to the editor: Glancy et al. (4) measure the rate of oxygen consumption (Jo) by mitochondria incubated with an ATP-consuming system in the presence of creatine kinase (CK), and they show that this in vitro model of aerobically exercising skeletal muscle conforms to Meyer's electrical analog of muscle oxidative phosphorylation in vivo (23). I want to suggest that the interpretation of these experiments is enhanced by distinguishing the specific features of this model from general properties of feedback control (see 1–8 below), and making explicit its relationship (see Fig. 1) to alternative models (2, 7, 10, 16, 20, 27–30).
All approaches start from the temporal buffering of ATP (3) by CK. Since CK is near-equilibrium, d[ATP]/d[PCr] ≈ (1/θ)([ADP]/[Cr]), where θ is [PCr]/[TCr], the phosphorylated fraction of the total creatine pool (TCr = PCr + Cr) (12, 14). Steady-state [ADP] is kept very low [i.e., ∼15 μM in resting human muscle (15)] by feedback mechanisms described below, and the high equilibrium constant1 (4) of CK permits θ, nevertheless, to be near 1 [∼0.8 at rest (15)]; so d[ATP]/d[PCr] ≈ 0, and [ATP] is buffered at the expense of [PCr] (3). Thus d[Pi]/d[PCr] ≈ −1, and since it happens (12) that [Pi] ≈ [Cr] at rest, this remains true during exercise (2). Thus when oxidative ATP synthesis JP (= ρJo, where ρ = P:O2 ratio) responds to a step increase from basal (resting) in ATP demand (say ΔJD), the kinetics of [PCr] and the increase in suprabasal JP (ΔJP) are given by ΔJP − d[PCr]/dt = ΔJD, and the (negative) change in [PCr] from rest is the time-integrated mismatch between ATP supply and use, −Δ[PCr] = ∫ (ΔJD − ΔJP)dt (12, 14). The mathematical implications are that if ΔJP and [PCr] follow exponential kinetics (4) (with rate constant k, say, and time constant τ = 1/k), the relationship between JP and [PCr] must be linear, ΔJP = − kΔ[PCr]; and, conversely, from such linearity (thick line, Fig. 1A), observed or postulated, exponential kinetics follow (12, 14, 20) (thick line, Fig. 1D). The metabolic control implication is that any causal relationship between JP and [PCr] or its near-correlates, which include [ADP] (2), [Cr], [Cr]/[PCr], [Pi] (27, 28), and ΔGATP (7, 23) (i.e., any to which, in the terminology of metabolic control analysis, JP shows appreciable elasticity), could serve as a negative feedback signal matching ATP supply to demand (2); in engineering terms, this is a form of integral feedback, which precludes steady-state error (12). The link between these implications is that exponential kinetics (and the required linearity of JP and [PCr]) can emerge from feedback mechanisms involving [ADP] (2) or ΔGATP (7, 23, 29) as the key signal, as well as more complicated models (30).
For argument's sake, assume the relationship of flux to [ADP] (Fig. 1B) is causally prior: something like the hyperbolic (Michaelis-Menten) JP-[ADP] curve seen with incubated mitochondria [no doubt also in Glancy et al. (4)] can be observed in exercising muscle in vivo (2, 16) (Fig. 1B), whether this reflects mainly the ADP dependence of the adenine nucleotide translocase (10) or summarizes several such interactions of correlated metabolites (30). If JP is some function f([ADP]), then for given ATP demand, steady-state [ADP] = f−1(JD). Given an appropriate f [for example, a hyperbolic function with micromolar [ADP]1/2 (2, 16)] this will, for submaximal JP, keep steady-state [ADP] very low, as ATP-buffering requires (see above). Furthermore, in the case where [ADP]1/2 ≈ 50 μM (see Fig. 1), this hyperbolic JP-[ADP] relationship implies (at constant pH) a linear JP-[PCr] (12, 14) and therefore monoexponential kinetics (thick lines, Fig. 1, A–D), as seen in aerobic exercise (23) as well as in vitro in Glancy et al. (4). If [ADP]1/2 is higher (lower) than 50 μM, JP-[PCr] will be concave upward (downward), with probably relatively little effect on kinetics (thin lines in Fig. 1, A–D). Making JP-[ADP] sigmoid (dashed line in Fig. 1B) makes JP-[PCr] sigmoid also (dashed line, Fig. 1A) and increases the midpoint slope (Fig. 1), yielding faster but less strictly monoexponential kinetics (dashed line, Fig. 1D): the relevance is, first, that modest sigmoidicity (Hill coefficient n ≈ 2) (10) may explain the dynamic range in muscle in vivo without recourse to feed-forward (parallel activation) mechanisms (18), and second, that a small degree of sigmoidicity (1 < n < 2) could (11, 13) result from effects of [PCr]/[Cr] on [ADP]1/2 demonstrated in vitro (28).
