Muscle cell volume and pH changes due to glycolytic ATP synthesis

This letter deals with a theoretical question raised by a recent paper (Usher-Smith et al. 2006) which, as well as experimental data, presents a kinetic simulation and a steady-state analysis of glycolytic cell swelling in muscle. Here I focus on the role of cellular buffer capacity, extending the steady-state analysis and connecting it with other work on cellular acid–base balance in order to show that one of the authors' main findings, that simulated cell swelling decreases with increasing cellular buffer capacity (Fig. 9B in Usher-Smith et al. (2006), hereinafter called simply Fig. 9B), must be due to complex changes in transmembrane fluxes rather than directly to altered buffering. I also briefly extend the analysis to include phosphocreatine (PCr) breakdown, an important feature of exercising muscle. Concentration, charge and volume Key: A, lactate (30 mm); C, cations; Cl, chloride; X and Y, titratable and non-titratable anions (neutral species are ignored). Concentrations sum to extracellular osmolarity (Π= 300 mm) and electroneutrality requires zero total (signed) area. The figure shows (a) rest (basal); and lactate accumulation (exercise) with zero cation flux (see section (ii) in the main text) and (b) normal or (c) 40% increased buffer capacity, and at normal buffer capacity with (d) equimolar and (e) more than equimolar cation efflux (see section (iii)) and (f) equimolar cation influx (see section (iv)). The volume ratio λ is the ratio of the lengths of white to black vertical arrows for all impermeant species: for clarity, concentrations of X + Y and impermeant cations are connected by dotted lines. Dashed lines and brackets show the effects of cation flux changes. For captioning and comparative purposes, b is considered to have ‘normal’ changes in pH and zX. [X] and pKX are from Leem et al. (1999), other parameters from Usher-Smith et al. (2006). As the source of basal zX=−1.65 in Usher-Smith et al. (2006), following Fraser & Huang (2004) and Fraser et al. (2005), is not clear (not being explicit in Maughan & Recchia (1985) and Maughan & Godt (2001), cited in support in Fraser & Huang (2004), and its effect on membrane potential is anyway fairly small (Fraser et al. 2005), I take minimum zX=−1 to avoid negative [Y]. I follow Usher-Smith et al. (2006) in assigning Y the same charge as X at rest. Note that cell volume could be imagined as an orthogonal third axis, along which area would represent total cell content, invariant for impermeant species. In the simple case above (eqn (2)), where μ= 1 and volume change is ignored, Δ[A]≈[A]=[X]ΔzX, independent of [X]: thus for given [A], the smaller [X] (and therefore βX) is made, the larger ΔzX (and therefore −ΔpH) becomes. In the full analysis λ is clearly independent of [X]0 if μ is constant: first because the two charge-change terms [X]0ΔzX and [X]ΔzX in eqs (6) bracket ∫[X]dzX=μ[A], which is independent of [X]0; and also because [X]0 does not occur in eqn (10). To account for the βX dependence of volume in Fig. 9B, μ must therefore be variable, depending somehow on [X]0. We must therefore consider what influences μ. For μ= 1 exactly (i.e. to make the buffered fraction unity, as in the simple analysis which ignores swelling and posits no transmembrane fluxes –eqn (2)), we actually need a small net cation accumulation (influx) of: For constant [C] despite the diluting effects of cell swelling (a simplification we might prefer to make), we would need cation accumulation of: This raises different issues. Efflux of equimolar H++ lactate ion, or of undissociated lactic acid, makes no difference to the relationships analysed above. Efflux of H+ without lactate is dealt with in (iv). Efflux as an anion of a fraction ω of lactate, without H+, does not define a value of μ, but multiplies by (1 +ω) whatever value is set by cation fluxes. With this background we can identify some processes contributing to the simulated steady-state buffer dependence of cell volume in Fig. 9B. (a) Changes in μ In Fig. 9B, λ decreases with increasing βX while βA remains constant (∼20 slykes = 17 mm lactate divided by 0.8 pH fall). The fall in λ suggests an