Thermodynamics of metabolic energy conversion under muscle load

Replacing the kinetic coefficients L_ij in equation (7) by the transport coefficients, $\sigma ,{\kappa }_{{J}_{M}}$ and α, yields the following expressions for the fluxes J_M and J_Em:

$$\begin{eqnarray}&&\left(\begin{array}{c}{J}_{M}\\ {J}_{{Em}}\end{array}\right)=\left(\begin{array}{cc}\sigma & \alpha \sigma \\ \alpha \sigma \mu & {\kappa }_{{{\rm{\Pi }}}_{M}}\end{array}\right)\left(\begin{array}{c}{F}_{M}\\ -\tfrac{{\rm{d}}\mu }{{\rm{d}}x}\end{array}\right).\end{eqnarray} \tag{ 17 }$$

Assuming constant parameters, the gradient of J_Em then reads

$$\begin{eqnarray}&&\displaystyle \frac{{\rm{d}}{J}_{{Em}}}{{\rm{d}}x}=\alpha {J}_{M}\displaystyle \frac{{\rm{d}}\mu }{{\rm{d}}x}-{\kappa }_{{J}_{M}}\displaystyle \frac{{{\rm{d}}}^{2}\mu }{{\rm{d}}{x}^{2}}.\end{eqnarray} \tag{ 18 }$$

Note that the transport coefficients are in general not constant, but the constant transport parameter assumption is routinely used when the potential gradients in a given system are sufficiently small so that the system remains close to equilibrium (or close to some effective quasi-equilibrium). For sake of simplicity the present work focusses on 'moderate' efforts, which imply small gradients of the intensive parameters and hence a negligible dependence of the transport coefficients on energy, space and time, as one analyses the production of mechanical force on the global scale. Also, we make the assumption of sufficiently short duration metabolic effort and hence secondary metabolites do not partake in the chemical-to-mechanical energy conversion and are simply considered as waste. In this case, the transport parameters are not affected by the presence of these secondary metabolites.

As energy conservation imposes a constant total energy flux J_U in equation (6), we get

$$\begin{eqnarray}&&\displaystyle \frac{{\rm{d}}{J}_{{Em}}}{{\rm{d}}x}={F}_{M}{J}_{M}\end{eqnarray} \tag{ 19 }$$

and combining equations (18) and (19), we finally obtain the local energy budget:

$$\begin{eqnarray}&&{\kappa }_{{J}_{M}}\displaystyle \frac{{{\rm{d}}}^{2}\mu }{{\rm{d}}{x}^{2}}=-\displaystyle \frac{{J}_{M}^{2}}{\sigma }.\end{eqnarray} \tag{ 20 }$$

One may now notice that although dissipation does not explicitly appear in the force-flux expressions, it does in the local budget equation through the term ${J}_{M}^{2}/\sigma$. In other words, the conservation of the energy and matter, ${\rm{\nabla }}{J}_{U}=0$ and ${\rm{\nabla }}{J}_{M}=0$ and the local working conditions of the metabolic device are totally defined by the local budget and the three transport coefficients $\sigma ,{\kappa }_{{J}_{M}}$ and α. Let us now extend the analysis to the management of this equivalent working fluid inside an organism or part of it.

Assuming that the metabolic processes take place in the volume A × l of the organism, the complete energy budget is obtained after integration of equation (20)

$$\begin{eqnarray}&&\displaystyle \frac{{\rm{d}}\mu }{{\rm{d}}x}=-\displaystyle \frac{{J}_{M}^{2}}{\sigma {\kappa }_{{J}_{M}}}x+C.\end{eqnarray} \tag{ 21 }$$

