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Cell pH and volume changes

Although this is sometimes disputed (see for the recent debate Robergs et al. 2004, 2005; Boning et al. 2005; Kemp, 2005; Kemp et al. 2006), glycolytic ATP synthesis matching ATP usage is, in acid–base terms, equivalent to cellular accumulation of lactic acid, or rather H⁺ + lactate anion (Kemp et al. 2001; Usher-Smith et al. 2006) (for simplicity I will call lactate A, and will generally not show charge as superscripts). In what could be called the traditional approach to pH (Kemp et al. 2006), the buffer capacity of anion X is _X = –[X]dz_X/dpH (Roos & Boron, 1981), where z_X is molar charge and [X] is total concentration. I define in vivo apparent buffer capacity as _A = –d[A]/dpH (Kemp et al. 2001), and µ = [X]dz_X/d[A], the fraction of ‘glycolytic’ H⁺ (i.e. the H⁺ accompanying lactate) which is buffered. Then:

(1)

In the simplest case, without volume change or transmembrane fluxes, µ = 1. Thus for an infinitesimal transition, electroneutrality requires:

(2)

from which, neglecting the very small [H⁺] and [OH^–] terms:

(3)

The relationship to the alternative ‘physicochemical’ approach to acid–base physiology (Lindinger et al. 2005) is that here the change in strong ion difference d[SID] = –d[A] (Kemp et al. 2006).

There are two complications in the system modelled by (Usher-Smith et al. 2006). First, volume change: to see why this is relevant, imagine that, in a cell containing only titratable anion X, as lactate accumulates cell volume is increased, e.g. by imposed decrease in external osmotic pressure (). Then the equivalent of eqn (2) is

(4)

where is the ratio of the new volume (V: for simplicity I omit the subscript from new steady-state values) to the basal volume (V₀). Thus the decrease in the charge contribution of X, which balances the increased contribution of lactate ion, is due to both titration and dilution (so _A > _X). The second complication is change in what the physicochemical approach calls strong cations (e.g. Na⁺, K⁺), and treats on the same footing as strong anions like lactate (Lindinger et al. 2005). Volume and cation changes are connected (Usher-Smith et al. 2006), and I show here how they interact in influencing cellular pH change.

In the charge-difference model which underpins (Usher-Smith et al. 2006), steady-state cell volume is constrained by osmotic balance and electroneutrality (Fraser & Huang, 2004; Fraser et al. 2005): in general, for osmotically active cytosolic constituents M:

(5)

(6)

(the latter actually only approximate, because the small charge imbalance gives rise to the membrane potential), from which:

(7)

After any perturbation the steady-state volume ratio = [S]₀/[S] for each impermeant species S, here taken to comprise titratable (X) and non-titratable (Y) anions, neutral species (N) and sometimes (see section (ii) below) monovalent cations (C) (Fig. 1). So from eqn (4):

(8)

where P are permeant fixed-charge anions, here comprising lactate (fixed by metabolism or by modeller's fiat) and chloride (in equilibrium with membrane potential, which we can take here as constant (Usher-Smith et al. 2006)). Equation (5) is a generalized version of the equation on p. 810 of Usher-Smith et al. (2006), and assuming zero resting lactate ([A]₀ = 0) for simplicity, can be rewritten in two ways which show the relationship of volume change to z_X, the overall charge change due to titration:

(9)

(10)

Monovalent cations, permeant or not, cancel out of eqn (5), but osmotic balance requires:

(11)

of which a component –[C](dln/d[A]) is dilutional, the remainder, say , being due to net cation influx (which may be positive, negative or zero):

(12)

In this model the electroneutrality condition is:

(13)

as [Cl^–] is taken here as constant. Like eqn (3), the right-hand side of eqn (9) contains both titration and dilution terms. Either by using eqn (8) to substitute for dln in eqn (9), or by directly differentiating eqn (6a), the lactate sensitivity of volume is given by:

(14)

from which (using eqn (1)) its pH sensitivity is:

(15)

View larger version (19K): [in this window] [in a new window]   Figure 1.  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 ( = 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 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.

