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We would like to thank Dr Kemp for his letter regarding our recent paper (Usher-Smith et al. 2006). His work is in some respects complimentary to our original paper, in that it attempts to provide an analytic study of some of the processes that we modelled using charge-difference modelling and investigated experimentally. However, we must argue that this analytic approach falls short of capturing the full gamut of interdependent processes that occur during cellular exposure to, or production of, lactic acid. In particular, we will show in this response that whilst the equations derived in Dr Kemp's letter are generally successful in describing what is physically possible to impose, they are not constrained by what is physiologically reasonable in that they fail to constrain ion fluxes to what is thermodynamically possible given known ion pumps and channels. In addition, some of the more minor points in the letter appear to reflect misunderstandings of the original paper which we hope to clarify here.

Dr Kemp states at the start of his letter that ‘one of the authors’ main findings, that simulated cell swelling decreases with increasing buffer capacity, must be due to complex changes in transmembrane fluxes rather than directly to altered buffering.' First, we would not describe the relationship depicted in Fig. 9B of our original paper as a central finding: it is a prediction derived from modelling. We use this to provide support for our hypotheses explaining the relationship we observed experimentally between intracellular lactate accumulation and cell volume.

Second, we clearly make the point that titration of intracellular buffers influences cell volume through the resultant changes in the balance of transmembrane ion fluxes, for example in the abstract to our paper where we state that volume changes result from ‘… osmotic effects resulting from the net cation efflux that would follow a titration of intracellular membrane-impermeant anions by the intracellular accumulation of protons.’ Thus it is not clear to us what Dr Kemp's conclusion that volume changes must be due to transmembrane fluxes adds to our analysis.

Third, because the figures in the Appendix of our original paper, including Fig. 9B, are all derived from computer modelling, it is strange that Dr Kemp seems to be suggesting a different interpretation of our results; in contrast, perhaps, to a theory describing experimental behaviour, model behaviour may be described absolutely and precisely. Thus, although it might be entirely reasonable to contend with our assumptions or the values used for parameters, this is not the focus of Dr Kemp's argument. Instead, he argues against our conclusion that intracellular buffering capacity is the chief determinant of the volume response of a cell to exposure to a membrane-permeant acid or alkali (such as lactic acid in this paper or ammonium in our previous work; Fraser et al. 2005), stating that ‘[the volume change] might be quite sensitive to detailed relationships between kinetic parameters not so far analysed in these terms.’ However, crucially, whilst in his analytic model ion fluxes are treated as parameters that can take any value, they are treated as dependent variables in our charge-difference model. Thus the ‘kinetic parameters’, which we take to mean ion fluxes, on which the volume response to lactate exposure does indeed depend are themselves constrained by ion permeabilities and vary according to the energetic favourability of ion movements.

Thus in Figs 7–10 of our paper, we demonstrate the influence of membrane permeabilities and transporter. This allowed us to identify the principal determinants of the volume response to lactate exposure as being the intracellular buffering capacity, the membrane permeability to the lactate anion (Lac) and Na+/H+-exchange (NHE) activity. These key parameters were kept constant in the simulations depicted in Fig. 9B whilst intracellular buffering capacity was varied, allowing the influence of buffering capacity upon transmembrane fluxes, and thereby upon cell volume, to be ascertained. In contrast to our treatment of transmembrane fluxes as dependent variables, Dr Kemp's analysis treats them as parameters that, in his words, may be ‘specified by fiat’. Thus his analysis shows a range of possible outcomes (see Fig. 1 in his letter), whereas we would contend that there is in fact only a single unique steady state for any given set of parameters. Furthermore, those parameters identified in our work as most important in determining the volume response to lactate exposure, such as the ratio of lactic acid to lactate ion permeability, have been measured experimentally (Woodbury & Miles, 1973; Wolosin & Ginsburg, 1975).

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Figure 1. Charge-difference modelling of transmembrane H+ fluxes following exposure to lactic acid Fluxes of LacH (long dashes), Lac (continuous line) and Na+/H+ exchange (NHE, short dashes) are shown over time following exposure to 40 mm lactate in normal Ringer at an extracellular pH of 7. Note the initial rapid influx of LacH and a much smaller influx of Lac. LacH influx then declines as intracellular LacH builds up, whilst the Lac flux becomes an efflux as [Lac]i builds up following dissociation of LacH within the cell and NHE activity increases greatly due to the resultant drop in intracellular pH (or increase in free [H+]i). Eventually a steady-state is reached where LacH influx and Lac efflux are precisely equal, and the resultant effective H+ influx is balanced by H+ efflux via the NHE.

