Abstract

The classic Boron & De Weer (1976) paper provided the first evidence of active regulation of pH\mathrm{pH} in cells by an energy-dependent acid-base transporter. These authors also developed a quantitative model — comprising passive fluxes of acid-base equivalents across the cell membrane, intracellular reactions, and an active transport mechanism in the cell membrane (modelled as a proton pump) — to help interpret their measurements of intracellular pH\mathrm{pH} under perturbations of both extracellular CO2/HCO3\mathrm{CO_2}/\mathrm{HCO_3^-} and extracellular NH3/NH4+\mathrm{NH_3}/\mathrm{NH_4^+}. This Physiome paper seeks to make that model, and the experimental conditions under which it was developed, available in a reproducible and well-documented form, along with a software implementation that makes the model easy to use and understand. We have also taken the opportunity to update some of the units used in the original paper, and to provide a few parameter values that were missing in the original paper. Finally, we provide an historical background to the Boron & De Weer (1976) proposal for active pH\mathrm{pH} regulation and a commentary on subsequent work that has enriched our understanding of this most basic aspect of cellular physiology.

Keywords:CO₂NH₃squid giant axonweak acidweak base

1Introduction

In 1976 Boron & De Weer published their landmark paper on “Intracellular pH\mathrm{pH} transients in squid giant axons caused by CO2\mathrm{CO_2}, NH3\mathrm{NH_3}, and metabolic inhibitors” Boron & De Weer, 1976. The authors used a squid giant axon preparation and a mathematical model of pH\mathrm{pH} buffering and the transport of protons, bicarbonate (HCO3\mathrm{HCO_3^-}) and CO2\mathrm{CO_2} to establish the experimental evidence for active regulation of intracellular pH\mathrm{pH} (pHi\mathrm{pH_i}) by a transporter in the plasma membrane that — at the expense of energy — either moves acid out of the cell, or base into the cell. Today, we refer to such a transporter generically as an acid-extrusion mechanism. For simplicity, Boron & De Weer modelled it as a proton pump, although the result would have been almost indistinguishable had they modelled it as the uptake of HCO3\mathrm{HCO_3^-} or carbonate (CO32\mathrm{CO_3^{2-}}). The paper reported on the consequences of adding and then removing extracellular CO2/HCO3\mathrm{CO_2}/\mathrm{HCO_3^-}, NH3/NH4+\mathrm{NH_3/NH_4^+} (where NH4+\mathrm{NH_4^+} is ammonium), or the metabolic inhibitors, cyanide, azide and dinitrophenol (DNP).

In the first experiment, following exposure of the cell to elevated CO2\mathrm{CO_2} and HCO3\mathrm{HCO_3^-}, CO2\mathrm{CO_2} rapidly enters the cell and intracellular CO2\mathrm{CO_2} equilibrates with the extracellular CO2\mathrm{CO_2}, and generates intracellular H+\mathrm{H^+} and HCO3\mathrm{HCO_3^-} via the CO2\mathrm{CO_2} hydration reaction (CO2+H2OH++HCO3\mathrm{CO_2}+\mathrm{H_2O} \longrightarrow \mathrm{H^+}+\mathrm{HCO^-_3}). The accumulating H+\mathrm{H^+} results in a rapid fall of pHi\mathrm{pH_i} (see Figure 1A & Figure 1B). To the extent that the membrane is permeable to HCO3\mathrm{HCO_3^-} as well as to CO2\mathrm{CO_2}, HCO3\mathrm{HCO_3^-} will initially enter the cell passively, along its electrochemical gradient. Soon, however, the accumulation of intracellular HCO3\mathrm{HCO_3^-} reverses the HCO3\mathrm{HCO_3^-} electrochemical gradient and would be expected to lead to the passive efflux of HCO3\mathrm{HCO_3^-}. This loss of cellular HCO3\mathrm{HCO_3^-} would tend to acidify the cell because — to replenish the lost HCO3\mathrm{HCO_3^-} — additional CO2\mathrm{CO_2} would enter the cell and form even more H+\mathrm{H^+} and HCO3\mathrm{HCO_3^-} (the passive CO2/HCO3\mathrm{CO_2}/\mathrm{HCO_3^-} shuttle). Thus, the expectation was that prolonged exposure to CO2\mathrm{CO_2} would cause pHi\mathrm{pH_i} to fall rapidly (passive influx of CO2\mathrm{CO_2}) and then to drift more slowly in the acidic direction (passive efflux of HCO3\mathrm{HCO_3^-}). In fact, Boron & De Weer observed an alkaline drift, leading to the postulate of active extrusion of H+\mathrm{H^+} — or an equivalent process[1] — at a rate that exceeds the passive shuttling by the CO2/HCO3\mathrm{CO_2}/\mathrm{HCO_3^-} couple (see Figure 1A & Figure 1C).

\mathrm{pH_i} changes caused by prolonged exposure of a squid giant axon to extracellular \mathrm{CO_2}/\mathrm{HCO_3^-} in the bulk solution. (A) Original \mathrm{pH_i} and V_\mathrm{m} traces from figure 1 of BDW. Exposing an axon to extracellular \mathrm{CO_2}/\mathrm{HCO_3^-} causes a rapid fall in \mathrm{pH_i} followed by a slow and sustained plateau-phase \mathrm{pH_i} recovery (i.e., \mathrm{pH_i} rises). Removing extracellular \mathrm{CO_2}/\mathrm{HCO_3^-} causes \mathrm{pH_i} to overshoot its initial resting value. Both the plateau-phase recovery (short arrow) and the overshoot (long arrow) are indicative of net acid extrusion during the period of \mathrm{CO_2}/\mathrm{HCO_3^-} exposure. (B) Cartoon illustrating the processes underlying the initial, rapid acidification phase in (A). The entry of \mathrm{CO_2} leads to the intracellular production of \mathrm{H^+} (and thus to the observed \mathrm{pH_i} decay) via the reaction \mathrm{CO_2}+\mathrm{H_2O} \longrightarrow \mathrm{H^+}+\mathrm{HCO^-_3}. (C) Cartoon illustrating the processes underlying the plateau-phase alkalinisation in (A). After \mathrm{CO_2} equilibration across the plasma membrane (\mathrm{pH_i} nadir in panel (A)), the slow entry of \mathrm{HCO_3^-} (or, equivalently, the slow exit of \mathrm{H^+}) — which has always been present but was overwhelmed by the influx of \mathrm{CO_2} — leads to the consumption of \mathrm{H^+} (and thus to the observed slow \mathrm{pH_i} rise) via the reaction \mathrm{H^+}+\mathrm{HCO^-_3} \longrightarrow \mathrm{CO_2}+\mathrm{H_2O}. The newly formed \mathrm{CO_2} then exits the cell. The observed \mathrm{pH_i} overshoot is the result of the accumulation of \mathrm{HCO_3^-} during exposure to extracellular \mathrm{CO_2}/\mathrm{HCO_3^-}. BDW used the mathematical model to postulate the presence of an active acid-extrusion mechanism that would explain both the observed plateau-phase \mathrm{pH_i} recovery and the \mathrm{pH_i} overshoot. (A), modified from . (B)-(C), modified from .