The energy flux Φ = AJ_Em can hence be expressed as

$$\begin{eqnarray}&&{\rm{\Phi }}=\alpha \mu {I}_{M}-{\kappa }_{{J}_{M}}A\displaystyle \frac{{\rm{d}}\mu }{{\rm{d}}x},\end{eqnarray} \tag{ 22 }$$

where I_M = AJ_M is the metabolic intensity. This latter is directly related to the average rate of production of the chemical reactions. The metabolic flux now reads from equation (19):

$$\begin{eqnarray}&&{\rm{\Phi }}(x)=\alpha \mu (x){I}_{M}+{R}_{M}{I}_{M}^{2}\displaystyle \frac{x}{l}+C.\end{eqnarray} \tag{ 23 }$$

The term C is determined by the boundary conditions at both locations x = 0 and x = l. Now, in order to complete the description of the system and the model, we turn to boundary conditions specification.

As a thermodynamic device, the metabolic conversion zone defined by the element of length l and section A is inserted in a global system. The boundary conditions impose the values of both energy and matter currents and consequently, the working conditions of the metabolic device. The complete thermodynamic system is represented on figure 2, which arguably outlines a quite general situation. Due to the presence of dissipative couplings between the conversion zone and the reservoirs, the reservoirs chemical potentials ${\mu }_{+}$ and ${\mu }_{-}$ are modified and become ${\mu }_{+M}$ and ${\mu }_{-M}$ as shown on figure 2.

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The metabolic converter is connected to a reservoir and a sink by two dissipative elements of resistances, R₊ and R₋. These resistances, together with the basal metabolic resistance ${R}_{E}=l/{\kappa }_{{J}_{M}}A$, allow considering mixed boundary conditions between two limit cases, namely (i) the Neumann conditions, where currents are imposed, considering high, diverging values of R₊ and R₋; (ii) the Dirichlet conditions, where potentials are imposed, considering instead vanishing values for R₊ and R₋. It is important to note that the crucial question of boundary conditions is imposed by the system, namely, a metabolic machine contained within a defined perimeter. From a theoretical point of view, it would be perfectly possible to define conditions at Neuman's limits by considering the organism within its biotope, as a complete system receiving a flow of matter and energy. However, the perimeter of the system, imposed by the body envelope is an unavoidable constraint in this model, and this constraint leads to consider irreducible mixed boundary conditions. From the general expression equation (23), we get the expressions of the incoming ${{\rm{\Phi }}}_{+}={\rm{\Phi }}(0)$ and outgoing ${{\rm{\Phi }}}_{-}={\rm{\Phi }}(l)$ fluxes:

$$\begin{eqnarray}&&{{\rm{\Phi }}}_{+}=\alpha {\mu }_{+M}{I}_{M}+\displaystyle \frac{{\rm{\Delta }}{\mu }_{M}}{{R}_{E}},\end{eqnarray} \tag{ 24 }$$

$$\begin{eqnarray}&&{{\rm{\Phi }}}_{-}=\alpha {\mu }_{-M}{I}_{M}+{R}_{M}{I}_{M}^{2}+\displaystyle \frac{{\rm{\Delta }}{\mu }_{M}}{{R}_{E}},\end{eqnarray} \tag{ 25 }$$

where ${\rm{\Delta }}{\mu }_{M}={\mu }_{+M}-{\mu }_{-M}$. The term $C={\rm{\Delta }}{\mu }_{M}/{R}_{E}$ comes from the flow balance at the incoming and outgoing borders of the conversion zone. Accounting for the connections to the reservoirs, we get two additional expressions for the fluxes:

$$\begin{eqnarray}&&{{\rm{\Phi }}}_{+}=\displaystyle \frac{{\mu }_{+}-{\mu }_{+M}}{{R}_{+}},\end{eqnarray} \tag{ 26 }$$

$$\begin{eqnarray}&&{{\rm{\Phi }}}_{-}=\displaystyle \frac{{\mu }_{-M}-{\mu }_{-}}{{R}_{-}}\end{eqnarray} \tag{ 27 }$$

that yield the following simple expression for the metabolic converter output power P:

$$\begin{eqnarray}&&P={{\rm{\Phi }}}_{+}-{{\rm{\Phi }}}_{-}=\left[\alpha {\rm{\Delta }}{\mu }_{M}-{R}_{M}{I}_{M}\right]{I}_{M}.\end{eqnarray} \tag{ 28 }$$

The expressions (24), (25) and (26), (27) now give a complete set of equations for a proper description and analysis of a given metabolic situation. The calculation of the general solution does not pose any specific difficulty but is quite tedious.