(16)

Also, for the charge-change terms in eqns (6), [X]₀z_X > [X]dz_X > [X]z_X if volume rises, in reverse order if volume falls, and equal if volume is constant (and anyway close if volume change is small). Furthermore, if µ is constant (see below), by integrating eqn (10) we have the following expression for volume as a function of [A] only:

(17)

and from the definitions:

(18)

Buffering and cation changes

The buffer dependence of cell volume (Fig. 9B) is tested, in present terms, by altering [X]₀ and adjusting [Y]₀ and [C]₀ for basal electroneutrality and osmotic balance:

(19)

(20)

(Actually it is not clear how buffer capacity is increased in Fig. 9B by changing the titratable proportion of X, as stated in Usher-Smith et al. (2006), because this is taken as normally 1 (Fraser et al. 2005).)

In the simple case above (eqn (2)), where µ = 1 and volume change is ignored, [A] [A] = [X]z_X, independent of [X]: thus for given [A], the smaller [X] (and therefore _X) is made, the larger z_X (and therefore –pH) becomes.

In the full analysis is clearly independent of [X]₀ if µ is constant: first because the two charge-change terms [X]₀z_X and [X]z_X in eqs (6) bracket [X]dz_X = µ[A], which is independent of [X]₀; and also because [X]₀ does not occur in eqn (10). To account for the _X dependence of volume in Fig. 9B, µ must therefore be variable, depending somehow on [X]₀. We must therefore consider what influences µ.

If cations adjust freely, µ is unconstrained by eqns (8) and (10): for any µ there is a which satisfies the electroneutrality and osmotic balance conditions, and vice versa. Formally, then, µ can be either specified by fiat or ‘set’ by cation flux such that

(21)

(and modified by lactate anion efflux: see section (v) below). To see this consider some hypothetical cases, analysed using eqns (1), (10), (11) and (12).

(i) No buffering

If [X]₀ = 0, or equivalently µ = 0, then = 1/{1 – 2[A]/( – 2[Cl])} 1 +2([A]/)and so dln/d[A] 2/. For electroneutrality this would require:

(22)

i.e. approximately equimolar cation influx, whence the factor 2 in dln/d[A]. This case is equivalent to the intercept in Fig. 9B (but see section (iv) below). An important conclusion from the charge-difference model is that titration of cell buffers (making z_X less negative) decreases the osmotic effect of lactate by reducing the need for charge-balancing cation accumulation (Fraser et al. 2005; Usher-Smith et al. 2006). However a direct comparison with the hypothetical no-buffering case cannot be made unless µ is defined. We must therefore examine the effects of cation flux changes.

(ii) No cation fluxes

Suppose first there are no net cation fluxes (Fig. 1, b and c), i.e. = 0 : then

(23)

(24)

Comparison with section (i) shows that titratable buffer halves the osmotic effect of lactate when charge-balancing cation accumulation is made zero by hypothesis (which is not quite the point made in Fraser et al. (2005) and Usher-Smith et al. (2006)). This effect is not a smooth function of [X] (as it appears in Fig. 9B); it depends only on whether [X] or µ is zero. Note these two closely related cases:

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:

(25)

representing a contribution to [SID] which cancels out the charge-balancing effect of dilution. and a flux to balance the small Cl influx, necessary when the cell swells (if membrane potential does not change).

For constant [C] despite the diluting effects of cell swelling (a simplification we might prefer to make), we would need cation accumulation of:

(26)

(iii) Cation efflux: osmotically balancing lactate

Suppose next that cation efflux matched lactate accumulation, i.e. = –1: then µ = 2, and this complete ‘osmotic compensation’ keeps constant, but halves _A, i.e. doubles the pH change for given lactate (Fig. 1, d). In this sense pH control and volume control are trade-offs. If such osmotic compensation is incomplete (say = – where is a parameter, unity for complete compensation) then:

(27)

Negative dln/d[A], i.e. cell contraction, as predicted at high (Usher-Smith et al. 2006), would require µ > 2 and < –1 ( > 1), i.e. more cation loss than lactate accumulation (Fig. 1, e).

