Physiological implications of linear kinetics of mitochondrial respiration in vitro.

TO THE EDITOR: Glancy et al. (4) measure the rate of oxygen consumption (Jo) by mitochondria incubated with an ATPconsuming 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/ )([ADP]/[Cr]), where 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 M in resting human muscle (15)] by feedback mechanisms described below, and the high equilibrium constant (4) of CK permits , 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 ( Jo, where P:O2 ratio) responds to a step increase from basal (resting) in ATP demand (say JD), the kinetics of [PCr] and the increase in suprabasal JP ( JP) are given by JP d[PCr]/dt JD, and the (negative) change in [PCr] from rest is the time-integrated mismatch between ATP supply and use, [PCr] ( JD JP)dt (12, 14). The mathematical implications are that if JP and [PCr] follow exponential kinetics (4) (with rate constant k, say, and time constant 1/k), the relationship between JP and [PCr] must be linear, JP k [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 GATP (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 GATP (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 (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 M (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 M, 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 GATP (J/mmol) is analogous to voltage, their product being power (kJ s 1 l ); changes in [PCr] (mmol/l) are analogous to charge. Capacitance is defined as C d[PCr]/ d GATP (mmol J 1 l ), and mitochondrial resistance is defined as Rm d GATP/dJP (J l 1 s 1 mmol ), and if both are constant, exponential kinetics follow with RmC (23). The CK equilibrium indeed constrains C to be approximately constant (23): over a midrange ( 0.2 0.7), C [TCr]/ (6RT) where R is the gas constant and T is temperature (23). Constancy of Rm is a midrange linear approximation to a sigmoid JPGATP (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 mid1 At this point, Glancy et al. also cite the near-linearity of [PCr] with GATP (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. Address for reprint requests and other correspondence: G. Kemp, School of Clinical Sciences, University of Liverpool, Liverpool L69 3GA, United Kingdom (e-mail: [email protected]). 2 2 Glancy et al. (4) calculate C both this way and as C /Rm (dJo/d GATP), and they take the agreement as confirming their assumed (4); this is equivalent to estimating (d[PCr]/dJo)/ , as Meyer did (23). Am J Physiol Cell Physiol 295: C844–C846, 2008; doi:10.1152/ajpcell.00264.2008.