For illustration purposes, let us consider the particular situation of an effort with reasonable (low) duration, neglecting the drawback effects of the waste fractions; in this case, the coupling resistance to the sink R₋ may be considered as negligible, and ${\mu }_{-M}\approx {\mu }_{-}$. We obtain the simplified expressions:

$$\begin{eqnarray}&&{{\rm{\Phi }}}_{+}=\displaystyle \frac{{F}_{\mathrm{iso}}{I}_{M}/{\eta }_{{\rm{C}}}+B}{{I}_{T}+{I}_{M}}\,{I}_{T},\end{eqnarray} \tag{ 29 }$$

$$\begin{eqnarray}&&{{\rm{\Phi }}}_{-}={R}_{M}{I}_{M}^{2}+\displaystyle \frac{{R}_{\mathrm{fb}}{I}_{M}^{2}+{F}_{\mathrm{iso}}\left(\tfrac{1}{{\eta }_{{\rm{C}}}}-1\right){I}_{M}+B}{{I}_{T}+{I}_{M}}\,{I}_{T},\end{eqnarray} \tag{ 30 }$$

where all the quantities are defined in table 2. In order to simplify the identification of the boundary conditions we define the parameter $r=\tfrac{{R}_{+}}{{R}_{+}+{R}_{E}}$. This parameter varies from r = 0 when the system is connected with potential boundary conditions, i.e. of the Dirichlet type, to r = 1 in the case of flux conditions, i.e. Neumann type.

Table 2.  Thermodynamic parameters characterizing the chemical-to-mechanical energy conversion.

Quantity	Definition
IM	Macroscopic (space integrated) metabolic intensity
IMFM	Available mechanical power
$B=\tfrac{{\rm{\Delta }}\mu }{{R}_{+}+{R}_{E}}=\tfrac{{\rm{\Delta }}\mu }{{R}_{E}}(1-r)$	Basal power consumed by the body at IM = 0, or F = Fiso
${I}_{T}=\tfrac{{R}_{E}+{R}_{+}}{\alpha {R}_{+}{R}_{E}}=\tfrac{1}{\alpha {{rR}}_{E}}$	Threshold metabolic intensity
${R}_{\mathrm{fb}}=\alpha {\mu }_{-}/{I}_{T}=r{\alpha }^{2}{R}_{E}{\mu }_{-}$	Feedback resistance
${F}_{\mathrm{iso}}=\alpha {R}_{E}B$=$\alpha {\rm{\Delta }}\mu (1-r)$	Isometric force
${\eta }_{{\rm{C}}}=\tfrac{{\mu }_{+}-{\mu }_{-}}{{\mu }_{+}}$	Overall chemical-mechanical machine efficiency

Note that the product or ratio of some couples of parameters result in constant quantities such as, e.g. ${F}_{\mathrm{iso}}/B=\alpha {R}_{E}$ and ${R}_{\mathrm{fb}}{I}_{T}=\alpha {\mu }_{-}$, which creates some particular constraints. This point will be developed further in the article.

Both R_fb and I_T are governed by the boundary conditions for the energy entering the system; in other words, they characterize the ability of the biological system to feed the conversion zone with chemical energy, this ability being limited by the coupling resistance R₊. Note that if the boundary conditions would only be of the potential type (Dirichlet conditions), the feedback resistance R_fb would be zero. We discuss the importance of the feedback resistance for muscle work in the frame of Hill's model in the next section, as I_T corresponds exactly to one of Hill's constants, and ${R}_{\mathrm{fb}}$ is instrumental to elucidate the question of the non-constancy of Hill's first 'constant' a (defined in the next section).

