LETTER

Circuit Models of Muscle Metabolism

The circuit model for phosphocreatine (PCr) resynthesis in skeletal muscle after exercise, which was proposed by Nevill et al. (4), was said to be "modified from that proposed by Meyer" (1) but "fits the PCr data better" after heavy exercise. Unfortunately, we can find no conceptual or mathematical relationship between the "modified" model of Nevill et al. (4) and the original model proposed by Meyer (1). Comparison of the two models shows that the original model is a direct analog of the physiology of ATP turnover and PCr metabolism in muscle, but the modified model bears little relationship to these processes.

In the original circuit model for PCr metabolism proposed by Meyer (Fig. 1A), all of the above rules of thumb for modeling are satisfied. First, each circuit element corresponds to a specific, measurable physiological parameter. The cytoplasmic adenosinetriphosphatase (ATPase) is represented by a controlled current element, which conceptually accounts for the fact that the cytoplasmic ATPase rate is switched in a controlled but variable fashion by a calcium signal during exercise. The battery represents the intramitochondrial potential for ATP synthesis, which can be measured from the steady-state relationships between muscle oxygen consumption and cytoplasmic phosphate metabolites under various conditions (5). The resistor (R) depends directly on the maximum oxidative capacity of the muscle, and the capacitance (C) bears a clear and precise relationship to the total creatine (TCr) content in the muscle (C = TCr/6RT; see Ref. 1). Second, the layout of elements matches the conceptual layout of the metabolic system being modeled. Thus the capacitor is placed in parallel with the mitochondrial elements, because during transitions between rest and exercise cytoplasmic ATP is supplied both by net PCr hydrolysis (the capacitive current) and by oxidative phosphorylation (the resistive current). In the steady state, all the ATP is supplied by the mitochondria, and there is no further change in PCr. Third, in the original publication (1), it was clearly stated that the model could only be applied during and after submaximal exercise, when several specific boundary conditions are likely to be valid (e.g., no substantial anaerobic ATP production or acidosis, constant intramitochondrial potential, and linear relationship between PCr and the free energy of ATP hydrolysis). Finally, the model correctly predicted that the time constant for PCr recovery after submaximal exercise is independent of workload (1) and depends linearly on both muscle mitochondrial content and creatine content (2, 5).

In contrast, the model of Nevill et al. (Fig. 1B) satisfies none of the above rules of thumb. First, the major new element in the model is an inductor (L). Unfortunately, no physiological or metabolic analog is proposed for this inductor. In electric circuits, an inductor is an element for which voltage depends on the rate of change in current (v = L*di/dt). Generally, the concept is used to model elements of a system that resist any rapid change in flux but have no effect in the steady state. In mechanical systems, the analogous concept is momentum, i.e., the tendency for something to keep going once set in motion. Although we do not exclude the possibility, we do have difficulty imagining the analog of this concept in the context of muscle oxidative metabolism. Second, the layout of the Nevill model bears no relationship to the metabolic system. The capacitor, which supposedly still represents the creatine kinase system, is in series with the battery and resistor, not in parallel. As a consequence, the steady-state current in this circuit must be zero and, therefore, unlike the original model, this circuit cannot model the change in PCr at the onset of exercise and leads to the obviously false conclusion that there is no cytoplasmic ATP turnover after exercise. In fact, in this series arrangement, the precise, fixed dependence of the capacitance on total creatine proposed in the original model disappears, and the capacitor becomes a completely arbitrary modeling parameter, just as for the inductor. This is illustrated by the fact that Nevill et al. (4) do not relate the quantitative results of their curve fitting back to any of the circuit's elements (R, C, or L) but, instead, report only the fitted kinetic rate constants. Similarly, no boundary conditions are imposed on the linearity of these elements, because they no longer represent specific, measurable metabolic properties of the muscle. Finally, the predictive power of the Nevill model is questionable. For example, a notable feature of inductive-capacitive circuits is that they can be driven into oscillation at some characteristic frequency. Therefore, this model seems to predict that a cyclic exercise/rest protocol can be devised that will drive PCr and muscle ATP turnover into oscillation so that the rate of ATP turnover during contraction, and the rate of PCr resynthesis during recovery, are each much higher than observed at lower cycle frequencies. There is no experimental support for this predicted behavior.

On the above grounds, we believe that, in contrast to the original model of Meyer (1), the model of Nevill et al. (4) has little relevance to the study of muscle metabolism. We agree that the arbitrary second-order differential equations used by Nevill et al. can fit some features of PCr recovery after intense exercise that the original model was never intended to address, i.e., the PCr overshoot sometimes observed and the multiexponential nature of PCr recovery in muscles with mixed fiber type. However, these phenomena can be considered without adding ad hoc circuit elements or rearranging the elements of the original model. For example, the PCr overshoot could result from increased intramitochondrial potential for ATP synthesis, due, e.g., to accumulation of NADH or acidosis (5) during intense exercise. This can be modeled by a variable-voltage source in place of the battery. The multiexponential nature of PCr recovery in human muscle can be explained by fiber type heterogeneity and, in fact, is predicted by the original model for muscles with mixed fiber type (3).