The Boron & De Weer Model of Intracellular pH Regulation
The classic Boron & De Weer (1976) paper provided the first evidence of active regulation of 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 under perturbations of both extracellular and extracellular . 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 regulation and a commentary on subsequent work that has enriched our understanding of this most basic aspect of cellular physiology.
1Introduction¶
In 1976 Boron & De Weer published their landmark paper on “Intracellular transients in squid giant axons caused by , , and metabolic inhibitors” Boron & De Weer, 1976. The authors used a squid giant axon preparation and a mathematical model of buffering and the transport of protons, bicarbonate () and to establish the experimental evidence for active regulation of intracellular () 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 or carbonate (). The paper reported on the consequences of adding and then removing extracellular , (where is ammonium), or the metabolic inhibitors, cyanide, azide and dinitrophenol (DNP).
In the first experiment, following exposure of the cell to elevated and , rapidly enters the cell and intracellular equilibrates with the extracellular , and generates intracellular and via the hydration reaction (). The accumulating results in a rapid fall of (see Figure 1A & Figure 1B). To the extent that the membrane is permeable to as well as to , will initially enter the cell passively, along its electrochemical gradient. Soon, however, the accumulation of intracellular reverses the electrochemical gradient and would be expected to lead to the passive efflux of . This loss of cellular would tend to acidify the cell because — to replenish the lost — additional would enter the cell and form even more and (the passive shuttle). Thus, the expectation was that prolonged exposure to would cause to fall rapidly (passive influx of ) and then to drift more slowly in the acidic direction (passive efflux of ). In fact, Boron & De Weer observed an alkaline drift, leading to the postulate of active extrusion of — or an equivalent process[1] — at a rate that exceeds the passive shuttling by the couple (see Figure 1A & Figure 1C).

Figure 1: changes caused by prolonged exposure of a squid giant axon to extracellular in the bulk solution. (A) Original and traces from figure 1 of BDW. Exposing an axon to extracellular causes a rapid fall in followed by a slow and sustained plateau-phase recovery (i.e., rises). Removing extracellular causes 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 exposure. (B) Cartoon illustrating the processes underlying the initial, rapid acidification phase in (A). The entry of leads to the intracellular production of (and thus to the observed decay) via the reaction . (C) Cartoon illustrating the processes underlying the plateau-phase alkalinisation in (A). After equilibration across the plasma membrane ( nadir in panel (A)), the slow entry of (or, equivalently, the slow exit of ) — which has always been present but was overwhelmed by the influx of — leads to the consumption of (and thus to the observed slow rise) via the reaction . The newly formed then exits the cell. The observed overshoot is the result of the accumulation of during exposure to extracellular . BDW used the mathematical model to postulate the presence of an active acid-extrusion mechanism that would explain both the observed plateau-phase recovery and the overshoot. (A), modified from Boron & De Weer (1976). (B)-(C), modified from Boron (2010).
Following removal of external , intracellular diffuses out, while intracellular combines with to leave the cell as . Thus, the entire intracellular load associated with entry would be removed, returning to its value before the addition of (or to a slightly lower value, to the extent that had passively exited during the exposure). In fact, Boron & De Weer observed that overshoots its resting value by an amount consistent with the net removal of by the active, acid-extrusion mechanism during the exposure (see Figure 1A & Figure 1C).
In the second experiment, following exposure of the cell to extracellular in the form of (ammonium chloride), the intracellular environment rapidly becomes alkaline as enters and combines with to form (equivalent to the hydration of to form and ). If this were the entire story, then would rise monotonically to a relatively alkaline value, and then the subsequent removal of would cause to fall to precisely its initial value. In fact, Boron & De Weer observed that the exposure to causes to rise rapidly and then fall slowly. Moreover, the subsequent removal of causes to undershoot its original value (see Figure 2A). Thus, Boron & De Weer postulated that, during the exposure, passively enters the cell down its electrochemical gradient. Early during the exposure, this influx would oppose the entry and slightly reduce the increase. Later during the exposure, after intracellular rises to match extracellular (), the continued passive influx of would generate intracellular and . The result would be a slow fall of and a rise in intracellular , the latter leading to the passive exit of (the passive shuttle, see Figure 2A & Figure 2B). The experimental data are consistent with the proposed mathematical model.