increase in μ (eqn (10)), roughly as follows: μ= 0 at ‘zero’βX, where cation influx/proton efflux must balance lactate accumulation: see section (iv) above and f in Fig. 1. μ= 1 at around half-normal βX, where net cation flux must be negligible: see section (ii) and b in Fig. 1. μ≈ 2 at normal βX, where volume is approximately constant because cation efflux balances lactate accumulation (Usher-Smith et al. 2006): see section (iii) and d in Fig. 1. μ≈ 4 at twice-normal βX, where volume falls due, e.g. to high cation efflux and/or low Na+,H+ exchange (so ρ−υ > 1): see section (iii) and e in Fig. 1. High lactate anion efflux (ω > 0) would help. This increase in μ explains the constancy of βA (eqn (1)): implied βX= 20 × 2 ≈ 40 slykes in the normal case, and (consistent with this) 20 × 4 ≈ 80 slykes at twice normal βX. (b) Changes in buffered proton load The argument can be presented another way. The buffered protons (∼[X]ΔzX) can be obtained from the charge difference equation in (Usher-Smith et al. 2006) or my version (eqn (6)), or indirectly as ∼μ[A]. In Fig. 9B, at normal βX, where volume is approximately constant, buffered protons = 2 × 17 ≈ 34 mm (Usher-Smith et al. 2006), and at twice normal βX, buffered protons = 4 × 17 ≈ 68 mm. Buffer capacity = (buffered protons)/(pH fall), and pH fall is ∼0.8 throughout, so implied normal βX= 34/0.8 ≈ 42 slykes in both cases. Consistent with this, parameters in Usher-Smith et al. (2006) are taken from Fraser et al. (2005) which bases a normal βX of 44 slykes on cardiac myocyte data (Leem et al. 1999); in that paper, though, basal β≈ 25 slykes (Leem et al. 1999), and furthermore the fraction of charge carried at rest by the more acid buffer component is, I calculate, ∼80–90% (depending on assumptions about zX), not 3/10 as assumed in Fraser et al. (2005). (c) Changes in buffer anion charge There is a puzzle about the changes in zX (ΔzX= 0.32 at normal βX, 0.64 at twice normal βX (Usher-Smith et al. 2006)). From the amount of buffered protons (see section (b) above) at normal βX, the implied [X]≈ 34/0.32 ≈ 110 mm, implying normal βX≈ 25 slykes using the pK values of Leem et al. (1999); at twice normal βX, implied [X]= 4 × 17/0.64 ≈ 110 mm, so normal [X]≈ 110/2 ≈ 55 mm, implying normal βX≈ 12 slykes. Apart from a discrepancy in normal βX between this and sections (a) and (b) above, the problem is that if −ΔpH does not change with βX then, as −∫βXdpH =∫[X]dzX, neither should ΔzX. The reason for this discrepancy is not clear. It cannot be because Fig. 9B refers to dynamic steady-state with persisting transmembrane fluxes, as the present analysis depends only on electroneutrality constraints, which apply almost exactly moment-to-moment, and osmotic constraints, which apply whenever cell volume achieves a stable value. (d) Direction of causation Because complex changes in ion handling underlie the apparently simple buffer dependence of simulated cell swelling in Fig. 9B, I argue that statements such as ‘… intracellular accumulation of the lactate … does not significantly influence steady-state cell volume because the accompanying increase in intracellular H+ content produces a large decrease in the magnitude of the mean charge valency (zX) of membrane-impermeant intracellular anions (X), provoking an equivalent net cation efflux’ (p. 811 in Usher-Smith et al. (2006)) miss the point that μ is not constrained if cations adjust freely, and that causality must therefore be the other way round: cation handling determines μ. Volume decreases with buffer capacity in Fig. 9B because μ increases, which I argue must be the result of increasing net cation efflux, although the mechanism is not obvious (since, for example, pH change is unaffected). Thus Fig. 9B is at least somewhat counterintuitive, arising in an indirect way which means that the conclusion that ‘… an intracellular buffering capacity that was significantly greater than known values would produce a cell that would actually shrink in response to the application of extracellular lactate or intracellular lactic acid production’ (p. 814 in Usher-Smith et al. (2006)) might be quite sensitive to detailed relationships between kinetic parameters not so far analysed in these terms.