## Abstract

The following is the abstract of the article discussed in the subsequent letter:

**Nevill, Alan M., David A. Jones, David McIntyre, Gregory C. Bogdanis, and Mary E. Nevill.** A model for phosphocreatine resynthesis. *J. Appl. Physiol.* 82(1): 329–335, 1997.—A model for phosphocreatine (PCr) resynthesis is proposed based on a simple electric circuit, where the PCr store in muscle is likened to the stored charge on the capacitor. The solution to the second-order differential equation that describes the potential around the circuit suggests the model for PCr resynthesis is given by PCr(*t*) = R − [*d*
_{1} ⋅ exp(−*k*
_{1} ⋅*t*) ±*d*
_{2} ⋅ exp(−*k*
_{2} ⋅ *t*)], where R is PCr concentration at rest,*d*
_{1}, *d*
_{2}, *k*
_{1}, and *k*
_{2} are constants, and *t* is time. By using nonlinear least squares regression, this double-exponential model was shown to fit the PCr recovery data taken from two studies involving maximal exercise accurately. In *study 1,* when the muscle was electrically stimulated while occluded, PCr concentrations rose during the recovery phase to a level above that observed at rest. In *study 2,* after intensive dynamic exercise, PCr recovered monotonically to resting concentrations. The second exponential term in the double-exponential model was found to make a significant additional contribution to the quality of fit in both *study 1*(*P* < 0.05) and *study 2* (*P* < 0.01).

## Circuit Models of Muscle Metabolism

*To the Editor*: Lumped-element analog circuit models such as the one recently published by Nevill et al. (4) are potentially useful tools for modeling physiological systems, provided that a few conceptual rules of thumb are satisfied. First, there should be a clear correspondence between the elements and parameters of the physiological system (e.g., blood vessels and blood flow in a cardiovascular model) and the elements and parameters of the circuit analog (e.g., resistors and currents). Second, the arrangement of the model elements in the circuit should not be arbitrary but should correspond in a meaningful fashion to the physiology (e.g., resistance elements representing blood vessels to various organs should be in parallel). Third, the boundary conditions within which the idealized, linear circuit elements are expected to approximate the physiological behavior should be clearly defined. (For example, the development of turbulence in blood vessels at high flow rates could not be modeled by an ideal resistor, which obeys Ohm’s law at any current.) Fourth, and perhaps most importantly, the circuit model should predict new behavior that can be tested by new observations of the physiological system.

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. 1
*A*), 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/6*RT*; 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. 1
*B*) 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**d*i*/d*t*). 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).

- Copyright © 1997 the American Physiological Society

## REFERENCES

## REPLY

*To the Editor*: In a recent paper, Nevill et al. (1-3) demonstrated empirically, using nonlinear least squares, that a double-exponential (second-order) model provided a significantly better fit to PCr resynthesis data than did a monoexponential (first-order) model. Not only did the double-exponential model provide a significantly better fit to PCr resynthesis data in the two examples studied but also the model was capable of describing an overshoot in PCr resynthesis observed in one such study, an impossible characteristic of a monoexponential (first-order) model. The paper proposed a model based on a simple electric analogy, which, by introducing an inductance (*L*), was able to provide the second order behavior. It is well known that a simple electric circuit is incapable of providing a precise interpretation of the complex physiological and biochemical changes that occur in the human body after exercise. However, the model was given to provide an analogy by which the study of the physiological responses to dynamic exercise could be advanced. A similar double-exponential model was anticipated for PCr resynthesis by Morton (1-2). Although this work was brought to my attention after the publication of Nevill et al. (1-3), Morton’s second-order differential equation was based on a model from fluid dynamics rather than the electric circuitry, as proposed in Nevill et al. (1-3).

Nevertheless, there are a number of possible physiological origins for the second-order behavior. In an electric circuit, the function of an inductance is to store energy in the form of a magnetic field. This is a dynamic store of energy that differs from the capacitive storage in a first-order model in which the storage is essentially static. Instead of an electric model, it is possible to imagine a mechanical model in which the storage of energy in the inductance is replaced by the storage of energy in mechanical inertia such as a flywheel or moving mass, not dissimilar to the changes in fluid dynamics proposed by Morton (1-2). Mathematically, this mechanical model is identical to the electric model proposed by Nevill et al. (1-3), and some readers may find this mechanical analogy of an inductance (a dynamic change in inertia) easier to interpret.

Meyer and his associates draw attention to problems in the proposed electric model, with the existence of a steady-state current, which they claim must be zero. In fact, an *LCR* electric circuit can have a steady current present and, unless the inductance has a magnetic core that can saturate because of the steady current, there will be a negligible effect on the exponential characteristics of the circuit. However, the steady current will eventually recharge the capacitor, a process that Nevill et al. (1-3) likened to PCr resynthesis in muscle.

Meyer and his associates also raise the point that any system with second-order behavior, whether electric or mechanical, can display oscillation at a resonant frequency. However, such behavior would only be manifest if the resistive term is relatively small. It is likely that in biological systems the energy losses, which are equivalent to the resistive losses in an electric circuit, will dominate, and, therefore, oscillatory behavior is extremely unlikely to be manifest.

The progress of science has frequently involved the transfer of mathematics and modeling from relatively simpler disciplines, such as the physical sciences and engineering, through to their more complex equivalents in the biological and environmental sciences. In making such transfers, it has been necessary to find equivalents or analogies for the simple physical concepts of, for example, inductance, capacitance, and resistance or the mechanical concepts of inertia, spring constants, and damping. The purpose of the paper by Nevill et al. (1-3) was to confirm the work of Harris et al. (1-1) [supported by the study of Morton (1-2)] that PCr resynthesis is best described by a double-exponential model and not to provide a precise source for the second-order behavior. It is to be hoped that the paper will stimulate physiological research to examine possible biological equivalents that will involve the storage of energy in a dynamic form.

## Acknowledgments

I acknowledge the valuable advice of Prof. Roger Morgan, Director of the School of Electric and Electronic Engineering at Liverpool John Moores University, UK.

- Copyright © 1997 the American Physiological Society