The metabolic power delivered during a physical effort is expressed as:

$$\begin{eqnarray}&&P={{\rm{\Phi }}}_{+}-{{\rm{\Phi }}}_{-}={F}_{M}{I}_{M}=\left[{F}_{\mathrm{iso}}-\left({R}_{M}+{R}_{H}({I}_{M})\right){I}_{M}\right]{I}_{M},\end{eqnarray} \tag{ 31 }$$

where F_iso is the so-called isometric force [32], which describes the situation of a muscle tension under load but with no motion, and where the resistance ${R}_{H}({I}_{M})=\tfrac{{F}_{\mathrm{iso}}+{R}_{\mathrm{fb}}{I}_{T}}{{I}_{T}+{I}_{M}}$ is introduced as a metabolic intensity-dependent resistance showing clearly that the organism cannot be seen as a passive system in the sense that it does not merely represent a system component that merely dissipates energy as a standard resistor would in an electrical circuit. In the present model, R_M acts as such a passive component. On the contrary, the definition of R_H contains R_fb, and both can be expected to stem from feedback effects. They are thus tightly related, notably through the threshold metabolic intensity.

With equation (31), we also recover the classical expression of a force-intensity response, with the isometric force ${F}_{\mathrm{iso}}$ and the composite internal resistance ${R}_{H}({I}_{M})+{R}_{M}$. As expected, the isometric force is proportional to the converted fraction of the chemical energy difference Δμ modulated by the resistance bridge value: $\tfrac{{R}_{E}}{{R}_{+}+{R}_{E}}$. Furthermore, as the composite resistance ${R}_{H}({I}_{M})+{R}_{M}$ depends on the metabolic intensity, we see that it may govern the shape of the response curves of a muscle, which is quite a central result, as will be shown later.

The output power accounts for the power spent to sustain the physical effort balanced by the dissipated power. It is important to note that the notion of 'physical effort' may encompass a quite large variety of efforts. In the case of an animal in motion, at velocity v, the physical effort simply corresponds to the production of the mechanical power necessary for actual motion. In this case, the metabolic intensity can be expressed directly from the rate of production of chemicals [33] and, consequently, the mechanical velocity of the system as I_M ≡ kv, where k is a dimensional constant. But the metabolic intensity is also non-zero for animals carrying heavy loads, while standing still. Further, the power P has two zeros, corresponding to two specific values for the intensity: I_M = 0 in the absence of any effort, and I_M = I_X, the maximum value of intensity at P = 0, in a situation of exhaustion with ${I}_{X}=\tfrac{{F}_{\mathrm{iso}}}{{R}_{H}+{R}_{M}}$. Therefore, the concept of metabolic intensity accounts for various types of efforts, that a purely mechanical description can not.

Turning to the figure of merit, we see that it now reads

$$\begin{eqnarray}&&{f}_{M}=\displaystyle \frac{{R}_{\mathrm{fb}}{F}_{\mathrm{iso}}{I}_{T}}{{R}_{M}B}.\end{eqnarray} \tag{ 32 }$$

We thus obtain a compact expression which, while preserving its thermodynamic structure, can be read in a form directly accessible to the understanding of metabolic performance. Let us analyse what are the compromises and expectations at stake for a given value of this merit factor. First of all, it is clear that mechanical power is only available as much as the mechanical viscosity R_M makes it possible. Similarly, we can notice the presence of the ${F}_{\mathrm{iso}}/B$ ratio which reflects the necessary compromise between the availability of force and the 'basal' dimension of the machine that leads to this availability. Last, the product ${R}_{\mathrm{fb}}{I}_{T}=\alpha {\mu }_{-}$ which can be considered constant, shows us that the metabolic intensity is necessarily contingent on the feedback resistance, and that high metabolic intensities can only be achieved by a significant reduction in this resistance.