Figure 2: changes caused by a short and a long exposure of a squid giant axon to extracellular in the bulk solution. (A) Original and traces from figure 2 of BDW. A short exposure of the axon to extracellular causes a rapid rise in , followed by a decay that modestly undershoots (lower short arrow) its initial resting value upon removal of extracellular . A longer exposure of squid giant axons to extracellular causes a rapid rise in , followed by a slow and sustained decay. Removing extracellular causes 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 exposure. (B) Cartoon illustrating the processes underlying the initial alkalinisation phase in (A) for both short and long exposures to extracellular . The initial entry of leads to the intracellular consumption of (and thus to the observed rise) via the reaction . (C) Cartoon illustrating the processes underlying the plateau-phase acidification during the long exposure in (A). After equilibration across the plasma membrane ( peak in panel (A)), the slow entry of — which has always been present but overwhelmed by the influx of — leads to the production of (and thus to the observed slow decay during the plateau phase) via the reaction . The newly formed then exits the cell. The undershoots observed upon removal of extracellular , during both short and long exposures, are the result of the accumulation of during exposure to extracellular . BDW used the mathematical model to postulate the above sequence of events, including both the plateau-phase acidification and the 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 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 . Conversely, a base is any substance that can accept a . A buffer is any substance that can reversibly consume or produce , thereby minimising changes in .
The dissociation of the uncharged weak acid () to the anionic weak base () is described by the equilibrium reaction:
which is governed by the equilibrium constant[4]
An example is the carbonic acid () dissociation reaction,
The total weak acid concentration, , is the sum of and . Note that is one of the two main unknowns in the BDW model for weak acids.
The dissociation of the cationic weak acid () to the uncharged weak base () is described by the equilibrium reaction,
where the equilibrium constant is
An example is the dissociation reaction,
The total weak base concentration, , is the sum of and . Note that is one of the two main unknowns in the BDW model for weak bases.
The buffer pair.¶
The formation of and from by hydration is given by the equilibrium reaction
where the equilibrium constant is
Taking logarithms of both sides of (5), and recognising from Henry’s law that
where is the solubility coefficient for and is the partial pressure of , we obtain the familiar Henderson-Hasselbalch equation
Here, [5] and .
In terms of the nomenclature above, one might regard as the weak acid [6], and as its conjugate base .
Buffering power ().¶
By definition, is the amount of strong base (e.g., ), or the negative of the amount of strong acid (e.g., ), that one must add to of solution to raise by one unit:
The units of are . For additional details, refer to Roos & Boron (1981)Boron & Boulpaep (2016). Note that BDW defined 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 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 () and the other the time-course of the intracellular free concentration (which is related to ). BDW derived these two equations for the general cases in which any buffer pair , or any buffer pair , 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 or to equilibrated extracellular .
Here, following BDW’s approach, we begin by deriving the equations for . In the next section, we apply the same general formalism to .
Derivation for weak acids.¶
Imagine that a cell is exposed to a solution containing equilibrated and that both and initially move into the cell — because of the chemical gradient in the case of , and because of the electrochemical gradient in the case of .
An integrated form of Fick’s first law of diffusion describes the net passive influx[7] of ()
where is flux () and () is the membrane permeability to the uncharged weak acid . Note that this is a passive diffusion equation because 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 ():
where () is the membrane permeability to the charged conjugate base , is the membrane potential (intracellular relative to extracellular potential), and is a shorthand for . Note that and have units of .
