Reply from James A. Fraser, Juliet A. Usher‐Smith and Christopher L.‐H. Huang

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.