktails recommended for use in H⁺-selective microelectrodes: Ionophore I/Cocktail A (cat. no. 95291, Sigma Aldrich, St. Louis, MO, USA), Ionophore I/Cocktail B (cat. no. 95293, Sigma), and Ionophore II/Cocktail A (cat. no. 95297, Sigma). All three cocktails respond linearly to changes in pH over the range 5.5–9.0, which covers most physiological applications. Outside of this range, care must be taken with cocktail choice; cocktails incorporating ionophore I can also be used at more alkaline pH values (up to pH 11, but not below pH 4.5) while cocktails incorporating ionophore II can be used at more acidic pH values (as low as pH 2.0 not above pH 9.0).

Ionophore I is tridodecylamine (TDDA), a lipophilic amine that acts as a proton carrier. In our laboratory we routinely use ISMs containing Ionophore I/Cocktail B (see Table 1) to monitor intracellular and extracellular pH. Cocktails A and B differ only in the identity of the cocktail additive inasmuch as Cocktail A contains sodium tetraphenylborate (NaTPB) rather than potassium tetrakis (4-chlorophenyl)borate (KTCPB). Both cocktails behave essentially the same, both being extremely selective for H⁺ over other ions such as Na⁺ in the physiological range (greater than 10¹²-fold preference) and both exhibiting similar response times. The authors who first reported the use of ISMs based on TDDA recommended overnight incubation of cocktail A with 100% CO₂ prior to use [16]. This procedure was presumably a precaution to minimize drift of potential due to CO₂ interference when the ISM is used in biological systems. However, CO₂ does not interfere appreciably with Cocktail B due to the substitution of the NaTPB additive with KTCPB [12]. Notably, a modified ISM that is fabricated with a CO₂-permeable column of Ionophore I and a backfill that contains carbonic anhydrase can be used in concert with a H⁺-selective ISM to monitor pCO₂ [17].

Ionophore II/Cocktail A is composed of the same solvent and additive as Ionophore I/Cocktail B, albeit in a slightly different ratio, but is based on a different H⁺ ionophore, namely 4-nonadecylpyridine (CAS No. 70268-36-9), which has a slightly poorer cation selectivity.

There are two commercially-available cocktails recommended for use in Na⁺-selective microelectrodes: Ionophore I/Cocktail A (cat. no. 99314; Sigma, see Table 2) and Ionophore II/Cocktail A (cat. no. 99357; Sigma).

Ionophore I is N,N′,N″-triheptyl-N,N′,N″-trimethyl-4,4′,4″-propylidynetris(3-oxabutyramide)—also known as ETH227—is reasonably selective for Na⁺ over K⁺ (more than 50-fold preference, see reference [19]) but not well selective for Na⁺ over Ca²⁺ (Ionophore I actually exhibits a 1.6-fold preference for Ca²⁺ over Na⁺). Thus Ionophore I/Cocktail A is the component of choice for monitoring of Na⁺ in the cytoplasm [19] in which compartment [Na⁺] is ∼4 mM, [K⁺]∼100 mM, and [Ca²⁺] is <1 μM.

Ionophore II is N,N′-dibenzyl-N,N′-diphenyl-1,2-phenylenedioxydiacetamide, which is also known as ETH157 exhibits a worse selectivity for Na⁺ over K⁺ (5-fold) than Ionophore I, but a better selectivity for Na⁺ over Ca²⁺ (20-fold). Ionophore II/cocktail A is better suited for inclusion in ISMs designed for extracellular measurements in which compartment [Na⁺] may be ∼100 mM, [K⁺]∼3 mM, and [Ca²⁺] ∼2 mM [20].

There are two commercially-available cocktails recommended for use in K⁺-selective microelectrodes: Ionophore I/Cocktail A (cat. no. 60031; Sigma, see Table 3) and Ionophore I/Cocktail B (cat. no. 60398; Sigma). Ionophore I is valinomycin, which carries K⁺ across membranes [21]. Both cocktails are similar in composition but cocktail A contains the solvent dibutyl sebacate, which confers upon the cocktail an improved selectivity for K⁺ over Na⁺.

At the time of writing, there is only one commercially available cocktail recommended for use in Cl⁻-selective microelectrodes: Ionophore I/Cocktail A (cat. no. 99408; Sigma, see Table 4). This cocktail is about 30-fold more selective for Cl⁻ over HCO₃⁻ [25], but is considered poorly selective with regard to its target ion compared to ion-selective cocktails for H⁺, Na⁺, and K⁺. The presence of HCO₃⁻ at close-to-physiological concentration does not substantially interfere with the Nernstian response of the electrode to Cl⁻, but if the electrode is to be used in the presence of HCO₃⁻, it should be calibrated in the presence of HCO₃⁻ in order to correct for interference [26]. Great care should be taken to account for interference by HCO₃⁻ when designing and interpreting the results of experiments that employ this cocktail, especially (1) if used in an assay space with an alkaline pH (above pH 7.6, see reference [25]), (2) if used in compartments such as the cytoplasm that could contain interfering anions that are beyond the investigators control, or (3) if [HCO₃⁻] is expected to change substantially over the assay period.