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Nevertheless, we freely acknowledge that an analytic approach can have considerable advantages over iterative modelling. In particular, it may allow a clearer demonstration of the precise causal relationships between variables. However, we employed an iterative modelling approach in our paper because we found that the relationships between the relevant variables are too complex to permit the formulation of an analytic relationship. Thus we were able to consider ion fluxes as variables that are dependent on their respective permeabilities, electrochemical equilibria, and the membrane potential. In contrast, Dr Kemp's analysis represents all transmembrane cation fluxes with a term ‘φ’ that appears to be allowed to take any value. We remain confident that our approach captured the relevant physiology well, and are concerned that the analytic approach presented in this letter represents a severe oversimplification. In particular, charge-difference modelling was designed ab initio from robust conservation principles allowing it to accurately replicate predictions from analytic approaches, such as Gibbs–Donnan analysis, when employing identical assumptions (Fraser & Huang, 2004, 2007). Thus we would not expect a simplified analytic approach to contribute findings that were not uncovered through charge-difference modelling.

Nevertheless, it is perhaps useful to restate the findings of the charge-difference model in relatively simple terms. For simplicity, the relevant determinants of the new cellular steady state during exposure to extracellular lactate may be divided into two separate processes: first, the influence of extracellular lactate concentration upon intracellular lactate, hence intracellular pH, and hence the mean charge-valency (zX) of intracellularly sequestered anions (X); and second, the resultant influence of zX and intracellular lactate upon cell volume.

Transmembrane fluxes of lactate and H+ following exposure to extracellular lactate

  1. Top of page
  2. Transmembrane fluxes of lactate and H+ following exposure to extracellular lactate
  3. Other transmembrane ion fluxes following exposure to extracellular lactate
  4. Minor points raised by Dr Kemp's letter
  5. Summary
  6. References

Upon exposure to extracellular lactate, the lactate fluxes reach a steady state via the following sequence of events, as illustrated in Fig. 1.

(1) There is initially an inward gradient of both the acid form (denoted LacH) and the anion form (denoted Lac). This produces a small inward flux of Lac and a large inward flux of LacH, as the permeability of the latter acid form is very much greater.

(2) The resultant increase in [LacH] within the cell favours dissociation of LacH [RIGHTWARDS ARROW] Lac+ H+. This dissociation keeps [LacH]i < [LacH]e and thus favours further LacH influx. In contrast, the equilibrium value of [Lac]i is low due to the negative membrane potential (Em). From the Nernst equation, [Lac]i=[Lac]e/(e(−EmF/(RT))). Thus for exposure to 40 mm lactate the equilibrium value of [Lac]i is approximately 1 mm.

(3) Increased [Lac]i, primarily due to LacHi dissociation, shifts the electrochemical gradient of Lac to an outward one. Thus the initially inward Lac flux reverses. Note that because the only non-electroneutral flux following exposure to extracellular lactate is directed outward, exposure to extracellular lactate provides a surprisingly good model of intracellular lactic acid production, as shown in Figs 7 and 8 of Usher-Smith et al. (2006).

(4) Intracellular acidification shifts the equilibrium of the lactic acid dissociation reaction more towards the acid form, LacH. Thus intracellular acidification slows the inward LacH flux.

(5) In order for this situation to reach steady state, either LacH influx and Lac efflux must cease, or their fluxes must be equal and opposite and accompanied by an H+ efflux equal to the net gain in H+ that results from the difference in LacH influx and Lac efflux. Net LacH fluxes would cease if [LacH]i=[LacH]e, whereas net Lac fluxes would cease if [Lac]i=[Lac]e/(e(−EmF/(RT))). As a numerical example, for exposure to 40 mm extracellular lactate at Em=− 85 mV the equilibrium values of [LacH]i≈ 0.028 mm and [Lac]i≈ 1.37 mm. However, to satisfy the Henderson-Hasselbalch equation such that the reaction LacHi[LEFT RIGHT ARROW] Laci+ H+i is at equilibrium would require a pH of just under 5.5. This is > 1 unit lower than experimentally observed values, suggesting a dynamic equilibrium rather than a steady-state with no fluxes.

(6) Both charge-difference modelling (Fig. 1 below) and experimental measurements of pH support the idea of a dynamic equilibrium at steady state in which JLacH=−JLac−=JNHE. This represents a net efflux of H+ via the NHE that balances the gain in intracellular H+ due to LacH entry, dissociation, and efflux in anionic Lac form.

This explains why intracellular buffer capacity does not influence steady-state pHi following lactate exposure, a key point for which Kemp states that ‘the mechanism is not obvious’. Thus pHi determines the ratio of LacH to Lac and NHE activity and therefore, ceteris paribus, there is a unique value of pHi that allows JLacH=−JLac−=JNHE. Whilst intracellular buffering capacity has an important influence on the rate of pHi change for a given change in intracellular H+ content, it does not influence the steady-state pHi because cause and effect is such that it is the value of pHi that determines the steady state. Thus greater buffering capacity permits a greater LacH influx before the steady-state pHi is attained, but does not influence this pHi.