Figure 1:pHi\mathrm{pH_i} changes caused by prolonged exposure of a squid giant axon to extracellular CO2/HCO3\mathrm{CO_2}/\mathrm{HCO_3^-} in the bulk solution. (A) Original pHi\mathrm{pH_i} and VmV_\mathrm{m} traces from figure 1 of BDW. Exposing an axon to extracellular CO2/HCO3\mathrm{CO_2}/\mathrm{HCO_3^-} causes a rapid fall in pHi\mathrm{pH_i} followed by a slow and sustained plateau-phase pHi\mathrm{pH_i} recovery (i.e., pHi\mathrm{pH_i} rises). Removing extracellular CO2/HCO3\mathrm{CO_2}/\mathrm{HCO_3^-} causes pHi\mathrm{pH_i} to overshoot its initial resting value. Both the plateau-phase recovery (short arrow) and the overshoot (long arrow) are indicative of net acid extrusion during the period of CO2/HCO3\mathrm{CO_2}/\mathrm{HCO_3^-} exposure. (B) Cartoon illustrating the processes underlying the initial, rapid acidification phase in (A). The entry of CO2\mathrm{CO_2} leads to the intracellular production of H+\mathrm{H^+} (and thus to the observed pHi\mathrm{pH_i} decay) via the reaction CO2+H2OH++HCO3\mathrm{CO_2}+\mathrm{H_2O} \longrightarrow \mathrm{H^+}+\mathrm{HCO^-_3}. (C) Cartoon illustrating the processes underlying the plateau-phase alkalinisation in (A). After CO2\mathrm{CO_2} equilibration across the plasma membrane (pHi\mathrm{pH_i} nadir in panel (A)), the slow entry of HCO3\mathrm{HCO_3^-} (or, equivalently, the slow exit of H+\mathrm{H^+}) — which has always been present but was overwhelmed by the influx of CO2\mathrm{CO_2} — leads to the consumption of H+\mathrm{H^+} (and thus to the observed slow pHi\mathrm{pH_i} rise) via the reaction H++HCO3CO2+H2O\mathrm{H^+}+\mathrm{HCO^-_3} \longrightarrow \mathrm{CO_2}+\mathrm{H_2O}. The newly formed CO2\mathrm{CO_2} then exits the cell. The observed pHi\mathrm{pH_i} overshoot is the result of the accumulation of HCO3\mathrm{HCO_3^-} during exposure to extracellular CO2/HCO3\mathrm{CO_2}/\mathrm{HCO_3^-}. BDW used the mathematical model to postulate the presence of an active acid-extrusion mechanism that would explain both the observed plateau-phase pHi\mathrm{pH_i} recovery and the pHi\mathrm{pH_i} overshoot. (A), modified from Boron & De Weer (1976). (B)-(C), modified from Boron (2010).

Following removal of external CO2\mathrm{CO_2}, intracellular CO2\mathrm{CO_2} diffuses out, while intracellular HCO3\mathrm{HCO_3^-} combines with H+\mathrm{H^+} to leave the cell as CO2\mathrm{CO_2}. Thus, the entire intracellular H+\mathrm{H^+} load associated with CO2\mathrm{CO_2} entry would be removed, returning pHi\mathrm{pH_i} to its value before the addition of CO2/HCO3\mathrm{CO_2}/\mathrm{HCO_3^-} (or to a slightly lower value, to the extent that HCO3\mathrm{HCO_3^-} had passively exited during the CO2/HCO3\mathrm{CO_2}/\mathrm{HCO_3^-} exposure). In fact, Boron & De Weer observed that pHi\mathrm{pH_i} overshoots its resting value by an amount consistent with the net removal of H+\mathrm{H^+} by the active, acid-extrusion mechanism during the CO2/HCO3\mathrm{CO_2}/\mathrm{HCO_3^-} exposure (see Figure 1A & Figure 1C).

In the second experiment, following exposure of the cell to extracellular NH3/NH4+\mathrm{NH_3}/\mathrm{NH_4^+} in the form of NH4Cl\mathrm{NH_4Cl} (ammonium chloride), the intracellular environment rapidly becomes alkaline as NH3\mathrm{NH_3} enters and combines with H+\mathrm{H^+} to form NH4+\mathrm{NH_4^+} (equivalent to the hydration of NH3\mathrm{NH_3} to form NH4+\mathrm{NH_4^+} and OH\mathrm{OH^-}). If this were the entire story, then pHi\mathrm{pH_i} would rise monotonically to a relatively alkaline value, and then the subsequent removal of NH4Cl\mathrm{NH_4Cl} would cause pHi\mathrm{pH_i} to fall to precisely its initial value. In fact, Boron & De Weer observed that the exposure to NH4Cl\mathrm{NH_4Cl} causes pHi\mathrm{pH_i} to rise rapidly and then fall slowly. Moreover, the subsequent removal of NH3/NH4+\mathrm{NH_3}/\mathrm{NH_4^+} causes pHi\mathrm{pH_i} to undershoot its original value (see Figure 2A). Thus, Boron & De Weer postulated that, during the NH3/NH4+\mathrm{NH_3}/\mathrm{NH_4^+} exposure, NH4+\mathrm{NH_4^+} passively enters the cell down its electrochemical gradient. Early during the exposure, this NH4+\mathrm{NH_4^+} influx would oppose the NH3\mathrm{NH_3} entry and slightly reduce the pHi\mathrm{pH_i} increase. Later during the NH4Cl\mathrm{NH_4Cl} exposure, after intracellular [NH3]\mathrm{[NH_3]} rises to match extracellular [NH3]\mathrm{[NH_3]} ([NH3]o\mathrm{[NH_3]_o}), the continued passive influx of NH4+\mathrm{NH_4^+} would generate intracellular H+\mathrm{H^+} and NH3\mathrm{NH_3}. The result would be a slow fall of pHi\mathrm{pH_i} and a rise in intracellular [NH3]\mathrm{[NH_3]}, the latter leading to the passive exit of NH3\mathrm{NH_3} (the passive NH3/NH4+\mathrm{NH_3}/\mathrm{NH_4^+} shuttle, see Figure 2A & Figure 2B). The experimental data are consistent with the proposed mathematical model.

\mathrm{pH_i} changes caused by a short and a long exposure of a squid giant axon to extracellular \mathrm{NH_3}/\mathrm{NH_4^+} in the bulk solution. (A) Original \mathrm{pH_i} and V_\mathrm{m} traces from figure 2 of BDW. A short exposure of the axon to extracellular \mathrm{NH_3}/\mathrm{NH_4^+} causes a rapid rise in \mathrm{pH_i}, followed by a \mathrm{pH_i} decay that modestly undershoots (lower short arrow) its initial resting value upon removal of extracellular \mathrm{NH_3}/\mathrm{NH_4^+}. A longer exposure of squid giant axons to extracellular \mathrm{NH_3}/\mathrm{NH_4^+} causes a rapid rise in \mathrm{pH_i}, followed by a slow and sustained \mathrm{pH_i} decay. Removing extracellular \mathrm{NH_3}/\mathrm{NH_4^+} causes \mathrm{pH_i} to undershoot substantially its initial resting value (long arrow). Both the plateau-phase acidification (upper short arrow) and the undershoot (long arrow) are indicative of net acid loading during the period of \mathrm{NH_3}/\mathrm{NH_4^+} exposure. (B) Cartoon illustrating the processes underlying the initial alkalinisation phase in (A) for both short and long exposures to extracellular \mathrm{NH_3}/\mathrm{NH_4^+}. The initial entry of \mathrm{NH_3} leads to the intracellular consumption of \mathrm{H^+} (and thus to the observed \mathrm{pH_i} rise) via the reaction \mathrm{NH_3}+\mathrm{H^+} \longrightarrow \mathrm{NH^+_4}. (C) Cartoon illustrating the processes underlying the plateau-phase acidification during the long \mathrm{NH_3}/\mathrm{NH_4^+} exposure in (A). After \mathrm{NH_3} equilibration across the plasma membrane (\mathrm{pH_i} peak in panel (A)), the slow entry of \mathrm{NH_4^+} — which has always been present but overwhelmed by the influx of \mathrm{NH_3} — leads to the production of \mathrm{H^+} (and thus to the observed slow \mathrm{pH_i} decay during the plateau phase) via the reaction \mathrm{NH^+_4} \longrightarrow \mathrm{NH_3}+\mathrm{H^+}. The newly formed \mathrm{NH_3} then exits the cell. The \mathrm{pH_i} undershoots observed upon removal of extracellular \mathrm{NH_3}/\mathrm{NH_4^+}, during both short and long \mathrm{NH_3}/\mathrm{NH_4^+} exposures, are the result of the accumulation of \mathrm{NH_4^+} during exposure to extracellular \mathrm{NH_3}/\mathrm{NH_4^+}. BDW used the mathematical model to postulate the above sequence of events, including both the plateau-phase acidification and the \mathrm{pH_i} undershoot. (A), modified from . (B)-(C), modified from .