Glass microelectrode can be fabricated with either a sharp end (tip diameter∼1 μm) for intracellular recordings or a ‘blunt’ end (tip diameter∼20 μm) for cell-surface/extracellular recordings. We pull our microelectrodes from borosilicate capillary glass using a micropipette puller (for example, we use a Model P-97 Flaming/Brown from Sutter Instrument, Novato, CA, USA). An example of each type of electrode is shown in Figure 3.

Sharp-ended intracellular microelectrodes can be pulled from thin-walled borosilicate glass capillary that contains a filament (we use OD = 2 mm, ID = 1.56 mm; Part No. 30-0077; Harvard Apparatus, Holliston, MA, USA). The usefulness of a batch of pulled microelectrodes can be tested by filling one sample electrode with saturated KCl; the resistance of this KCl-filled microelectrode should be ∼0.5 MΩ. Smaller tip diameters can be achieved by controlled tip-breakage of occluded microelectrodes [29].

Blunt-ended extracellular microelectrodes can be pulled from thick-walled borosilicate glass capillary that contains a filament (we use OD = 2 mm, ID = 1.16 mm; Part No. G200F-4; Warner Instruments, Hamden, CT, USA). The tips of these pulled capillaries are fire polished using a microforge to eliminate sharp edges. The use of thick-walled, fire polished electrodes is important if the ISM is intended to be pushed up against a cell surface or into a tissue slice without cell impalement.

Prior to filling with ionophore cocktail, the empty glass microelectrodes must be baked, silanized, and cured [30]. This process makes the inside surface of the microelectrode hydrophobic, which encourages an unbroken interface between the hydrophobic ionophore cocktail and the glass, eliminating shunt pathways between the assay space and backfill solution that circumvent the cocktail and thereby reduce the selectivity of the ISM.

Empty microelectrodes are mounted in a metal rack with their tips pointing upwards and baked overnight in an oven at 200 °C to remove moisture. The following day, still at 200 °C, the microelectrode-containing rack is transferred to a glass petri-dish and covered with an inverted glass-jar to create a loosely sealed glass chamber around the baked microelectrodes. 90 μL of bis(dimethylamino)dimethylsilane (an organic silicon compound; cat. no. 14755; Sigma) is introduced under the jar such that the microelectrodes are exposed to “silane” vapor for 30-50 min within the chamber. Finally the jar is removed and the silane-coated microelectrodes are left to cure at 200 °C overnight. Cooling the cured electrodes to room temperature can be performed in a nitrogen-filled desiccation-chamber containing phosphorous pentoxide, to prevent condensation forming inside the microelectrode tip that could adversely affect ISM performance.

A short column (∼1 mm) of ionophore-cocktail can be introduced into the silanized microelectrode tip either by backfilling (for this purpose we use a 34 Gauge MicroFil™ flexible plastic syringe needle, cat. no. MF34G-5, World Precision Instruments Inc., Sarasota, FL, USA). The filament in each microelectrode draws the ionophore-cocktail into the extremity of the tip (the meniscus of the column of cocktail in each is indicated with cyan arrows in Figure 3). The backfill is introduced beneath the surface of the ionophore cocktail—to avoid introducing an air gap between the backfill and the cocktail—using a wider gauge needle (we use a 28 Gauge MicroFil™ needle, cat. no. MF28G-5, World Precision Instruments Inc.). An alternative approach is to frontfill the electrode, using suction to draw “backfill” solution into the electrode barrel through the tip and then to frontfill the electrode with a short column of ionophore cocktail (e.g., see reference [31]).

The microelectrode can be mounted in a straight holder with a female connector jack, a silver wire (Ag/AgCl) half-cell that maintains a constant electrochemical potential within the ISM backfill, and a vent or pressure port (e.g., cat. no. MEH2SFW, World Precision Instruments Inc.).

In the final assembly, the ionophore-cocktail column should be a few hundred μm deep and the backfill should be reasonably shallow (perhaps 1 cm), but still deep enough to make contact, when mounted, with the Ag/AgCl half-cell, as shown in Figure 4. The connector jack enables the ISM assembly to be mounted into the electrometer headstage probe (see Section 5.2).