Although and can interconvert in the cytosol, BDW assumed that the intracellular concentration of total weak acid only can change due to the transmembrane fluxes of and (see Figure 3). Thus, the time rate of change of is
where () is the area-to-volume ratio for the cell, and converts the transmembrane flux per unit area (in units of ) to a time rate of change per unit cell volume ( or ). (11) is the first of two ODEs of the BDW model for the buffer pair .
![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^+}).](https://physiome.curvenote.com/S000002/d4/public/_static/Figure 3-73c27f0c2fe4c79d60d871eacb50375d.png)
Figure 3:Cartoon illustrating the main assumptions in the BDW model of permeating uncharged weak acid and its conjugate anionic weak base . The BDW model consists of two time-dependent ODEs. The first one describes the time-course of the intracellular concentration of total weak acid , and the second one describes the time-course of . BDW assumed that changes in time because of the transmembrane fluxes of () — modelled according to Fick’s first law of diffusion — and () — modelled according to the Goldman, Hodgkin, Katz (GHK) equation. According to BDW, the time rate of change of depends on the net rate at which acids are added into the cytosol. BDW assumed that depends on (i) the release of by some fraction of the entering (i.e., ), (ii) the consumption of by some fraction of the entering (i.e., ), and (iii) the additional rate of intracellular consumption via metabolism or active acid extrusion ().
Later, Bevensee and Boron defined the time rate of change per unit volume (e.g., ) as a ‘pseudoflux’ , with the area-to-volume ratio folded into the value of 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, , depends on the rate at which acids are added into the cytosol per unit volume and per unit time — denoted () where is the total intracellular acid content. Like , both and are pseudofluxes.
In their simple system of a squid giant axon exposed to (i.e., ) and (i.e., ), BDW assumed that only three general processes affect : (i) the release of by some fraction () of the entering (i.e., ), (ii) the consumption of by some fraction of the entering (i.e., ), and (iii) the “additional” rate of intracellular consumption or active extrusion of (; see Figure 3) above the fixed background rate of extrusion necessary to balance the fixed background rate of acid loading (i.e., addition of or equivalent acid, or removal of or equivalent base) in the absence of . Thus,
A critical insight by BDW is that during each infinitesimal increment in time during which a bolus of enters the cell, the entering redistributes itself between (and ) vs , according to the pre-existing ratios and . Thus, the fraction of entering that remains is
This is also the fraction of entering that combines with and becomes . Combining the above expression with (2),
which BDW defined as . Conversely, the fraction of that remains is
This is also the fraction of entering that dissociates to form and . Combining the above expression with (2),
In summary, (12) becomes:
BDW modelled () as the additional proton-extrusion rate above the fixed background rate
where () is the proton-pumping rate constant, is the additional flux of above the background flux of , which occurs at the resting of (i.e., resting of ). Note that has units of (), consistent with the membrane permeability terms and in (9) and (10).[8]
The BDW authors used the definition of buffering power, in its infinitesimal form, to derive the relation between and , as shown in the following steps.
Our first goal is to obtain an expression for in terms of . According to the chain rule:
By definition (see (8)), (), or equivalently
Our next goal is to obtain an expression for in terms of . According to the chain rule:
By definition, , so that:
Combining (22) and (23), we have
or equivalently,
Substituting (21) into (25), we obtain
Finally, substituting (17) into (26),
which is the second equation of the BDW model.
Substituting for (from (14)), , in (11) and (27), we obtain the two ODEs of the BDW model in terms of and :
where (from (9)), and (from (10)) are given by:
and is given by (18).
The numerical solution of the above two equations yields the time courses of and , which in turn yield the time-courses of and via:
where
Simulation for experiments.¶
BDW employed (28) and (29) to simulate the experiments in which they exposed a squid giant axon to a solution containing equilibrated . Their simulation protocol was a step change in (a) extracellular from to ( or, with , ) and (b) extracellular from to (the value that has in a solution containing at of )[9]. The step change is applied for () at constant .
Table 1 and Table 2 report the parameter values used by BDW. Table 1 provides parameter values that are common to both the and the experiments. Table 2 provides parameter values exclusive to the experiments only.