Figure 2:pHi\mathrm{pH_i} changes caused by a short and a long exposure of a squid giant axon to extracellular NH3/NH4+\mathrm{NH_3}/\mathrm{NH_4^+} in the bulk solution. (A) Original pHi\mathrm{pH_i} and VmV_\mathrm{m} traces from figure 2 of BDW. A short exposure of the axon to extracellular NH3/NH4+\mathrm{NH_3}/\mathrm{NH_4^+} causes a rapid rise in pHi\mathrm{pH_i}, followed by a pHi\mathrm{pH_i} decay that modestly undershoots (lower short arrow) its initial resting value upon removal of extracellular NH3/NH4+\mathrm{NH_3}/\mathrm{NH_4^+}. A longer exposure of squid giant axons to extracellular NH3/NH4+\mathrm{NH_3}/\mathrm{NH_4^+} causes a rapid rise in pHi\mathrm{pH_i}, followed by a slow and sustained pHi\mathrm{pH_i} decay. Removing extracellular NH3/NH4+\mathrm{NH_3}/\mathrm{NH_4^+} causes pHi\mathrm{pH_i} to undershoot substantially its initial resting value (long arrow). Both the plateau-phase acidification (upper short arrow) and the undershoot (long arrow) are indicative of net acid loading during the period of NH3/NH4+\mathrm{NH_3}/\mathrm{NH_4^+} exposure. (B) Cartoon illustrating the processes underlying the initial alkalinisation phase in (A) for both short and long exposures to extracellular NH3/NH4+\mathrm{NH_3}/\mathrm{NH_4^+}. The initial entry of NH3\mathrm{NH_3} leads to the intracellular consumption of H+\mathrm{H^+} (and thus to the observed pHi\mathrm{pH_i} rise) via the reaction NH3+H+NH4+\mathrm{NH_3}+\mathrm{H^+} \longrightarrow \mathrm{NH^+_4}. (C) Cartoon illustrating the processes underlying the plateau-phase acidification during the long NH3/NH4+\mathrm{NH_3}/\mathrm{NH_4^+} exposure in (A). After NH3\mathrm{NH_3} equilibration across the plasma membrane (pHi\mathrm{pH_i} peak in panel (A)), the slow entry of NH4+\mathrm{NH_4^+} — which has always been present but overwhelmed by the influx of NH3\mathrm{NH_3} — leads to the production of H+\mathrm{H^+} (and thus to the observed slow pHi\mathrm{pH_i} decay during the plateau phase) via the reaction NH4+NH3+H+\mathrm{NH^+_4} \longrightarrow \mathrm{NH_3}+\mathrm{H^+}. The newly formed NH3\mathrm{NH_3} then exits the cell. The pHi\mathrm{pH_i} undershoots observed upon removal of extracellular NH3/NH4+\mathrm{NH_3}/\mathrm{NH_4^+}, during both short and long NH3/NH4+\mathrm{NH_3}/\mathrm{NH_4^+} exposures, are the result of the accumulation of NH4+\mathrm{NH_4^+} during exposure to extracellular NH3/NH4+\mathrm{NH_3}/\mathrm{NH_4^+}. BDW used the mathematical model to postulate the above sequence of events, including both the plateau-phase acidification and the pHi\mathrm{pH_i} undershoot. (A), modified from Boron & De Weer (1976). (B)-(C), modified from Boron (2010).

Finally, exposure of the cells — in turn — to cyanide, DNP and azide resulted in intracellular acidosis, consistent with the accumulation of acid metabolites.

In the present paper, we re-formulate the models from Boron & De Weer (1976), henceforth referred to as ‘BDW’, and specify the simulation using the Physiome modelling standards CellML Cuellar et al., 2003 and SED-ML Bergmann et al., 2017 in order to ensure that the model reproduces the graphs in the original paper and that the model is fully curated.[2] Note that this effort requires the specification of some parameters used in BDW’s simulations, but not described in the BDW paper. The curated and annotated model is made available in a form that users can run with OpenCOR[3] to understand the model and to explore the effect of changes in parameter values.

2pH Buffering by Weak Acids and Weak Bases

We begin by reviewing a few rudimentary concepts of pH\mathrm{pH} buffering by weak acids and bases Roos & Boron, 1981Bevensee & Boron, 2013Boron & Boulpaep, 2016 to provide the background for understanding the derivation and implementation of the BDW model.

Buffers.

According to Brönsted’s definition Brönsted, 1923, an acid is any substance that can donate a H+\mathrm{H^+}. Conversely, a base is any substance that can accept a H+\mathrm{H^+}. A buffer is any substance that can reversibly consume or produce H+\mathrm{H^+}, thereby minimising changes in pH\mathrm{pH}.

The dissociation of the uncharged weak acid (HA\mathrm{HA}) to the anionic weak base (A\mathrm{A^-}) is described by the equilibrium reaction:

HAA+H+\mathrm{HA \rightleftharpoons A^- + H^+}

which is governed by the equilibrium constant[4]

KHA=[A][H+][HA].K_\mathrm{HA}=\dfrac{\mathrm{[A^-][H^+]}}{\mathrm{[HA]}}.

An example is the carbonic acid (H2CO3\mathrm{H_2CO_3}) dissociation reaction,

H2CO3HCO3+H+.\mathrm{H_2CO_3 \rightleftharpoons HCO_3^- + H^+}.

The total weak acid concentration, [TA]\mathrm{[TA]}, is the sum of [HA]\mathrm{[HA]} and [A]\mathrm{[A^-]}. Note that [TA]\mathrm{[TA]} is one of the two main unknowns in the BDW model for weak acids.

The dissociation of the cationic weak acid (BH+\mathrm{BH^+}) to the uncharged weak base (B\mathrm{B}) is described by the equilibrium reaction,

BH+B+H+,\mathrm{BH^+ \rightleftharpoons B + H^+},

where the equilibrium constant is

KBH=[B][H+][BH+].K_\mathrm{BH}=\dfrac{\mathrm{[B][H^+]}}{\mathrm{[BH^+]}}.

An example is the NH4+\mathrm{NH_4^+} dissociation reaction,

NH4+NH3+H+.\mathrm{NH_4^+ \rightleftharpoons NH_3 + H^+}.

The total weak base concentration, [TB]\mathrm{[TB]}, is the sum of [BH+]\mathrm{[BH^+]} and [B]\mathrm{[B]}. Note that [TB]\mathrm{[TB]} is one of the two main unknowns in the BDW model for weak bases.

The CO2/HCO3\mathrm{CO_2}/\mathrm{HCO_3^-} buffer pair.

The formation of HCO3\mathrm{HCO_3^-} and H+\mathrm{H^+} from CO2\mathrm{CO_2} by hydration is given by the equilibrium reaction

CO2+H2OHCO3+H+,\mathrm{CO_2 + H_2O \rightleftharpoons HCO_3^- + H^+},

where the equilibrium constant is

KCO2=[HCO3][H+][CO2].K_\mathrm{CO_2}=\dfrac{\mathrm{[HCO_3^-][H^+]}}{\mathrm{[CO_2]}}.

Taking logarithms of both sides of (5), and recognising from Henry’s law that

[CO2]=spCO2,\mathrm{[CO_2]}=s\cdot p_\mathrm{CO_2},

where ss is the solubility coefficient for CO2\mathrm{CO_2} and pCO2p_\mathrm{CO_2} is the partial pressure of CO2\mathrm{CO_2}, we obtain the familiar Henderson-Hasselbalch equation

pH=pKCO2+log[HCO3]spCO2.\mathrm{pH}=\mathrm{pK_{CO_2}}+\log{\dfrac{\mathrm{[HCO_3^-]}}{s\cdot p_\mathrm{CO_2}}}.

Here, pH=log[H+]\mathrm{pH}=-\log[\mathrm{H^+}][5] and pKCO2=log(KCO2)\mathrm{pK_{CO_2}}=-\log(K_\mathrm{CO_2}).

In terms of the nomenclature above, one might regard CO2\mathrm{CO_2} as the weak acid HA\mathrm{HA}[6], and HCO3\mathrm{HCO_3^-} as its conjugate base A\mathrm{A^-}.

Buffering power (β\beta).

By definition, β\beta is the amount of strong base (e.g., NaOH\mathrm{NaOH}), or the negative of the amount of strong acid (e.g., HCl\mathrm{HCl}), that one must add to 1 L1~\mathrm{L} of solution to raise pH\mathrm{pH} by one pH\mathrm{pH} unit:

β=Δ[Strong Base]ΔpH=Δ[Strong Acid]ΔpH.\beta=\dfrac{\Delta\text{[Strong~Base]}}{\Delta \mathrm{pH}}=-\dfrac{\Delta\text{[Strong~Acid]}}{\Delta \mathrm{pH}}.