It is important to note that f_M does not depend on the ratio $r={R}_{E}/({R}_{+}+{R}_{E});$ in fact, it is a quantity that intrinsically characterizes the thermodynamic conversion capacity, regardless of the coupling conditions to the reservoirs. Finally, using the set of equations (29), (30) and (31), we can now plot the generic response of an organism to an effort as shown on figure 3(a) where the input (Φ₊), output (${{\rm{\Phi }}}_{-}$) and mechanical ($P={{\rm{\Phi }}}_{+}-{{\rm{\Phi }}}_{-}$) powers are reported. We also plot on figure 3(b) the efficiency $\eta =\tfrac{{{\rm{\Phi }}}_{+}-{{\rm{\Phi }}}_{-}}{{{\rm{\Phi }}}_{+}}$ versus power curve: the maximal efficiency is obtained for a metabolic intensity substantially below that required for the production of maximal mechanical power, while the mechanical output is only slightly below the maximal one. The latter is reached at the cost of efficiency. It can easily be shown that the higher the f_M factor, the further away the maximum efficiency and maximum power points shall be apart. This confirms that there is no optimal operating point in absolute terms, let alone an underlying general variational principle. Depending on the metabolic parameters, each muscle is subjected to a trade-off between its maximal efficiency and its maximal power, or minimum production of waste. We shall discuss the question of the variational principle in section 5. We show that the biological system becomes obviously deterministic at the condition that the system definition properly includes the boundary conditions.

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The principal elements of Hill's model are summarized in the appendix; the model shows that a chemical contribution to dissipation during muscle contraction was a central hypothesis of the muscle model [9], and as such, had been added to the energy budget. This point had first been raised by Fenn who claimed that the fact that an active muscle shortens more slowly under a greater force, is not due to mechanical viscosity but to how the chemical energy release is regulated [38, 39] (see also the critical analysis of Fenn's works in [40]). Both Fenn and Hill did not make use of any feedback effect between chemical input and mechanical output. However, considering the expressions of R_fb and I_T, the chemical origin of a is evident. In 1966, Caplan conducted a similar analysis, but the work done considered the local feedback to be an exogenous additional process, without any hint to a change in boundary conditions applied to a macroscopic size system [41]. The influence of modified chemical conditions can also be found in [42] and [43].

In his Nobel lecture Hill put forth the central question of the fast and slow muscle fibers [11]:

'The difference in the time-scales of the two types of muscle makes one regard it as improbable that physical viscosity alone is the determining factor. One cannot see why viscosity should have ten times the effect in a human muscle than it has in a frog's, and probably one hundred or one thousand times as much as it has in a fly's.'

From a mechanical point of view there is no reason why a machine should be constrained by limitations as long as it is able to receive and convert energy, such as chemical energy, into mechanical energy. On the other hand, from the point of view of thermodynamics, the situation is quite different. Indeed, since the performance of the conversion machine is defined both by the thermodynamic fluid and by the machine that uses this fluid, it follows that the maximum performance is limited by the intrinsic capacity of the fluid to carry out the energy conversion. The constant figure of merit ${f}_{M}=\tfrac{{R}_{\mathrm{fb}}{F}_{\mathrm{iso}}{I}_{T}}{{R}_{M}B}$ reflects these intrinsic performances. For a general thermodynamic system there is a figure of merit that defines the upper limit of the performance of the conversion to useful work. This limit can be approached but never exceeded. The situation here is strictly similar and there is therefore a trade-off as to the values of the different parameters. The performance of a muscle is then bounded from above by the figure of merit, that defines the maximum thermodynamic conversion capacity, and consequently, the maximum intrinsic efficiency. To this must be added the constraints that link some of the parameters that make up the figure of merit, namely: F_iso/B = α R_E and ${R}_{\mathrm{fb}}{I}_{T}=\alpha {\mu }_{-}$, which are considered constant. Indeed, in the present case of the response to a time-limited effort, the intrinsic parameters ${R}_{E},{\mu }_{-}$ and α are constant.