Table 1:Parameter values used in both simulations of squid-axon experiments and experiments.
Symbol | Name | BDW Value | Unit | New Value | Unit |
---|---|---|---|---|---|
temperature | () | º (º ) | |||
gas constant | |||||
Faraday constant | |||||
area/volume ratio | [10] | ||||
extracellular |
Table 2:Parameter values for simulations of squid-axon experiments.
Symbol | Name | BDW Value | Unit | New Value | Unit |
---|---|---|---|---|---|
buffering power | |||||
solubility constant for | [11] | ||||
partial pressure of | |||||
partial pressure of | |||||
extracellular | |||||
extracellular | |||||
membrane permeability | |||||
membrane permeability | |||||
acid dissociation constant | |||||
acid dissociation constant | |||||
membrane potential | [12] | ||||
pump rate constant | [13] | ||||
intracellular | |||||
basal | [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 and . Note that Figure 4 illustrates the time courses not only of — 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 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.](https://physiome.curvenote.com/S000002/d4/public/_static/Figure 4-b527c01fcde0bf138feda7fc2762a7fc.png)
Figure 4:Solution of the BDW model during and following a period of externally applied . In these simulations and is determined from the equilibrium with and (footnote 9). Note that, during the plateau phase, continues to rise as rises at a constant (the proton pumping rate is set to , thus ). Note also that, after the removal of , rises to a higher value () than its starting value (), indicating the net extrusion of acid from the cell during the 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 (), and the other the time-course of the intracellular free , for any buffer pair .
Derivation for weak bases.¶
Imagine that a cell is exposed to a solution containing equilibrated , and that both and initially move into the cell — because of the chemical gradient in the case of , and because of the electrochemical gradient in the case of .
![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^+}).](https://physiome.curvenote.com/S000002/d4/public/_static/Figure 5-60f01f6992b5873c6bc8e57ceafb9194.png)
Figure 5:Cartoon illustrating the main assumptions in the BDW model of permeating uncharged weak base and its conjugate anionic weak acid . The BDW model consists of two time-dependent ODEs. The first one describes the time-course of the intracellular concentration of total weak base , and the second one describes the time-course of . BDW assumed that changes in time because of the transmembrane fluxes of () — modelled according to Fick’s first law of diffusion — and () — modelled according to the Goldman, Hodgkin, Katz (GHK) equation. According to BDW, the time rate of change of depends on the net rate at which acids are added into the cytosol. BDW assumed that depends on (i) the release of by some fraction of the entering (i.e., ), (ii) the consumption of by some fraction of the entering (i.e., ), and (iii) the additional rate of intracellular consumption via metabolism or active acid extrusion ().
Assuming, as in Figure 5, that only can change due to the transmembrane fluxes of () and (), the time rate of change of — analogous to (11) above — is
where () is again the area-to-volume ratio for the cell. The equation
is an integrated form of Fick’s first law of diffusion that describes the net passive flux of B, and
describes the net passive influx of according to the GHK equation. In the previous two equations, () is the membrane permeability to the uncharged weak base , () is the membrane permeability to the charged conjugate weak acid , and is a shorthand for . (33) is the first of two ODEs of the BDW model for the buffer pair .
The second equation of the BDW model for a weak base — analogous to (27) above — is
where is the same as in (18) and
and
Substituting for , , , in (33) and (36), we obtain the two ODEs of the BDW model in terms of and
where
and is given by (18).
Numerically integrating the above two equations yields the time courses of and , from which we can compute the time-courses of and from
where
Simulation for experiments.¶
BDW employed (39) and (40) to simulate the experiments in which they exposed a squid giant axon to equilibrated . Their simulation protocol was a step change in extracellular from to (that is, a step change in from to , and in from to ) applied for () at constant .[15]
Table 1 and Table 3 report the parameter values used by BDW. Note that in the simulations, is always zero, that is, does not affect these processes.
Table 3:Parameter values for simulations of squid-axon