(iv) Cation influx: charge-balancing lactate

Suppose instead that cation influx matches lactate accumulation, i.e. = 1; then µ = 0, and this complete ‘charge compensation’ holds pH constant, but increases volume change, dln/d[A] 2/ (Fig. 1, f). This is just as if X were non-titratable or absent (see section (i) above), i.e. ‘a cell in which pH_i has no influence on cell volume due to a relative buffering capacity of zero’ (Usher-Smith et al. 2006), but the direction of causation in my example (viz µ is constrained by cation influx, to use the terms of the physicochemical model, or by the efflux of H⁺ in traditional terms) explains the otherwise puzzling feature of Fig. 9B, that a cell with no buffering can have finite, indeed normal, pH change. More physiologically, if Na⁺/H⁺ exchange extruded only a fraction of ‘glycolytic protons’, i.e. = then:

(28)

only the residual protons titrate buffers, so apparent buffering is increased, but volume control is impaired, as:

(29)

(v) Lactate and H⁺-efflux

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 of lactate, without H⁺, does not define a value of µ, but multiplies by (1 + ) whatever value is set by cation fluxes.

(vi) A combination of processes

If cation efflux (see section (iii) above), cation influx (see section (iv)) and lactate efflux (see section (v)) coexist, then volume change and buffering are governed by:

(30)

(31)

The overall effects thus depend on the parameters , and , and how they change with, e.g. pH, lactate, cation content and time. From the physicochemical point of view, focusing on electroneutrality, the system works as follows. The negative charge of lactate can be balanced by buffering, dilution and cation influx (eqn (9)). The rise in lactate contributes a fraction, say = 1/{µ + (2 – µ)([C]/)}, to the fall in SID which is responsible for acidification; this fraction is half if osmotic compensation is complete (see section (iii) above), and 1/{1 + ([C]/)} in the absence of cation fluxes (see section (ii)). The remainder of the change in SID is due to cation concentration changes, which are the result of cell volume change and of cation fluxes, if any (cation efflux tends to make SID more negative, and pH change consequently larger). The fraction of the SID change which is charge-balanced by titration of buffers is µ; this is 1 if volume is constant (see section (iii)), obviously, and it is 1/{1 + ([C]/)} in the absence of cation fluxes (see section (ii)). The remainder of the SID change is charge-balanced by dilution. Without cation fluxes (see section (ii)), an increase in _X decreases z_X and –pH proportionately, with no change in (Fig. 1, c). Otherwise, the response depends on what happens to cation fluxes: cation influx restrains pH change, but exacerbates swelling (see section (iv)) (Usher-Smith et al. 2006); cation efflux restrains swelling but worsen pH control (see section (iii)).

Interpretation of Fig. 9B in Usher-Smith et al. (2006)

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, decreases with increasing _X while _A remains constant (20 slykes = 17 mM lactate divided by 0.8 pH fall). The fall in suggests an increase in µ (eqn (10)), roughly as follows:

µ = 0 at ‘zero’ _X, where cation influx/proton efflux must balance lactate accumulation: see section (iv) above and f in Fig. 1.

µ = 1 at around half-normal _X, where net cation flux must be negligible: see section (ii) and b in Fig. 1.

µ 2 at normal _X, 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 _X, where volume falls due, e.g. to high cation efflux and/or low Na⁺,H⁺ exchange (so – > 1): see section (iii) and e in Fig. 1. High lactate anion efflux ( > 0) would help.

This increase in µ explains the constancy of _A (eqn (1)): implied _X = 20 x 2 40 slykes in the normal case, and (consistent with this) 20 x 4 80 slykes at twice normal _X.