Let us now consider the specific case of a so-called fast muscle and observe the constraints imposed by the metabolic model. In this case it is expected that the force does not collapse at high speed v, i.e. for high values of metabolic intensity close to I_T. In view of the above remarks, this implies that R_fb is minimal, and then so is R₊. This leads to an increase in the basal power B and therefore an increase in the isometric force F_iso by the same amount. Therefore, it appears that fast fibres are also those with the highest isometric forces, and as such are the most powerful fibers, which benefit from little-constrained access to the resource since $r\ll 1$. This double signature is indeed encountered in the case of fast fibres, which are highly energy consuming, even at low speeds, and in particular at the basal level [44, 45]. Then the performance is achieved at the cost of maintaining a substantial basal power that has a definite energy cost which in turn reduces the overall efficiency. On the contrary, in the case of slow fibres, the stress on large I_T values is released which leads to a higher R_fb resistance and finally a moderate basal power. Note that at birth, some mammals have a very high proportion of slow muscles [44], which tends to decrease as they grow. Such a signature at birth is in accordance with the main property of slow muscles which is their low basal consumption. The same conclusions can be found in the works of Ruegg [46] and Clinch [47].

The difference in the respective values of the basal consumption of fibres and their contraction rates is commonly reported in the literature. As an example we cite the case of the muscles 'extensor digitorum longus' and 'soleus', which in mice have contraction rate ratios of 5.9/2 ≃ 2.95 and basal consumption in the ratio 4/1.3 ≃ 3.08 (a value quite close to 2.95) [20, 48, 49]. Since the basal power does not contribute to the production of mechanical power, the choice of slow fibres, when they are sufficient to achieve the function, becomes obvious. From the point of view of glucose combustion, the glycolitic pathway is incomplete since it does not include a Krebbs cycle. The similarity with complete or incomplete combustion of alkanes in internal combustion engines, is quite natural, the efficiency ceiling being in both cases set by the figure of merit of the fuel under total combustion conditions. In other words, one may observe that the overall performance of the muscle is, as in the case of any thermodynamic system, the result of a trade-off between the intrinsic combustion properties of the fuel on the one hand, and the implementation of this combustion according to a particular path imposed by the machine, on the other hand. Similarly, we can see that in the case of muscles of the same nature, (r constant) the ratio a₀/F_iso is predicted to be constant, which is actually reported in [44]. Although temperature is not explicitly present in our model, the influence of a muscle temperature during exercise is easily visible on the measurements [50]. The general shape of the curves is not globally modified, but the parameters have a sizable temperature dependence.

It is known that slow and fast fibres do not have the same force–speed response. If the classic form of the force–speed response according to Hill is that of a hyperbola, mention has been made of responses with a very shallow, or even totally rectilinear, character [42, 44, 45]. The existence of linear characteristics means that in these cases the dissipation resistance R_H(I_M) + R_M is almost constant. As a result, the term R_H(I_M) is no longer a function of the current I_M. This amounts to considering that I_T ≫ I_M. It follows that the predominance of fast fibres leads to more linear characteristics. This was indeed observed for the human in the force–speed response of the arms [51]. Physiological analyses reveal the predominance of fast fibres in the arms, unlike legs which have more slow fibres. This is true in general for the untrained individual. In the case of sprint-trained runners [52] or cyclists [53], there is a strong tendency towards the linear form, which in these cases reflects the predominance of fast fibres, as predicted by the model. The same signature is obtained if the pedalling is done with the arms [54].

