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Analyzed the data: AP. Wrote the paper: AP. Formulated all of the mathematical derivations: AP.

The author has declared that no competing interests exist.

The first-degree power-law polynomial function is frequently used to describe activity metabolism for steady swimming animals. This function has been used in hydrodynamics-based metabolic studies to evaluate important parameters of energetic costs, such as the standard metabolic rate and the drag power indices. In theory, however, the power-law polynomial function of any degree greater than one can be used to describe activity metabolism for steady swimming animals. In fact, activity metabolism has been described by the conventional exponential function and the cubic polynomial function, although only the power-law polynomial function models drag power since it conforms to hydrodynamic laws. Consequently, the first-degree power-law polynomial function yields incorrect parameter values of energetic costs if activity metabolism is governed by the power-law polynomial function of any degree greater than one. This issue is important in bioenergetics because correct comparisons of energetic costs among different steady swimming animals cannot be made unless the degree of the power-law polynomial function derives from activity metabolism. In other words, a hydrodynamics-based functional form of activity metabolism is a power-law polynomial function of any degree greater than or equal to one. Therefore, the degree of the power-law polynomial function should be treated as a parameter, not as a constant. This new treatment not only conforms to hydrodynamic laws, but also ensures correct comparisons of energetic costs among different steady swimming animals. Furthermore, the exponential power-law function, which is a new hydrodynamics-based functional form of activity metabolism, is a special case of the power-law polynomial function. Hence, the link between the hydrodynamics of steady swimming and the exponential-based metabolic model is defined.

Activity metabolism represents the relationship between metabolic rate and steady speed. Any functional form that is an interpolant of activity metabolism can be used to describe activity metabolism. For example, the conventional exponential function and the cubic polynomial function have been used to describe activity metabolism for steady swimming animals

The first-degree power-law polynomial function, which is the standard functional form used in hydrodynamics-based metabolic studies, is

Equation (2) is the log-linear form of equation (1). Thus, like any log-linear function, equation (2) has an intercept ( = ln

Equation (1) tacitly assumes a constant ψ of 1 and thus does not take into account the differences in the power conversion efficiency, which is usually different for individuals within species and almost always different for individuals among species. The Second Law of Thermodynamics explicitly states that the power conversion efficiency is always less (never greater) than 1

If ψ is treated as a parameter, then the power-law polynomial function derives from the antilogarithm-transformed curvilinear form of equation (2) for ψ greater than or equal to 1:

The four parameters (

Equation (3) links the total metabolic rate (

According to equation (5), the degree of the power-law polynomial function (or the value of

A. The data are described by the power-law polynomial function (equation 3). All three curves represent different hydrodynamics-based functional forms of

The parameters

I show that fitting different hydrodynamics-based functional forms of

Only the standard metabolic rate (

If

A. Hypothetical representation of actual observed data, where circles represent

ψ | ||||

actual values from |
0.90 | 3.3 | 1.9 | 1.6 |

incorrect values from |
0.95 | 9.2 | 2.4 | 1.0 |

deviation from |
0.050 | 5.9 | 0.50 | −0.60 |

ψ | ||||

actual values from |
0.90 | 3.3 | 1.9 | 2.5 |

incorrect values from |
1.0 | 36 | 3.1 | 1.0 |

deviation from |
0.10 | 33 | 1.2 | −1.5 |

ψ | ||||

actual values from |
0.90 | 3.3 | 1.9 | 3.3 |

incorrect values from |
1.2 | 115 | 3.8 | 1.0 |

deviation from |
0.30 | 112 | 1.9 | −2.3 |

How would one interpret the association between the energetics and the hydrodynamics of steady swimming from the observed data in

For over 30 years, the first-degree power-law polynomial function (equation 1) has been used to describe

Two methods can be used to formulate equation (3): in the first method, equation (1) is factored into two multiplicative parts,

Like all hydrodynamics-based functional forms of

In conclusion, equation (3) describes many hydrodynamics-based functional forms of

The following derivation of the exponential power-law function (equation 4) is an adaptation of the definition of the conventional exponential function

Parameter

Start with equation (3):

I thank Sean H. Rice and two anonymous reviewers for providing comments that improved the manuscript.