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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). Figure 1 Charge-difference modelling of transmembrane H+ fluxes following exposure to lactic acid 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.

The combined effects of intracellular lactate and proton accumulation on cell volume, Vc, were investigated in resting Rana temporaria striated muscle fibres. Intracellular lactate and H+ concentrations were simultaneously increased by exposing resting muscle fibres to extracellular solutions that contained 20–80 mm sodium lactate. Cellular H+ and lactate entry was confirmed using pH-sensitive electrodes and 1H-NMR, respectively, and effects on Vc were measured using confocal microscope xz-scanning. Exposure to extracellular lactate up to 80 mm produced significant changes in pH and intracellular lactate (from a pH of 7.24 ± 0.03, n = 8, and 4.65 ± 1.07 mm, n = 6, respectively, in control fibres, to 6.59 ± 0.03, n = 4, and 26.41 ± 0.92 mm, n = 3, respectively) that were comparable to those observed following fatiguing stimulation (6.30–6.70 and 18.04 ± 1.78 mm, n = 6, respectively). Yet, the increase in intracellular osmolarity expected from such an increase in intracellular lactate did not significantly alter Vc. Simulation of these experimental results, modified from the charge difference model of Fraser & Huang, demonstrated that such experimental manoeuvres produced changes in intracellular [H+] and [lactate] comparable to those observed during muscle fatigue, and accounted for this paradoxical conservation of Vc through balancing negative osmotic effects resulting from the net cation efflux that would follow a titration of intracellular membrane-impermeant anions by the intracellular accumulation of protons. It demonstrated that with established physiological values for intracellular buffering capacity and the permeability ratio of lactic acid and anionic lactate, PLacH: PLac−, this would provide a mechanism that precisely balanced any effect on cell volume resulting from lactate accumulation during exercise.

The influence of extracellular hypotonicity on the relationship between cell volume (Vc) and resting membrane potential (Em) was investigated in Rana temporaria skeletal muscle. Vc was measured by confocal microscope imaging of fibres through their transverse (xz) planes, and Em was determined using standard microelectrode techniques. Hypotonic solutions first elicited a rapid increase in fibre volume, ΔVR+ that fulfilled expectations of simple osmotic behaviour described in earlier reports. However, this was consistently followed by a slow increase in Vc (ΔVS+) to 10–15% above osmotic predictions. Longer (>1 h) exposures to hypotonic solutions permitted a subsequent slow decrease in Vc (ΔVS−), the eventual magnitude of which exceeded that of the preceding ΔVS+. Restoration of isotonic conditions elicited a prompt recovery in Vc that matched simple osmotic predictions and thus left a net change in Vc. Such alterations in Vc attributable to ΔVS+ then gradually reversed, while those due to ΔVS− persisted. Both ΔVS+ and ΔVS− persisted under conditions of Cl− deprivation. The depolarization of Em that accompanied ΔVR+ was consistent with dilution of intracellular [K+]. Em did not significantly alter during the subsequent ΔVS transients. These empirical features of ΔVS+ and ΔVS− were analysed using the quantitative charge-difference model of Fraser and Huang, published in 2004. This attributed the ΔVS+ to an electroneutral increase in the effective osmotic activity of normally membrane-impermeant intracellular anions. In contrast, the ΔVS− could only be explained by an efflux of such anions and was accordingly comparable to organic anion-dependent regulatory volume decreases reported in other cell types.

The relationship between cell volume (Vc) and membrane potential (Em) in Rana temporaria striated muscle fibres was investigated under different conditions of intracellular acidification. Confocal microscope xz-scanning monitored the changes in Vc, whilst conventional KCl and pH-sensitive microelectrodes measured Em and intracellular pH (pHi), respectively. Applications of Ringer solutions with added NH4Cl induced rapid reductions in Vc that rapidly reversed upon their withdrawal. These could be directly attributed to the related alterations in extracellular tonicity. However: (1) a slower and persistent decrease in Vc followed the NH4Cl withdrawal, leaving Vc up to 10% below its resting value; (2) similar sustained decreases in resting Vc were produced by the addition and subsequent withdrawal of extracellular solutions in which NaCl was isosmotically replaced with NH4Cl; (3) the same manoeuvres also produced a marked intracellular acidification, that depended upon the duration of the preceding exposure to NH4Cl, of up to 0.53 ± 0.10 pH units; and (4) the corresponding reductions in Vc similarly increased with this exposure time. These reductions in Vc persisted and became more rapid with Cl− deprivation, thus excluding mechanisms involving either direct or indirect actions of pHi upon Cl−-dependent membrane transport. However they were abolished by the Na+,K+-ATPase inhibitor ouabain. The Em changes that accompanied the addition and withdrawal of NH4+ conformed to a Nernst equation modified to include realistic NH4+ permeability terms, and thus the withdrawal of NH4+ restored Em to close to control values despite a persistent change in Vc. Finally these Em changes persisted and assumed faster kinetics with Cl− deprivation. The relative changes in Vc, Em and pHi were compared to predictions from the recent model of Fraser and Huang published in 2004 that related steady-state values of Vc and Em to the mean charge valency (zx) of intracellular membrane-impermeant anions, X−i. By assuming accepted values of intracellular buffering capacity (βi), intracellular acidification was shown to produce quantitatively predictable decreases in Vc. These findings thus provide experimental evidence that titration of the anionic zx by increased intracellular [H+] causes cellular volume decrease in the presence of normal Na+,K+- ATPase activity, with Cl−-dependent membrane fluxes only influencing the kinetics of such changes.