The units of β\beta are mM\mathrm{mM}. For additional details, refer to Roos & Boron (1981)Boron & Boulpaep (2016). Note that BDW defined β\beta as a negative number, as did Koppel and Spiro in their original definition of buffering Koppel, 1914Roos & Boron, 1980, rather than as a now-conventional positive quantity, as did Van Slyke in his later work Van Slyke, 1922. BDW’s definition, which they consistently applied, has no effect on the outcome of their simulations. In the present paper, we will follow the definition of Van Slyke — defining β\beta as a positive number — and make appropriate sign changes to the derived equations.

3The Boron & De Weer Model for the Permeation by an Uncharged Weak Acid and its Conjugate, Anionic Weak Base

The BDW mathematical model consists of two time-dependent ordinary differential equations (ODEs), one describing the time-course of the concentration of total intracellular buffer ([TA]i=[HA]i+[A]i\mathrm{[TA]_i} = \mathrm{[HA]_i}+\mathrm{[A^-]_i}) and the other the time-course of the intracellular free H+\mathrm{H^+} concentration (which is related to pHi\mathrm{pH_i}). BDW derived these two equations for the general cases in which any buffer pair HA/A\mathrm{HA}/\mathrm{A^-}, or any buffer pair B/BH+\mathrm{B}/\mathrm{BH^+}, can move passively across the plasma membrane of a prototype cell. Then, they applied these two general equations to their specific experimental conditions, namely exposure of a cell (a squid giant axon) to equilibrated extracellular CO2/HCO3\mathrm{CO_2}/\mathrm{HCO_3^-} or to equilibrated extracellular NH3/NH4+\mathrm{NH_3}/\mathrm{NH_4^+}.

Here, following BDW’s approach, we begin by deriving the equations for HA/A\mathrm{HA}/\mathrm{A^-}. In the next section, we apply the same general formalism to B/BH+\mathrm{B}/\mathrm{BH^+}.

Derivation for weak acids.

Imagine that a cell is exposed to a solution containing equilibrated HA/A\mathrm{HA}/\mathrm{A^-} and that both HA\mathrm{HA} and A\mathrm{A^-} initially move into the cell — because of the chemical gradient in the case of HA\mathrm{HA}, and because of the electrochemical gradient in the case of A\mathrm{A^-}.

An integrated form of Fick’s first law of diffusion describes the net passive influx[7] of HA\mathrm{HA} (JHAJ_\mathrm{HA})

JHA=PHA([HA]o[HA]i),J_\mathrm{HA}=P_\mathrm{HA}\bigg(\mathrm{[HA]_o}-\mathrm{[HA]_i}\bigg),

where JHAJ_\mathrm{HA} is flux (molm2s1\mathrm{mol\cdot m^{-2}\cdot s^{-1}}) and PHAP_\mathrm{HA} (ms1\mathrm{m\cdot s^{-1}}) is the membrane permeability to the uncharged weak acid HA\mathrm{HA}. Note that this is a passive diffusion equation because HA\mathrm{HA} is uncharged.

The constant field equation — also known as the Goldman, Hodgkin, Katz (GHK) Goldman, 1943Hodgkin & Huxley, 1952 equation — describes the net passive influx of A\mathrm{A^-} (JAJ_\mathrm{A^-}):

JA=PA(VmFRT)([A]oϵ[A]i1ϵ),J_\mathrm{A^-}=P_\mathrm{A^-}\left(\dfrac{V_\mathrm{m}F}{RT}\right)\left(\dfrac{\mathrm{[A^-]_o}-\epsilon\mathrm{[A^-]_i}}{1-\epsilon}\right),

where PAP_\mathrm{A^-} (ms1\mathrm{m\cdot s^{-1}}) is the membrane permeability to the charged conjugate base A\mathrm{A^-}, VmV_\mathrm{m} is the membrane potential (intracellular relative to extracellular potential), and ϵ\epsilon is a shorthand for eVmF/RTe^{{-V_\mathrm{m}F}/{RT}}. Note that JHAJ_\mathrm{HA} and JAJ_\mathrm{A^-} have units of molm2s1\mathrm{mol\cdot m^{-2}\cdot s^{-1}}.

Although HA\mathrm{HA} and A\mathrm{A^-} can interconvert in the cytosol, BDW assumed that the intracellular concentration of total weak acid [TA]i\mathrm{[TA]_i} only can change due to the transmembrane fluxes of HA\mathrm{HA} and A\mathrm{A^-} (see Figure 3). Thus, the time rate of change of [TA]i\mathrm{[TA]_i} is

d[TA]idt=ρ(JHA+JA),\dfrac{\mathrm{d[TA]_i}}{\mathrm{d}t}=\rho\left(J_\mathrm{HA}+J_\mathrm{A^-}\right),

where ρ\rho (m1\mathrm{m^{-1}}) is the area-to-volume ratio for the cell, and converts the transmembrane flux per unit area (in units of molm2s1\mathrm{mol\cdot m^{-2}\cdot s^{-1}}) to a time rate of change per unit cell volume (molm3s1\mathrm{mol\cdot m^{-3}\cdot s^{-1}} or mMs1\mathrm{mM\cdot s^{-1}}). (11) is the first of two ODEs of the BDW model for the buffer pair HA/A\mathrm{HA}/\mathrm{A^-}.

Cartoon illustrating the main assumptions in the BDW model of permeating uncharged weak acid \mathrm{HA} and its conjugate anionic weak base \mathrm{A^-}. The BDW model consists of two time-dependent ODEs. The first one describes the time-course of the intracellular concentration of total weak acid \mathrm{[TA]_i}, and the second one describes the time-course of \mathrm{[H^+]_i}. BDW assumed that \mathrm{[TA]_i} changes in time because of the transmembrane fluxes of \mathrm{HA} (J_\mathrm{HA}) — modelled according to Fick’s first law of diffusion — and \mathrm{A^-} (J_\mathrm{A^-}) — modelled according to the Goldman, Hodgkin, Katz (GHK) equation. According to BDW, the time rate of change of \mathrm{[H^+]_i} depends on the net rate \mathrm{d}Q/\mathrm{d}t at which acids are added into the cytosol. BDW assumed that \mathrm{d}Q/\mathrm{d}t depends on (i) the release of \mathrm{H^+} by some fraction x of the entering \mathrm{HA} (i.e., xJ_\mathrm{HA}), (ii) the consumption of \mathrm{H^+} by some fraction y of the entering \mathrm{A^-} (i.e., yJ_\mathrm{A^-}), and (iii) the additional rate of intracellular \mathrm{H^+} consumption via metabolism or active acid extrusion (J_\mathrm{H^+}).

Figure 3:Cartoon illustrating the main assumptions in the BDW model of permeating uncharged weak acid HA\mathrm{HA} and its conjugate anionic weak base A\mathrm{A^-}. The BDW model consists of two time-dependent ODEs. The first one describes the time-course of the intracellular concentration of total weak acid [TA]i\mathrm{[TA]_i}, and the second one describes the time-course of [H+]i\mathrm{[H^+]_i}. BDW assumed that [TA]i\mathrm{[TA]_i} changes in time because of the transmembrane fluxes of HA\mathrm{HA} (JHAJ_\mathrm{HA}) — modelled according to Fick’s first law of diffusion — and A\mathrm{A^-} (JAJ_\mathrm{A^-}) — modelled according to the Goldman, Hodgkin, Katz (GHK) equation. According to BDW, the time rate of change of [H+]i\mathrm{[H^+]_i} depends on the net rate dQ/dt\mathrm{d}Q/\mathrm{d}t at which acids are added into the cytosol. BDW assumed that dQ/dt\mathrm{d}Q/\mathrm{d}t depends on (i) the release of H+\mathrm{H^+} by some fraction xx of the entering HA\mathrm{HA} (i.e., xJHAxJ_\mathrm{HA}), (ii) the consumption of H+\mathrm{H^+} by some fraction yy of the entering A\mathrm{A^-} (i.e., yJAyJ_\mathrm{A^-}), and (iii) the additional rate of intracellular H+\mathrm{H^+} consumption via metabolism or active acid extrusion (JH+J_\mathrm{H^+}).