(b) Changes in buffered proton load.  The argument can be presented another way. The buffered protons ([X]z_X) 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 _X, where volume is approximately constant, buffered protons = 2 x 17 34 mM (Usher-Smith et al. 2006), and at twice normal _X, buffered protons = 4 x 17 68 mM. Buffer capacity = (buffered protons)/(pH fall), and pH fall is 0.8 throughout, so implied normal _X = 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 _X of 44 slykes on cardiac myocyte data (Leem et al. 1999); in that paper, though, basal 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 z_X), not 3/10 as assumed in Fraser et al. (2005).

(c) Changes in buffer anion charge.  There is a puzzle about the changes in z_X (z_X = 0.32 at normal _X, 0.64 at twice normal _X (Usher-Smith et al. 2006)). From the amount of buffered protons (see section (b) above) at normal _X, the implied [X] 34/0.32 110 mM, implying normal _X 25 slykes using the pK values of Leem et al. (1999); at twice normal _X, implied [X] = 4 x 17/0.64 110 mM, so normal [X] 110/2 55 mM, implying normal _X 12 slykes. Apart from a discrepancy in normal _X between this and sections (a) and (b) above, the problem is that if –pH does not change with _X then, as –_XdpH = [X]dz_X, neither should z_X. 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 (z_X) 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.

The Lohmann ‘reaction’

Lastly, missing from this model is a notable feature of exercising skeletal muscle, the PCr ‘splitting’ which, via the creatine kinase equilibrium, buffers ATP against temporary mismatch of production and use, and which tends to oppose acidification (Kemp et al. 2001). There is debate about whether this is appropriately called the Lohmann reaction (Robergs et al. 2004, 2005; Kemp, 2005; Kemp et al. 2006). In charge-balance terms this ‘replaces’ PCr by less-negative orthophosphate (P_i), contributing (along with titration of buffers) to the decrease in the negative charge on membrane-impermeant anions which balances the negative charge of lactate (Kemp et al. 2006). It has an additional osmotic effect because of the increase in creatine (Cr) (Usher-Smith et al. 2006). Taking account of both, it can be shown that eqn (6a) becomes:

(32)

where [T] = [PCr] + [Cr], here taken as [PCr] + [P_i], = [Cr]/[T] and z_T = z_PCr + (1 – )z_Pi, the average charge of P_i + PCr (which is related to the ‘Lohmann coefficient’ = z_PCr – z_Pi, which represents the H⁺ generated per mole of PCr broken down (Kushmerick, 1997; Kemp et al. 2001)). Eqn (10) becomes:

(33)

where d/d[A] describes covariation of lactate and PCr (Kemp et al. 2001), and eqn (12) becomes:

(34)

Neatly, for zero cation flux (see section (ii)) _{A Pi} + _X and:

(35)

thus (compared to eqn (10) with µ 1) PCr breakdown increases glycolytic swelling by a positive fraction (1 + )(–d[PCr]/d[A]), representing the osmotic effect of Cr accumulation less (because is negative) the effect of ‘proton consumption’ by the Lohmann ‘reaction’ (Kemp, 2005; Kemp et al. 2006), which will ‘spare’ cation accumulation in the same way as titration (Usher-Smith et al. 2006). Note that for PCr breakdown without lactate accumulation, cell swelling would be purely osmotic, –dln/d[PCr] 1/, requiring –d[C]/d[PCr] [Cl]/, a small cation influx to balance Cl^–influx.

Graham Kemp¹

¹ Division of Metabolic & Cellular MedicineFaculty of Medicine, University of LiverpoolLiverpool L69 3GA, UKEmail: gkemp{at}liv.ac.uk

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This Article Full Text (PDF) All Versions of this Article: 582/1/461    most recent jphysiol.2007.134643v1 Services Email this article to a friend Similar articles in this journal Similar articles in PubMed Alert me to new issues of the journal Download to citation manager Citing Articles Citing Articles via Google Scholar Google Scholar Articles by Kemp, G. Search for Related Content PubMed PubMed Citation Articles by Kemp, G. Related Collections Letters