As discussed in Section 2.5, the ion-selective potential measured by an ISM is described by the equation: (18) Δ Ψ solutionB − solutionA = 0.058 z ln [ X z + ] solutionB [ X z + ] solutionA

Implicit in Equation (18) is the assumption that solutions A and B have equal electrical potentials. In practice, ΔΨ_{solutionB−solutionA} measured by the ISM can also include a contribution from the difference in electrical potential between solution A and solution B (for example if solution A is the cytoplasm and solution B is extracellular). For this reason a saturated-KCl-filled reference microelectrode (a sharp-ended microelectrode, see Section 4.1) is used to measure the electrical potential of solutions A and B in order to calculate corrections to the potentials measured by the ISM. Because K⁺ and Cl⁻ have equal mobilities, junction-potential errors introduced by the reference electrode are not a major concern. When the reference-electrode signals are subtracted from the measured ΔΨ_{solutionB−solutionA}, we obtain the true ion-selective ΔΨ_{solutionB−solutionA}. The reference microelectrode signal is also used to correct for differences in electrical potential between assay compartments. For example, if the ISM is used to impale a cell for intracellular measurements, the reference electrode must also be impaled into the cell so that Ψ does not include a contribution from the membrane potential of the cell (i.e., the electrical potential difference between the cell cytoplasm and the extracellular compartment where the ISM was calibrated). The electrical configuration required to monitor the response of an ISM is represented in Figure 5.

In brief, the sample to be assayed (a solution or a cell in a solution) is placed in a recording chamber (such as those available from Warner Instruments). We use an FD223 dual-channel electrometer (World Precision Instruments Inc.) to monitor voltage signals. Both channels are connected to microelectrode holder/microelectrode assemblies via headstage probes that contain unity-gain amplifiers. The first probe is connected to an ISM and the second probe is connected to a saturated-KCl-filled reference electrode that senses the electrical potential of the assay space. Both probes should be mounted onto micromanipulators to allow fine positioning of the electrode tips.

Three features of the electrical set up allow accurate monitoring of the voltage signal from each electrode. Firstly, the amplifiers have an input-resistance/impedance (Z_FD223 > 10¹⁵ Ω) that is many times greater than that of the electrode tip (for sharp-ended ISMs, Z_ISM is typically ∼10¹¹ Ω). This is an important consideration because the current that relays the input voltage signal (V_in) from the assay space passes through the ISM tip as well as the amplifier and thus is influenced by the sum of their impedances (Z_FD223 + Z_ISM), whereas the current that relays the output voltage signal (V_out) is influenced only by the impedance of the amplifier (Z_FD223). A restatement of Ohm's law relates these quantities as: (19) V out = V i n × Z F D 223 Z ISM + Z F D 223 .

Thus, the relatively high impedance of the amplifier compared to that of the ISM tip minimizes the loss of signal amplitude by ensuring that both V_in and V_out are influenced by total impedances of similar magnitude. Furthermore the higher impedance of the amplifier reduces phase shift which could distort the relationship between V_in and V_out.

Secondly, a signal-driven shield built into each probe, feeds a duplicate V_in signal into the cable shielding, a maneuver that prevents stray capacitance between the signal-carrying wire and cable-shielding, increasing the responsiveness of the electrode. The shield also reduces electrical noise.

Thirdly, the dual-channel electrometer generates an ion-selective potential signal by subtracting the reference electrode signal (V_ref) from the ISM signal (V_ISM) with a high common-mode rejection ratio, which effectively eliminates noise common to both channels such that the true differential (i.e., ion-selective) signal is accurately obtained, as depicted in Figure 5.

Note that, for intracellular recordings, the reference-electrode signal can be used to monitor membrane potential (V_m) by clamping the electrical potential of the recording chamber to 0 mV using voltage-clamp circuitry (e.g., the OC-725C Oocyte Clamp from Warner Instruments), in which case V_m = V_ref. Alternatively the potential of the extracellular space (V_chamber) can be monitored with a calomel electrode or a second saturated-KCl-filled microelectrode (connected to a separate amplifier) and subtracted from V_ref using a separate subtraction amplifier, in which case V_m = V_ref − V_chamber. Note that V_ref = V_chamber when the reference electrode is not impaled into a cell.

The theory that underlies the calibration of ISMs is presented in Sections 2.5 and 5.2. An example calibration plot for a Na⁺-selective microelectrode is presented in Figure 2. A calibration slope of 58 mV/decade tells us that the ISM is ideally ion-selective with respect to its target ion. We use in-house software to convert the dual-channel electrometer output into a measure of ion concentration.

The series of solutions used to calibrate an ISM should have a composition that is close to that of the experimental sample to be measured (including potentially interfering ions), should cover the entire range of values expected to be encountered during the experiment, and should differ only in the concentration of target ion. For example, for calibration of a H⁺-selective ISM, we use a pH 6 buffer, a pH 7.5 buffer, and a pH 8 buffer. For ISMs that are less-ideally ion selective, such as Na⁺-selective microelectrodes, more calibration points may be necessary (e.g., a five-point calibration is shown in Figure 2).