Meyer's electrical analog (23) is a linear model in which JP (in vivo, mmol·l−1·s−1 or equivalent units) is analogous to current and ΔGATP (J/mmol) is analogous to voltage, their product being power (kJ·s−1·l−1); changes in [PCr] (mmol/l) are analogous to charge. Capacitance is defined as C = d[PCr]/dΔGATP (mmol2·J−1·l−1), and mitochondrial resistance is defined as Rm = dΔGATP/dJP (J·l−1·s−1·mmol−2), and if both are constant, exponential kinetics follow with τ = RmC (23). The CK equilibrium indeed constrains C to be approximately constant (23): over a midrange (θ ≈ 0.2 − 0.7), C ≈ [TCr]/(6RT) where R is the gas constant and T is temperature2 (23). Constancy of Rm is a midrange linear approximation to a sigmoid JP-ΔGATP (5, 24) (inset, Fig. 1C). This could be seen as an epiphenomenon of a causal [ADP] dependence (Fig. 1B): for hyperbolic JP-[ADP] (n = 1; solid line, Fig. 1C), Rm ≈ 6RT/JP,MAX (12, 14); more generally, 1/Rm, the actual midpoint slope of JP-ΔGATP, increases with n (Fig. 1), which accelerates the kinetics (dashed line, Fig. 1D). Alternative approaches using nonequilibrium thermodynamic formalism (7) or an empirical kinetic fit (9, 10) also describe a sigmoid JP − ΔGATP approximated by the linear relationship on which the electrical model is based (inset, Fig. 1C). Figure 1 summarizes how the JP-ΔGATP fit parameters relate to the (hypothetically) causal parameters of JP-[ADP], but if JP-ΔGATP were taken as causally prior, the relationships to [PCr] and [ADP] (Fig. 1, A and B) would be epiphenomena: the literature contains both perspectives, which cannot be distinguished at present.
Against this background, we can see that some of the results of Glancy et al. (4) are general properties of all CK-mediated feedback-controlled supply-demand mechanisms:
1) Fixed mitochondrial capacity. That measures of mitochondrial capacity (like 1/Rm) are unaffected by [TCr] but proportional to mitochondrial content (4) [as in vivo (24)] follows from, and justifies, the conceptual separation of the mitochondrion from the cytosolic CK system. The role of mitochondrial CK in this is heavily debated (14) but is probably small (11, 13).
2) Demand-drive. The low elasticity of the ATP-consumer toward CK-related signals [as in vivo, where ATP turnover is largely demand-driven (8)] explains why Jo is unaffected by changes in JP,MAX and [TCr] (4).
3) Changes in “feedback signals.” Increasing demand ΔJD tends to increase the (let us assume stimulatory) signal X. Furthermore, increasing mitochondrial capacity JP,MAX increases open-loop gain (12), the absolute sensitivity of output to the error signal dJP/dX, which decreases the ΔX for given ΔJD. Whatever X really is, this explains why increasing Jo increases, and increasing JP,MAX decreases (4), changes in CK-correlated metabolites (3).
4) Constancy of “feedback signals.” Attempts to perturb a feedback signal X must fail, at steady state, unless demand is also changed. In principle this could distinguish the causal role of an X from near-correlates: for example, that acute respiratory acidosis in vivo increases [ADP] but leaves ΔGATP unchanged (25) is evidence for Meyer's model and against ADP-control [although the effects of acidosis are complicated (5)]. Meyer's model also explains why ΔGATP, is, for a given Jo, unaffected by changing [TCr] (4) [also in vivo (1)], although a richer data set, covering a range of pH allowing dissociation between, e.g., [Cr] and [Pi] (3), would be needed to exclude independent effects of its near-correlates [and developments in detailed mitochondrial modeling (30) perhaps make this whole approach too simple].
Other findings (4) are particular, although not exclusive, properties of the subset of “linear” models:
5) Linear relationships. That some metabolite changes (ΔGATP and [PCr]) are linear with Jo (4) [also in vivo (24)] is likely a midrange approximation (22), compatible (algebraically and causally) with some nonlinear JP-[ADP] relationships (Fig. 1B).
7) Kinetics and [TCr]. The time constant τ is proportional to [TCr] in vitro (4) [also in vivo (22)] because, in Meyer's model, increasing [TCr] increases C, increasing the charge (Δ[PCr]) required to reach the required voltage (ΔGATP) (22); in ADP-control models, equivalently, it takes a bigger Δ[PCr] to reach the required [ADP].
8) Kinetics and mitochondrial capacity. The validity of 1/Rm and (given the approximate constancy of C) of k as measures of mitochondrial capacity, i.e., proportional to JP,MAX (4) [as also in vivo (24)]3 , rests on the validity of the linear-model assumptions in the absence of pH change (5). That τ is similar in vivo and in vitro when the ratio of mitochondrial volume to [TCr] is matched (4) makes sense in Meyer's model (4) and also in typical ADP-control models, where τ ≈ [TCr]/JP,MAX (Fig. 1).
In summary, that in a simple system “the robust energetic forces and flows maintained by the isolated mitochondria are similar to those observed in vivo” does indeed “reveal important features of mitochondrial function” (4), notably, that it is likely a demand-driven CK-mediated feedback mechanism not obviously in need of parallel activation mechanisms (18) to explain responses typical of aerobic exercise, but the data also make sense in several other perspectives than Meyer's elegant electrical analog model (23).
↵1 At this point, Glancy et al. also cite the near-linearity of [PCr] with ΔGATP (23), but this is a consequence, not a cause, of ATP buffering; also “the kinetics of PCr breakdown and Jo rise are virtually identical at the onset of exercise” (4) because of ATP buffering, not “because of a relatively constant phosphorylation ratio” (4), which is not obtained.
↵2 2 Glancy et al. (4) calculate C both this way and as C = τ/Rm = τρ(dJo/dΔGATP), and they take the agreement as confirming their assumed ρ (4); this is equivalent to estimating ρ = (d[PCr]/dJo)/τ, as Meyer did (23).
↵3 How well estimates of actual JP,MAX (9, 17) perform is not established, although they largely avoid (19) the “artifactual” pH dependence of k (6). If causal primacy is conceded to ADP feedback (Fig 1B), 1/Rm increases with n, responding to the shape as well as the maximum of JP-[ADP], appropriately reflecting faster kinetics (Fig 1D).
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