Later, Bevensee and Boron defined the time rate of change per unit volume (e.g., d[TA]i/dt\mathrm{d[TA]_i/dt}) as a ‘pseudoflux’ ϕ\phi, with the area-to-volume ratio folded into the value of ϕ\phi Bevensee & Boron, 2013. Physiologists sometimes prefer to present experimental data in terms of pseudoflux because most mammalian cells often have complex geometries that make it difficult to estimate surface area.

In deriving the second ODE of their model, BDW started by noting that the time rate of change of free protons, d[H+]i/dt\mathrm{d[H^+]_i}/\mathrm{d}t, depends on the rate at which acids are added into the cytosol per unit volume and per unit time — denoted dQ/dt\mathrm{d}Q/\mathrm{d}t (molm3s1\mathrm{mol\cdot m^{-3}\cdot s^{-1}}) where QQ is the total intracellular acid content. Like d[TA]i/dt\mathrm{d[TA]_i}/\mathrm{d}t, both d[H+]i/dt\mathrm{d[H^+]_i}/\mathrm{d}t and dQ/dt\mathrm{d}Q/\mathrm{d}t are pseudofluxes.

In their simple system of a squid giant axon exposed to CO2\mathrm{CO_2} (i.e., HA\mathrm{HA}) and HCO3\mathrm{HCO_3^-} (i.e., A\mathrm{A^-}), BDW assumed that only three general processes affect dQ/dt\mathrm{d}Q/\mathrm{d}t: (i) the release of H+\mathrm{H^+} by some fraction (xx) of the entering HA\mathrm{HA} (i.e., xJHAxJ_\mathrm{HA}), (ii) the consumption of H+\mathrm{H^+} by some fraction (y)(y) of the entering A\mathrm{A^-} (i.e., yJAyJ_\mathrm{A^-}), and (iii) the “additional” rate of intracellular consumption or active extrusion of H+\mathrm{H^+} (JH+J_\mathrm{H^+}; see Figure 3) above the fixed background rate of H+\mathrm{H^+} extrusion necessary to balance the fixed background rate of acid loading (i.e., addition of H+\mathrm{H^+} or equivalent acid, or removal of OH\mathrm{OH^-} or equivalent base) in the absence of HA/A\mathrm{HA}/\mathrm{A^-}. Thus,

dQdt=ρ(xJHAyJAJH+).\dfrac{\mathrm{d}Q}{\mathrm{d}t}=\rho\left(xJ_\mathrm{HA}-yJ_\mathrm{A^-}-J_\mathrm{H^+}\right).

A critical insight by BDW is that during each infinitesimal increment in time during which a bolus of HA\mathrm{HA} enters the cell, the entering HA\mathrm{HA} redistributes itself between HA\mathrm{HA} (and H+\mathrm{H^+}) vs A\mathrm{A^-}, according to the pre-existing ratios [HA]i/[TA]i\mathrm{[HA]_i}/\mathrm{[TA]_i} and [A]i/[TA]i\mathrm{[A^-]_i}/\mathrm{[TA]_i}. Thus, the fraction yy of entering HA\mathrm{HA} that remains HA\mathrm{HA} is

y=[HA]i[TA]i=[HA]i[HA]i+[A]i.y=\dfrac{\mathrm{[HA]_i}}{\mathrm{[TA]_i}}=\dfrac{\mathrm{[HA]_i}}{\mathrm{[HA]_i}+\mathrm{[A^-]_i}}.

This is also the fraction of entering A\mathrm{A^-} that combines with H+\mathrm{H^+} and becomes HA\mathrm{HA}. Combining the above expression with (2),

y=[HA]i[HA]i+[A]i=[H+]i[H+]i+KHA=α,y=\dfrac{\mathrm{[HA]_i}}{\mathrm{[HA]_i}+\mathrm{[A^-]_i}}=\dfrac{\mathrm{[H^+]_i}}{\mathrm{[H^+]_i}+K_\mathrm{HA}}=\alpha,

which BDW defined as α\alpha. Conversely, the fraction xx of A\mathrm{A^-} that remains A\mathrm{A^-} is

x=[A]i[HA]i+[A]i.x=\dfrac{\mathrm{[A^-]_i}}{\mathrm{[HA]_i}+\mathrm{[A^-]_i}}.

This is also the fraction of entering HA\mathrm{HA} that dissociates to form A\mathrm{A^-} and H+\mathrm{H^+}. Combining the above expression with (2),

x=[A]i[HA]i+[A]i=KHA[H+]i+KHA=1α.x=\dfrac{\mathrm{[A^-]_i}}{\mathrm{[HA]_i}+\mathrm{[A^-]_i}}=\dfrac{K_\mathrm{HA}}{\mathrm{[H^+]_i}+K_\mathrm{HA}}=1-\alpha.

In summary, (12) becomes:

dQdt=ρ((1α)JHAαJAJH+).\dfrac{\mathrm{d}Q}{\mathrm{d}t}=\rho\bigg((1-\alpha)J_\mathrm{HA}-\alpha J_\mathrm{A^-}-J_\mathrm{H^+}\bigg).

BDW modelled JH+J_\mathrm{H^+} (molm2s1\mathrm{mol\cdot m^{-2}\cdot s^{-1}}) as the additional proton-extrusion rate above the fixed background rate

JH+={kρ([H+]i[H+]i)pHi<pHi,0otherwise,J_\mathrm{H^+}= \begin{cases} \dfrac{k}{\rho}\bigg(\mathrm{[H^+]_i}-\mathrm{[H^+]'_i}\bigg) & \mathrm{pH_i} < \mathrm{{pH}'_i}, \\ 0 & \mathrm{otherwise}, \end{cases}

where kk (s1\mathrm{s^{-1}}) is the proton-pumping rate constant, (k/ρ)[H+]i(k/\rho)\mathrm{[H^+]_i} is the additional flux of H+\mathrm{H^+} above the background H+\mathrm{H^+} flux of (k/ρ)[H+]i(k/\rho)\mathrm{[H^+]'_i}, which occurs at the resting [H+]i\mathrm{[H^+]_i} of [H+]i\mathrm{[H^+]'_i} (i.e., resting pHi\mathrm{pH_i} of pHi\mathrm{{pH}'_i}). Note that k/ρk/\rho has units of (ms1\mathrm{m\cdot s^{-1}}), consistent with the membrane permeability terms PHAP_\mathrm{HA} and PAP_\mathrm{A^-} in (9) and (10).[8]

The BDW authors used the definition of buffering power, in its infinitesimal form, to derive the relation between d[H+]i/dt\mathrm{d}{\mathrm{[H^+]_i}}/\mathrm{d}t and dQ/dt\mathrm{d}Q/\mathrm{d}t, as shown in the following steps.

Our first goal is to obtain an expression for dpH/dt\mathrm{d}{\mathrm{pH}}/\mathrm{d}t in terms of dQ/dt\mathrm{d}Q/\mathrm{d}t. According to the chain rule:

dpHdt=(dpHdQ)(dQdt).\dfrac{\mathrm{dpH}}{\mathrm{d}t}=\left(\dfrac{\mathrm{dpH}}{\mathrm{d}Q}\right) \left(\dfrac{\mathrm{d}Q}{\mathrm{d}t}\right).

By definition (see (8)), β=dQ/dpH\beta=-{\mathrm{d}Q}/{\mathrm{dpH}} (molm3\mathrm{mol\cdot m^{-3}}), or equivalently

dpHdQ=1β.\dfrac{\mathrm{dpH}}{\mathrm{d}Q}=-\dfrac{1}{\beta}.

Combining (19) and (20)

dpHdt=(1β)(dQdt).\dfrac{\mathrm{dpH}}{\mathrm{d}t}=\left(-\dfrac{1}{\beta}\right)\left(\dfrac{\mathrm{d}Q}{\mathrm{d}t}\right).

