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letters to the editor
The following is the abstract of the article discussed
in the subsequent letter:
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
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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) 5 R 2
[d1 · exp(2k1 · t) 6 d2 · exp(2k2 · t)], where R is PCr concentration at rest, d1, d2, k1, and k2 are constants, and t is time. By
using nonlinear least squares regression, this doubleexponential 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).
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 5 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 5 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
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LETTERS TO THE EDITOR
Fig. 1. A: circuit analog model for phosphocreatine changes in
skeletal muscle. Vb, free energy potential; Rm, resistor; Icy, current
representing cystolic adenosinetriphosphatase rate; Vo, cyotolic free
energy of ATP hydrolysis; C, capacitance. (Reproduced from Ref. 1.)
B: modified circuit model proposed by Nevill et al., E, energy source;
q, stored charge on the capacitor; i, electric current; A, switch;
R, resistance; L, inductance. (Reproduced from Ref. 4.)
capacitor, which supposedly still represents the creatine kinase system, is in series with the battery and
resistor, not in parallel. As a consequence, the steadystate 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.
REFERENCES
1. Meyer, R. A. A linear model of muscle respiration explains
monoexponential phosphocreatine changes. Am. J. Physiol. 254
(Cell Physiol. 23): C548–C553, 1988.
2. Meyer, R. A. Linear dependence of muscle phosphocreatine
kinetics on total creatine content. Am. J. Physiol. 257 (Cell
Physiol. 26): C1149–C1157, 1989.
3. Meyer, R. A., A. T. Paganini, R. Stoyanova, and T. R. Brown.
Non-negative least squares (NNLS) analysis of PCr recovery in
skeletal muscles with mixed fiber type (Abstract). Proc. Int. Soc.
Magnetic Resonance Med. 5: 1309, 1997.
4. Nevill, A. M., D. A. Jones, D. McIntyre, G. C. Bogdanis, and
M. E. Nevill. A model for phosphocreatine resynthesis. J. Appl.
Physiol. 82: 329–335, 1997.
5. Paganini, A. T., J. M. Foley, and R. A. Meyer. Linear dependence of muscle phosphocreatine kinetics on oxidative capacity.
Am. J. Physiol. 272 (Cell Physiol. 41): C501–C510, 1997.
Ronald A. Meyer
Dept. of Physiology and Radiology
Michigan State University
East Lansing, Michigan 48824
Robert W. Wiseman
Jeroen A. L. Jeneson
Dept. of Radiology
University of Washington
Seattle, Washington 98105
REPLY
To the Editor: In a recent paper, Nevill et al. (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 mono-
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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).
LETTERS TO THE EDITOR
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. (3) was to confirm the work of Harris et al.
(1) [supported by the study of Morton (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.
I acknowledge the valuable advice of Prof. Roger Morgan, Director
of the School of Electric and Electronic Engineering at Liverpool John
Moores University, UK.
REFERENCES
1. Harris, R. C., R. H. T. Edwards, E. Hultman, L. O. Nordesjo,
B. Nylind, and K. Sahlin. The time course of phosphorylcreatine
resynthesis during recovery of the quadriceps muscle in man.
Pflügers Arch. 367: 137–142, 1976.
2. Morton, R. H. A three-component model of human bioenergetics.
Math. Biol. 24: 451–466, 1986.
3. Nevill, A. M., D. A. Jones, D. McIntyre, G. C. Bogdanis, and
M. E. Nevill. A model for phosphocreatine resynthesis. J. Appl.
Physiol. 82: 329–335, 1997.
Alan M. Nevill
David A. Jones
David McIntyre
Gregory C. Bogdanis
Mary E. Nevill
School of Human Sciences
Liverpool John Moores University
Liverpool L3 3AF, United Kingdom
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exponential (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 (2). Although this work was
brought to my attention after the publication of Nevill
et al. (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. (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 (2). Mathematically, this
mechanical model is identical to the electric model
proposed by Nevill et al. (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. (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
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