Other transmembrane ion fluxes following exposure to extracellular lactate

  1. Top of page
  2. Transmembrane fluxes of lactate and H+ following exposure to extracellular lactate
  3. Other transmembrane ion fluxes following exposure to extracellular lactate
  4. Minor points raised by Dr Kemp's letter
  5. Summary
  6. References

The influence of buffered H+ upon cell volume is significant, as shown both experimentally and using charge-difference modelling (Fraser et al. 2005). We have explained this influence as being due to the titration, and hence a change in the mean charge valency, zX, of intracellularly sequestered anions, X. zX is an important determinant of cell volume and, to a lesser extent, Em, at steady-state (Fraser & Huang, 2004). Of course, this influence of zX upon cell volume and Em is due to changes in the balance of transmembrane fluxes, both active and passive. In addition, Lac has both a direct osmotic influence, and an osmotic influence through an increase in associated cellular cation content. The resultant complex effects on transmembrane ion fluxes of exposure to extracellular lactate are shown in Fig. 2.

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Figure 2. Charge-difference modelling of the influence of lactate exposure upon transmembrane ion fluxes Exposure to 40 mm lactate as in Figure 1 produces a complex sequence of changes in the total transmembrane ion fluxes (shown as Jtotal), that produce first a net influx of ions followed by a net efflux of ions, before settling on a new steady state with no net fluxes. See also Fig. 6 in Usher-Smith et al. (2006). The initial net ion influx and resultant volume increase essentially results from LacH influx. Volume then more slowly settles due primarily to K+ loss secondary to a reduction in zX, with a smaller component of the volume recovery resulting from decreased [Lac]i as intracellular acidification and increased [LacH]i slows JLacH. Note that Jtotal includes the lactate fluxes depicted in Figure 1 as well as the other fluxes shown on this figure.

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Although these dynamic changes towards steady state are complex, it is possible to explain the influence of zX upon steady-state cell volume in simple terms by considering the cell as a double Gibbs–Donnan equilibrium. We have laid this out mathematically in a recent review article (Fraser & Huang, 2007), but for convenience will restate the key points here. The Gibbs–Donnan approach is valuable because although it cannot accurately predict the absolute membrane potential as it must assume Na+ is membrane-impermeant, it correctly predicts the direction and approximate relative magnitude of changes in cell volume and Em that result from changes in zX. Indeed, it should not be surprising that the cell is close to a Gibbs–Donnan equilibrium since such an equilibrium is, by definition, a stable state that does not require any energy input, and therefore cellular homeostasis requires less energy the closer the cell approximates to a Gibbs–Donnan equilibrium.

The key point is that if Cl and K+ are passively distributed, they must have transmembrane gradients that are equal in magnitude but opposite in direction. Ignoring the influence of transient fluxes, since fluxes must reach zero at steady state, a reduction in the magnitude of zX must cause cellular shrinkage when [K+]i/[K+]e=[Cl]i/[Cl]e >> 1, as [K+]i must decrease by considerably more than [Cl]i increases to maintain the equality of these ratios. A more physiological model cell, with a non-zero Na+ permeability and Na+/K+-ATPase activity follows these predictions reasonably closely because although passive and active Na+ fluxes are influenced by the membrane potential, the effect of small changes in zX upon the membrane potential is small around physiological values of zX (Fraser & Huang, 2004).

This also explains why cell volume is predicted to be influenced by intracellular buffering capacity even though steady-state pHi is not. Thus buffering capacity influences the amount of buffered H+ at any given pHi, and therefore influences the change in zX that is induced by any change in pHi.

It would be difficult to generate an analytic relationship that could be used to calculate the unique value of pHi at which a steady state would be reached, however. Dr Kemp begins the formulation of an analytic relationship by introducing a term, μ, that he defines as the ‘fraction of “glycolytic” H+ which is buffered’. However, he then seems to treat this term as if it is constant or even easily predictable, whereas we must argue that it varies with changes in several other interdependent variables including pHi, NHE activity, intracellular [Na+]i, Em, Lac and LacH fluxes, sodium pump activity, cell volume, intracellular buffering capacity, etc. Similarly, Dr Kemp groups cation fluxes in a term, φ. However, this term is not constrained by the thermodynamic favourabilities of the various component fluxes. Moreover, we would argue that electroneutral fluxes (e.g. JLacH and JNHE) and anion fluxes (e.g. JCl and JLac−) have critical influence upon the cellular steady state that do not appear to be included in his analysis.

Dr Kemp's conclusion is not entirely clear to us, but it appears to be that lactate production could cause any combination of transmembrane fluxes and cell volume change. The conclusions of our paper (Usher-Smith et al. 2006) are in sharp contrast to this. By modelling all relevant fluxes according to the membrane potential, equilibrium potentials and permeabilities, we were able to specify precisely the determinants of the volume response to lactate exposure/production. Briefly, in approximate order of importance the major parameters that influence the volume and pHi response on exposure to lactate are as in Table 1.