Our next goal is to obtain an expression for dpHdt\dfrac{\mathrm{dpH}}{\mathrm{d}t} in terms of d[H+]idt\dfrac{\mathrm{d[H^+]_i}}{\mathrm{d}t}. According to the chain rule:

dpHdt=(dpHd[H+]i)(d[H+]idt).\dfrac{\mathrm{dpH}}{\mathrm{d}t}=\left(\dfrac{\mathrm{dpH}}{\mathrm{d[H^+]_i}}\right)\left( \dfrac{\mathrm{d[H^+]_i}}{\mathrm{d}t}\right).

By definition, pH=ln[H+]i/2.303\mathrm{pH}={-\ln{\mathrm{[H^+]_i}}}/{2.303}, so that:

dpHd[H+]i=12.303[H+]i.\dfrac{\mathrm{dpH}}{\mathrm{d}{\mathrm{[H^+]_i}}}=-\dfrac{1}{2.303\mathrm{[H^+]_i}}.

Combining (22) and (23), we have

dpHdt=(12.303[H+]i)(d[H+]idt),\dfrac{\mathrm{dpH}}{\mathrm{d}t}=\left(-\dfrac{1}{2.303\mathrm{[H^+]_i}}\right)\left(\dfrac{\mathrm{d}\mathrm{[H^+]_i}}{\mathrm{d}t}\right),

or equivalently,

d[H+]idt=2.303[H+]i(dpHdt).\dfrac{\mathrm{d[H^+]_i}}{\mathrm{dt}}=-2.303\mathrm{[H^+]_i}\left(\dfrac{\mathrm{dpH}}{\mathrm{d}t}\right).

Substituting (21) into (25), we obtain

d[H+]idt=(2.303[H+]iβ)(dQdt).\dfrac{\mathrm{d[H^+]_i}}{\mathrm{dt}}=\left(\dfrac{2.303\mathrm{[H^+]_i}}{\beta}\right)\left(\dfrac{\mathrm{d}Q}{\mathrm{d}t}\right).

Finally, substituting (17) into (26),

d[H+]idt=(2.303[H+]iβ)ρ((1α)JHAαJAJH+),\dfrac{\mathrm{d[H^+]_i}}{\mathrm{dt}}=\left(\dfrac{2.303\mathrm{[H^+]_i}}{\beta}\right) \rho\bigg((1-\alpha)J_\mathrm{HA}-\alpha J_\mathrm{A^-}-J_\mathrm{H^+}\bigg),

which is the second equation of the BDW model.

Substituting for α\alpha (from (14)), [HA]i=α[TA]i,[A]i=(1α)[TA]i[\mathrm{HA]_i} = \alpha [\mathrm{TA]_i}, [\mathrm{A^-]_i} = (1-\alpha) [\mathrm{TA]_i}, in (11) and (27), we obtain the two ODEs of the BDW model in terms of [TA]i[\mathrm{TA]_i} and [H+]i\mathrm{[H^+]_i}:

d[TA]idt=ρ(JHA+JA),\dfrac{\mathrm{d[TA]_i}}{\mathrm{d}t}=\rho\left(J_\mathrm{HA}+J_\mathrm{A^-}\right),
d[H+]idt=(2.303[H+]iβ)ρ((KHA[H+]i+KHA)JHA([H+]i[H+]i+KHA)JAJH+),\dfrac{\mathrm{d[H^+]_i}}{\mathrm{d}t}=\left(\dfrac{2.303\mathrm{[H^+]_i}}{\beta}\right)\rho\left(\left(\dfrac{K_\mathrm{HA}}{\mathrm{[H^+]_i}+K_\mathrm{HA}}\right)J_\mathrm{HA}-\left(\dfrac{\mathrm{[H^+]_i}}{\mathrm{[H^+]_i}+K_\mathrm{HA}}\right)J_\mathrm{A^-}-J_\mathrm{H^+} \right),

where JHAJ_\mathrm{HA} (from (9)), and JAJ_\mathrm{A^-} (from (10)) are given by:

JHA=PHA([HA]o[H+]i[H+]i+KHA[TA]i),J_\mathrm{HA}=P_\mathrm{HA}\left( \mathrm{[HA]_o}-\dfrac{\mathrm{[H^+]_i}}{\mathrm{[H^+]_i}+K_\mathrm{{HA}}}\mathrm{[TA]_i} \right),
JA=PA(VmFRT)([A]oKHA[H+]i+KHA[TA]iϵ1ϵ),J_\mathrm{A^-}=P_\mathrm{A^-}\left(\dfrac{V_\mathrm{m}F}{RT}\right)\left(\dfrac{\mathrm{[A^-]_o}-\dfrac{K_\mathrm{HA}}{\mathrm{[H^+]_i}+K_\mathrm{HA}}\mathrm{[TA]_i} \epsilon}{1-\epsilon}\right),

and JH+J_\mathrm{H^+} is given by (18).

The numerical solution of the above two equations yields the time courses of [TA]i[\mathrm{TA]_i} and [H+]i\mathrm{[H^+]_i}, which in turn yield the time-courses of [HA]i[\mathrm{HA]_i} and [A]i[\mathrm{A^-]_i} via:

[HA]i=α[TA]i,\mathrm{[HA]_i}=\alpha\mathrm{[TA]_i},
[A]i=(1α)[TA]i,\mathrm{[A^-]_i}=(1-\alpha)\mathrm{[TA]_i},

where

α=[H+]i[H+]i+KHA.\alpha=\dfrac{\mathrm{[H^+]_i}}{\mathrm{[H^+]_i}+K_\mathrm{HA}}.
Simulation for CO2/HCO3\mathrm{CO_2}/\mathrm{HCO_3^-} experiments.

BDW employed (28) and (29) to simulate the experiments in which they exposed a squid giant axon to a solution containing equilibrated CO2/HCO3\mathrm{CO_2}/\mathrm{HCO_3^-}. Their simulation protocol was a step change in (a) extracellular pCO2p_\mathrm{CO_2} from 00 to 5%5\% CO2\mathrm{CO_2} (3737 mmHg\mathrm{mmHg} or, with s=0.0321s=0.0321 mMmmHg1\mathrm{mM\cdot mmHg^{-1}}, [CO2]o=s.pCO2=1.1877 mM\mathrm{[CO_2]_o}=s.p_\mathrm{CO_2}=1.1877~\mathrm{mM}) and (b) extracellular [HCO3]\mathrm{[HCO_3^-]} from 00 to 59.526059.5260 mM\mathrm{mM} (the value that [HCO3]o\mathrm{[HCO_3^-]_o} has in a solution containing 5%5\% CO2\mathrm{CO_2} at pHo\mathrm{pH_o} of 7.707.70)[9]. The step change is applied for 27002700 s\mathrm{s} (45 min45~\mathrm{min}) at constant pHo=7.70\mathrm{pH_o}=7.70.

Table 1 and Table 2 report the parameter values used by BDW. Table 1 provides parameter values that are common to both the CO2/HCO3\mathrm{CO_2}/\mathrm{HCO_3^-} and the NH3/NH4+\mathrm{NH_3}/\mathrm{NH_4^+} experiments. Table 2 provides parameter values exclusive to the CO2/HCO3\mathrm{CO_2}/\mathrm{HCO_3^-} experiments only.

Table 1:Parameter values used in both simulations of squid-axon CO2/HCO3\mathrm{CO_2}/\mathrm{HCO_3^-} experiments and NH3/NH4+\mathrm{NH_3}/\mathrm{NH_4^+} experiments.

SymbolNameBDW ValueUnitNew ValueUnit
T\mathrm{T}temperature2323 (296.15296.15)º C\mathrm{C}K\mathrm{K})
R\mathrm{R}gas constant8.3148.314Jmol1K1\mathrm{J\cdot mol^{-1}\cdot K^{-1}}
F\mathrm{F}Faraday constant9648596485Cmol1\mathrm{C\cdot mol^{-1}}
ρ\rhoarea/volume ratio0.0080.008[10]μm1\mu\mathrm{m}^{-1}80008000m1\mathrm{m}^{-1}
pHo\mathrm{pH_o}extracellular pH\mathrm{pH}7.707.70

Table 2:Parameter values for simulations of squid-axon CO2/HCO3\mathrm{CO_2}/\mathrm{HCO_3^-} experiments.