We can illustrate the dependence of the boundary conditions connection quality (value of r) according to the fibre types by using data on EDL and soleus muscles [44]. These latter are distinguished by their fraction of fast and slow fibres, which in our formalism translates as a better coupling to the reservoirs, and hence: ${r}_{\mathrm{SOL}}\lt {r}_{\mathrm{EDL}}$. Further, by normalizing the force by the isometric force, F_iso, in equation (33), and the velocity by the short circuit velocity, v_X, we obtain the following expression:

$$\begin{eqnarray}&&\displaystyle \frac{F}{{F}_{\mathrm{iso}}}=\zeta (r)\displaystyle \frac{1-v/{v}_{X}}{v/{v}_{X}+\zeta (r)}-\displaystyle \frac{{R}_{M}{v}_{X}}{{F}_{\mathrm{iso}}}\displaystyle \frac{v}{{v}_{X}}\end{eqnarray} \tag{ 38 }$$

with $\zeta (r)=\tfrac{{a}_{0}}{{F}_{\mathrm{iso}}}=\tfrac{{a}_{0}}{\alpha {\rm{\Delta }}\mu }\tfrac{1}{1-r}$ and $\tfrac{{a}_{0}}{\alpha {\rm{\Delta }}\mu }$ constant. The first term corresponds to the force–velocity response in the absence of contribution of the mechanical dissipation, i.e. R_M = 0. We then obtain an expression whose only degree of freedom is r. In other words, one can only distinguish a fast fibre from a slow fibre, or two muscles composed of a different fraction of each other, by their dependence on boundary conditions. The presence of mechanical dissipation (${R}_{M}\ne 0$) results in a correction for the higher velocity: for a given velocity, the available force is reduced.

The panels on the right side of figure 5 show experimental data on the EDL and soleus muscles [44], in a force–velocity representation. The fits shown are based on Hill's equation without direct contribution of the mechanical dissipation, for which R_M = 0 and a = a₀; this hypothesis makes it possible to fit the experimental data with very good agreement. In the following, we limit the scope of our analysis to this configuration.

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The experimental data normalized by the isometric force, F_iso, and the short circuit velocity, v_X, obtained for each of the previous fit are reported on the panel left of figure 5. The fits of these data are also done with equation (38) under the assumption of mechanical dissipation. The uncertainty of the experimental data was estimated at 1/4 of the marker size. We observe in the inset of figure 5 that both relative and absolute uncertainties decrease sharply with the age of the muscle. Consequently the data obtained for the oldest ages are dramatically more weighted in the fitting process. The shaded regions represent the best fit of the normalized and aggregated data for each muscle at plus and minus one standard deviation. We then have ζ_EDL = 0.27 ± 0.02 and ζ_SOL = 0.23 ± 0.02, which verifies that r_SOL < r_EDL. By recognizing that $\tfrac{{\zeta }_{\mathrm{EDL}}}{{\zeta }_{\mathrm{SOL}}}=\tfrac{1-{r}_{\mathrm{SOL}}}{1-{r}_{\mathrm{EDL}}}$, we obtain the condition ${r}_{\mathrm{EDL}}\geqslant 1-\tfrac{{\zeta }_{\mathrm{SOL}}}{{\zeta }_{\mathrm{EDL}}}$. A strict equality in the latter expression would imply that ${r}_{\mathrm{SOL}}$ is exactly zero, i.e. the soleus muscle is under the Dirichlet boundary condition. Based on the experimental data, we can therefore conclude, with r_SOL > 0, that r_EDL > 0.14 ± 0.08.