Table 1. 
Increase in:Causes influence of lactate exposure upon volume to:Causes influence of lactate exposure upon pHi to:
Intracellular buffering capacityIncrease(No change)
Membrane Lac permeabilityIncreaseIncrease
Na+/H+ exchange activityDecreaseDecrease
Membrane LacH permeability(Increase, but normally supramaximal)(Increase, but normally supramaximal)

We show that normal physiological values for these parameters produce volume shrinkage in response to intracellular acidification similar to that resulting from lactate exposure that is close to the volume expansion that would otherwise be expected from intracellular lactate accumulation. Thus the overall influence of lactate exposure, or intracellular production of lactic acid, upon cell volume is expected to be small. This is arguably adaptive given that, quite apart from any adverse influence of cell volume upon skeletal muscle function, significant volume expansion of skeletal muscle in anaerobic exercise might seriously compromise the circulating volume, which contains just 7–8% of body water compared to the roughly 45% of body water that is contained within skeletal muscles (Gosmanov et al. 2003).

Minor points raised by Dr Kemp's letter

  1. Top of page
  2. Transmembrane fluxes of lactate and H+ following exposure to extracellular lactate
  3. Other transmembrane ion fluxes following exposure to extracellular lactate
  4. Minor points raised by Dr Kemp's letter
  5. Summary
  6. References

On the background of the above clarifications, we can now address some more minor points and queries contained in Dr Kemp's letter.

Points (b) and (c) in the section in Kemp's letter entitled ‘Interpretation of Fig. 9B’ contain several misunderstandings regarding details of how we modelled intracellular buffering, that whilst not significantly influencing the conclusions of either Kemp's analysis or our own, require clarification. However, it does lead him to calculate erroneous values for βi. Some variation βi is to be expected as βi does change with pHi (Leem et al. 1999; Fraser et al. 2005). Some of the discrepancy derives from Dr Kemp's estimation of values from the graphs in our paper. However, the wide variations in the values calculated in section (c) of his letter reflect a misunderstanding of the relationship between [X]i and intracellular buffering capacity in our original paper.

We express buffer concentrations as a fraction (n) of the charge carried by membrane impermeant anions, and thus each buffer species is present at a concentration of nzX[X]i. However, [X]i, which properly denotes the osmotic activity of membrane-impermeant anions, is not influenced by changes in intracellular buffer capacity. We modelled two components of intracellular buffer (after Leem et al. 1999), A and B, where pKa(A)= 6.03 and pKa(B)= 7.57. Under normal conditions,

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(Note that there is, unfortunately, a misprint in the appendix to Fraser et al. 2005 that swaps these proportions.) To halve buffer capacity, we simply halved [A] and [B]. As stated in our paper, and in contrast to Dr Kemp's assumptions, this does not change the initial values of zX and [X]. Instead, it changes the relationship between pHi and zX. When buffer capacity was double normal, [A]+[B] > zX[X]. This simply implies that more [H+] ions could bind to the intracellular buffer than there are negative charges on X (i.e. zX could become positive at extremely low values of pHi). However, in none of the simulations was pHi low enough to produce values of zX greater than −1.01. Dr Kemp's confusion in point (c) in his letter seems to stem from his belief that by changing intracellular buffer capacity we change [X]. We must stress that [X] is a variable, not a parameter, of our model. A further confusion appears to stem from Dr Kemp's failure to recognize that intracellular buffering capacity influences total lactate influx through its influence upon the relationship between lactate influx and pHi.

Summary

  1. Top of page
  2. Transmembrane fluxes of lactate and H+ following exposure to extracellular lactate
  3. Other transmembrane ion fluxes following exposure to extracellular lactate
  4. Minor points raised by Dr Kemp's letter
  5. Summary
  6. References

Dr Kemp states in his letter that in the charge-difference model ‘steady-state cell volume is constrained by osmotic balance and electroneutrality’. Whilst Dr Kemp's analytic approach is indeed based solely on these constraints, the steady-state of the charge difference model is additionally constrained by a requirement that all intracellular concentrations must be constant. As such concentrations are determined by relative transmembrane fluxes, the wide range of possible electroneutral and osmotically balanced states identified by Dr Kemp are, for a given set of transmembrane ion permeabilities and transporter kinetics, reduced to a unique cellular steady state (Usher-Smith et al. 2006).

References

  1. Top of page
  2. Transmembrane fluxes of lactate and H+ following exposure to extracellular lactate
  3. Other transmembrane ion fluxes following exposure to extracellular lactate
  4. Minor points raised by Dr Kemp's letter
  5. Summary
  6. References