SymbolNameBDW ValueUnitNew ValueUnit
βCO2\beta_\mathrm{CO_2}buffering power26-26mM\mathrm{mM}2626mM\mathrm{mM}
sssolubility constant for CO2\mathrm{CO_2}0.03210.0321 [11]mM/mmHg\mathrm{mM}/\mathrm{mmHg}0.2410.241mM/KPa\mathrm{mM}/\mathrm{KPa}
pCO2p_\mathrm{CO_2}partial pressure of CO2\mathrm{CO_2}3737mmHg\mathrm{mmHg}4.9334.933KPa\mathrm{KPa}
pCO2p_\mathrm{CO_2}partial pressure of CO2\mathrm{CO_2}3737mmHg\mathrm{mmHg}4.9334.933KPa\mathrm{KPa}
[CO2]o{\mathrm{[CO_2]_o}}extracellular CO2\mathrm{CO_2}1.18771.1877mM\mathrm{mM}
[HCO3]o{\mathrm{[HCO_3^-]_o}}extracellular HCO3\mathrm{HCO_3^-}59.526059.5260mM\mathrm{mM}
PCO2P_{\mathrm{CO_2}}membrane permeability6×1036\times 10^{-3}cms1\mathrm{cm\cdot s^{-1}}6×1056\times 10^{-5}ms1\mathrm{m\cdot s^{-1}}
PHCO3P_{\mathrm{HCO_3^-}}membrane permeability5×1075\times 10^{-7}cms1\mathrm{cm\cdot s^{-1}}5×1095\times 10^{-9}ms1\mathrm{m\cdot s^{-1}}
KCO2K_\mathrm{CO_2}acid dissociation constant10310^{-3}mM\mathrm{mM}
pKCO2\mathrm{pK_{CO_2}}acid dissociation constant6.06.0
VmV_\mathrm{m}membrane potential57-57 [12]mV\mathrm{mV}0.057-0.057V\mathrm{V}
kkH+\mathrm{H^+} pump rate constant03000-300 [13]s1\mathrm{s^{-1}}
pHi\mathrm{pH_i}intracellular pH\mathrm{pH}7.407.40
pHi\mathrm{{pH}'_i}basal pH\mathrm{pH}7.307.30 [14]

In the present work, the differential (28) and (29) — when coded in CellML and solved with OpenCOR — produce the plots in Figure 4. The simulation file Boron-CO2.sedml contains the computational setting for running the model. Open the .sedml file in OpenCOR and click Run Simulation. The initial conditions are [TA]i=0[\mathrm{TA]_i}=0 mM\mathrm{mM} and pHi=7.40\mathrm{pH_i}=7.40. Note that Figure 4 illustrates the time courses not only of pHi\mathrm{pH_i} — as presented by BDW — but also of quantities (e.g., various solute concentrations and fluxes) not displayed in the original paper; these values are useful for understanding the processes that contribute to the pHi\mathrm{pH_i} transient. Moreover, our curated and annotated version of the BDW model also allows one to alter the parameter values from those originally chosen by BDW, thereby extending the ability of the user to investigate the predictive power of the computational model.

Solution of the BDW model during and following a 2700~\mathrm{s} period of externally applied \mathrm{CO_2}. In these simulations \mathrm{pH_o}=7.70 and \mathrm{[HCO_3^-]_o} is determined from the equilibrium with \mathrm{[H^+]_o} and \mathrm{CO_2} (footnote 9). Note that, during the plateau phase, \mathrm{[HCO_3^-]_i} continues to rise as \mathrm{pH_i} rises at a constant \mathrm{[CO_2]_i} (the proton pumping rate k is set to 300 \mathrm{s^{-1}}, thus k/\rho= 0.0375 \mathrm{m\cdot s^{-1}}). Note also that, after the removal of \mathrm{CO_2}/\mathrm{HCO_3^-}, \mathrm{pH_i} rises to a higher value (\sim 8.15) than its starting value (\sim 7.4), indicating the net extrusion of acid from the cell during the \mathrm{CO_2}/\mathrm{HCO_3^-} exposure.

Figure 4:Solution of the BDW model during and following a 2700 s2700~\mathrm{s} period of externally applied CO2\mathrm{CO_2}. In these simulations pHo=7.70\mathrm{pH_o}=7.70 and [HCO3]o\mathrm{[HCO_3^-]_o} is determined from the equilibrium with [H+]o\mathrm{[H^+]_o} and CO2\mathrm{CO_2} (footnote 9). Note that, during the plateau phase, [HCO3]i\mathrm{[HCO_3^-]_i} continues to rise as pHi\mathrm{pH_i} rises at a constant [CO2]i\mathrm{[CO_2]_i} (the proton pumping rate kk is set to 300300 s1\mathrm{s^{-1}}, thus k/ρ=0.0375k/\rho= 0.0375 ms1\mathrm{m\cdot s^{-1}}). Note also that, after the removal of CO2/HCO3\mathrm{CO_2}/\mathrm{HCO_3^-}, pHi\mathrm{pH_i} rises to a higher value (8.15\sim 8.15) than its starting value (7.4\sim 7.4), indicating the net extrusion of acid from the cell during the CO2/HCO3\mathrm{CO_2}/\mathrm{HCO_3^-} exposure.

4The Boron & De Weer Model for the Permeation by an Uncharged Weak Base and its Conjugate, Cationic Weak Acid

Following an approach analogous to the one outlined above for weak acids, BDW derived two time-dependent ODEs. The first describes the time-course of the concentration of total intracellular buffer ([TB]i=[B]i+[BH+]i[\mathrm{TB]_i} = [\mathrm{B]_i}+[\mathrm{BH^+]_i}), and the other the time-course of the intracellular free [H+]i\mathrm{[H^+]_i}, for any buffer pair B/BH+\mathrm{B}/\mathrm{BH^+}.

Derivation for weak bases.

Imagine that a cell is exposed to a solution containing equilibrated B/BH+\mathrm{B}/\mathrm{BH^+}, and that both B\mathrm{B} and BH+\mathrm{BH^+} initially move into the cell — because of the chemical gradient in the case of B\mathrm{B}, and because of the electrochemical gradient in the case of BH+\mathrm{BH^+}.

Cartoon illustrating the main assumptions in the BDW model of permeating uncharged weak base \mathrm{B} and its conjugate anionic weak acid \mathrm{BH^+}. The BDW model consists of two time-dependent ODEs. The first one describes the time-course of the intracellular concentration of total weak base \mathrm{[TB]_i}, and the second one describes the time-course of \mathrm{[H^+]_i}. BDW assumed that \mathrm{[TB]_i} changes in time because of the transmembrane fluxes of \mathrm{HA} (J_\mathrm{B}) — modelled according to Fick’s first law of diffusion — and \mathrm{BH^+} (J_\mathrm{BH^+}) — modelled according to the Goldman, Hodgkin, Katz (GHK) equation. According to BDW, the time rate of change of \mathrm{[H^+]_i} depends on the net rate \mathrm{d}Q/\mathrm{d}t at which acids are added into the cytosol. BDW assumed that \mathrm{d}Q/\mathrm{d}t depends on (i) the release of \mathrm{H^+} by some fraction x of the entering \mathrm{BH^+} (i.e., xJ_\mathrm{BH^+}), (ii) the consumption of \mathrm{H^+} by some fraction y of the entering \mathrm{B} (i.e., yJ_\mathrm{B}), and (iii) the additional rate of intracellular \mathrm{H^+} consumption via metabolism or active acid extrusion (J_\mathrm{H^+}).

Figure 5:Cartoon illustrating the main assumptions in the BDW model of permeating uncharged weak base B\mathrm{B} and its conjugate anionic weak acid BH+\mathrm{BH^+}. The BDW model consists of two time-dependent ODEs. The first one describes the time-course of the intracellular concentration of total weak base [TB]i\mathrm{[TB]_i}, and the second one describes the time-course of [H+]i\mathrm{[H^+]_i}. BDW assumed that [TB]i\mathrm{[TB]_i} changes in time because of the transmembrane fluxes of HA\mathrm{HA} (JBJ_\mathrm{B}) — modelled according to Fick’s first law of diffusion — and BH+\mathrm{BH^+} (JBH+J_\mathrm{BH^+}) — modelled according to the Goldman, Hodgkin, Katz (GHK) equation. According to BDW, the time rate of change of [H+]i\mathrm{[H^+]_i} depends on the net rate dQ/dt\mathrm{d}Q/\mathrm{d}t at which acids are added into the cytosol. BDW assumed that dQ/dt\mathrm{d}Q/\mathrm{d}t depends on (i) the release of H+\mathrm{H^+} by some fraction xx of the entering BH+\mathrm{BH^+} (i.e., xJBH+xJ_\mathrm{BH^+}), (ii) the consumption of H+\mathrm{H^+} by some fraction yy of the entering B\mathrm{B} (i.e., yJByJ_\mathrm{B}), and (iii) the additional rate of intracellular H+\mathrm{H^+} consumption via metabolism or active acid extrusion (JH+J_\mathrm{H^+}).

Assuming, as in Figure 5, that [TB]i\mathrm{[TB]_i} only can change due to the transmembrane fluxes of B\mathrm{B} (JBJ_\mathrm{B}) and BH+\mathrm{BH^+} (JBH+J_{\mathrm{BH^+}}), the time rate of change of [TB]i]\mathrm{[TB]_i]} — analogous to (11) above — is

d[TB]idt=ρ(JB+JBH+),\dfrac{\mathrm{d[TB]_i}}{\mathrm{d}t}=\rho\left(J_\mathrm{B}+J_\mathrm{BH^+}\right),

where ρ\rho (m1\mathrm{m^{-1}}) is again the area-to-volume ratio for the cell. The equation

JB=PB([B]o[B]i),J_\mathrm{B}=P_\mathrm{B}\bigg(\mathrm{[B]_o}-\mathrm{[B]_i}\bigg),

is an integrated form of Fick’s first law of diffusion that describes the net passive flux of B, and

JBH+=PBH+(VmFRT)([BH+]oϵ[BH+]iϵ1),J_\mathrm{BH^+}=P_\mathrm{BH^+}\left(\dfrac{V_\mathrm{m}F}{RT}\right)\left(\dfrac{\mathrm{[BH^+]_o}-\epsilon '\mathrm{[BH^+]_i}}{\epsilon '-1}\right),

describes the net passive influx of BH+\mathrm{BH^+} according to the GHK equation. In the previous two equations, PBP_\mathrm{B} (ms1\mathrm{m\cdot s^{-1}}) is the membrane permeability to the uncharged weak base B\mathrm{B}, PBH+P_\mathrm{{BH^+}} (ms1\mathrm{m\cdot s^{-1}}) is the membrane permeability to the charged conjugate weak acid BH+\mathrm{BH^+}, and ϵ\epsilon ' is a shorthand for eVmF/RTe^{{V_\mathrm{m}F}/{RT}}. (33) is the first of two ODEs of the BDW model for the buffer pair B/BH+\mathrm{B}/\mathrm{BH^+}.

The second equation of the BDW model for a weak base — analogous to (27) above — is

d[H+]idt=(2.303[H+]iβ)ρ((1α)JBH+αJBJH+),\dfrac{\mathrm{d[H^+]_i}}{\mathrm{d}t}=\left(\dfrac{2.303\mathrm{[H^+]_i}}{\beta}\right) \rho\bigg((1-\alpha)J_\mathrm{BH^+}-\alpha J_\mathrm{B}-J_\mathrm{H^+}\bigg),

where JH+J_\mathrm{H+} is the same as in (18) and

α=[BH+]i[BH+]i+[B]i=[H+]i[H+]i+KBH+,\alpha=\dfrac{\mathrm{[BH^+]_i}}{\mathrm{[BH^+]_i}+\mathrm{[B]_i}}=\dfrac{\mathrm{[H^+]_i}}{\mathrm{[H^+]_i}+K_\mathrm{BH^+}},

and

1α=[B]i[BH+]i+[B]i=KBH+[H+]i+KBH+.1-\alpha=\dfrac{\mathrm{[B]_i}}{\mathrm{[BH^+]_i}+\mathrm{[B]_i}}=\dfrac{K_\mathrm{BH^+}}{\mathrm{[H^+]_i}+K_\mathrm{BH^+}}.

Substituting for α\alpha, [BH+]i=α[TB]i,[B]i=(1α)[TB]i[\mathrm{BH^+]_i} = \alpha [\mathrm{TB]_i}, [\mathrm{B]_i} = (1-\alpha) [\mathrm{TB]_i}, JBH+J_\mathrm{BH^+} , JBJ_\mathrm{B} in (33) and (36), we obtain the two ODEs of the BDW model in terms of [TB]i[\mathrm{TB]_i} and [H+]i\mathrm{[H^+]_i}

d[TB]idt=ρ(JB+JBH+),\dfrac{\mathrm{d[TB]_i}}{\mathrm{d}t}=\rho\left(J_\mathrm{B}+J_\mathrm{BH^+} \right),
d[H+]idt=2.303[H+]iβρ((KBH+[H+]i+KBH+)JBH+([H+]i[H+]i+KBH+)JBJH+),\dfrac{\mathrm{d[H^+]_i}}{\mathrm{d}t}=\dfrac{2.303\mathrm{[H^+]_i}}{\beta}\rho\left(\left(\dfrac{K_\mathrm{BH^+}}{\mathrm{[H^+]_i}+K_\mathrm{BH^+}}\right)J_\mathrm{BH^+}-\left(\dfrac{\mathrm{[H^+]_i}}{\mathrm{[H^+]_i}+K_\mathrm{BH^+}}\right)J_\mathrm{B}-J_\mathrm{H^+} \right),

where

JBH+=PBH+(VmFRT)([BH+]o[H+]i[H+]i+KBH+[TB]iϵϵ1),J_\mathrm{BH^+}=P_\mathrm{BH^+}\left(\dfrac{V_\mathrm{m}F}{RT}\right)\left(\dfrac{\mathrm{[BH^+]_o}-\dfrac{\mathrm{[H^+]_i}}{\mathrm{[H^+]_i}+K_\mathrm{BH^+}}\mathrm{[TB]_i} \epsilon '}{\epsilon '-1}\right),
JB=PB([B]oKBH+[H+]i+KBH+[TB]i),J_\mathrm{B}=P_\mathrm{B}\left( \mathrm{[B]_o}-\dfrac{K_\mathrm{BH^+}}{\mathrm{[H^+]_i}+K_\mathrm{BH^+}}\mathrm{[TB]_i} \right),

and JH+J_\mathrm{H^+} is given by (18).

Numerically integrating the above two equations yields the time courses of [TB]i[\mathrm{TB]_i} and [H+]i\mathrm{[H^+]_i}, from which we can compute the time-courses of [BH+]i[\mathrm{BH^+]_i} and [B]i[\mathrm{B]_i} from

[BH+]i=α[TB]i,\mathrm{[BH^+]_i}=\alpha\mathrm{[TB]_i},
[B]i=(1α)[TB]i,\mathrm{[B]_i}=(1-\alpha)\mathrm{[TB]_i},

where

α=[H+]i[H+]i+KBH+.\alpha=\dfrac{\mathrm{[H^+]_i}}{\mathrm{[H^+]_i}+K_\mathrm{BH^+}}.
Simulation for NH3/NH4+\mathrm{NH_3}/\mathrm{NH_4^+} experiments.

BDW employed (39) and (40) to simulate the experiments in which they exposed a squid giant axon to equilibrated NH3/NH4+\mathrm{NH_3}/\mathrm{NH_4^+}. Their simulation protocol was a step change in extracellular NH4Cl\mathrm{NH_4Cl} from 00 to 99 mM\mathrm{mM} (that is, a step change in [NH4+]o\mathrm{[NH_4^+]_o} from 00 to 8.868.86 mM\mathrm{mM}, and in [NH3]i\mathrm{[NH_3]_i} from 00 to 0.140.14 mM\mathrm{mM}) applied for 1500 s1500~\mathrm{s} (25 min25~\mathrm{min}) at constant pHo=7.70\mathrm{pH_o}=7.70.[15]

Table 1 and Table 3 report the parameter values used by BDW. Note that in the NH3/NH4+\mathrm{NH_3}/\mathrm{NH_4^+} simulations, kk is always zero, that is, JH+J_\mathrm{H^+} does not affect these processes.

Table 3:Parameter values for simulations of squid-axon