In his seminal work [9] Hill identified the three parameters as constants, following the viscoelastic origin of the muscle response proposed by Gasser and Hill [55]. As indicated in the appendix, Hill's 1938 model is based on two hypotheses, one of which concerns the heat released during muscle contraction, which Hill proposed to formulate as proportional to the rate of contraction, H = av, where a is the so-called extra heat coefficient considered to be constant. Interestingly enough, this dependence, initially ignored by Hill was considered by Fenn [38, 39, 56], and later reconsidered in experiments by Aubert [57, 58], and finally taken up by Hill himself [10]. A review of these effects can be found in [59, 60]. Indeed, the correction on this term is very generally small, which very often leads to not detecting it experimentally. However, if we consider the complete expression proposed in equation (35), we note that the corrective term involves the resistance to mechanical displacement R_M, which corresponds to the viscosity opposing the displacement of the strands within the muscle fibers. This results in a corrected expression of type $H={{aI}}_{M}+{R}_{M}{I}_{M}^{2}$. This correction, although small, is in no way incidental because neglecting it amounts to assimilating, from a mechanical point of view, the muscle fibre to a generator without internal resistance, which could therefore lead to an infinite speed displacement and an equally infinite power production. In practice this situation is not realized because this resistance is in series with the feedback resistance R_fb which overrides R_M in the mechanical viscous dissipation, which leads to masking the effect of R_M. From an experimental point of view, by relying on equation (35), it is easy to consider the situations for which this corrective term becomes visible. To do this, it is important that the metabolic intensity, and therefore the rate of contraction, can reach high values. This is only possible in cases where I_T itself is important, i.e. in the presence of a dominant presence of fast fibres. In this case, the hyperbolic form of the response of the force–speed response curve gives way to a much more rectilinear profile, especially at high velocities as previously described.

The above developments show that it is possible to describe the process of converting chemical energy into mechanical energy in the same terms as any thermodynamic machine composed of a working fluid with a figure of merit, on the one hand, and a device that implements this working fluid on the other hand. In this case, glucose degradation remains the central element to which a figure of merit can be associated. Depending on the coupling conditions to the reservoirs governed by ${R}_{+}$, this degradation can occur quickly or slowly (glycolytic or oxidative route). More precisely, the kinetic of the glucose combustion, fast or slow, requires the presence of direct (r ≈ 0), or reduced, (r ≈ 1) access to the resource. But r, hence ${R}_{+}$, does not account for glucose combustion kinetics rate itself, but the latter is compatible only with certain values of r. In other words, there is no use to have an efficient device if it is not correctly fed. It could be argued that in this case the condition of direct access to the resource (r ≈ 0) should apply systematically since it would ensure a sufficient energy start for any type of muscle, but this is not the case as for slow fibres it would lead to unnecessarily high basal consumption. Our present model does not claim to explain the detailed metabolic pathways, which have been the subject of much work, but rather to characterize the chemical-mechanical conversion through a systemic approach. If by its fast kinetics, inherited from Dirichlet coupling conditions, the glycolytic pathway allows both force and power production, its high basal consumption is nonetheless incompatible with a prolonged effort. On the other hand, due to its Neumann coupling condition, the oxidative pathway exhibits a strong reduction in basal power, allowing a prolonged muscle activity at low power. Therefore, the glycolytic and oxidative pathways are perfectly adapted to the boundary conditions to which they are subjected.

As mentioned above, the model only considers short-term efforts and, as such, does not take into account the influence of waste $({R}_{-}=0)$. This model can be applied to force–velocity response measurements that are mainly performed under these conditions. An extension of the model to the case of prolonged efforts is of course possible, but the variation of the parameter ${\mu }_{-}$ would have to be accounted for, which, for simplicity, has not been the case in this article. In figure 6 the force–velocity response is displayed for various ratios of the boundary conditions parameter r. For a given value of the figure of merit, the variation of r alone is sufficient to produce the slow or fast muscle response curves, while the second figure reproduces the shape of the power curves as they were estimated from Hill's initial 1938 model and then from his corrected 1963 model, which coincides with our model. Similar result can be found in [58]. The figure represents the relative influences of the composite resistance R_H(I_M) + R. In the most classic case R_fb ≫ R_M and the muscle response follows Hill's initial hyperbolic law. Since R_M has a very small value, the feedback term R_fb drives the overall response and the dissipation appears to be controlled by the boundary conditions. Otherwise, if R_fb < R_M then the response becomes linear. This case of high mechanical friction R_M would correspond to a particularly low figure of merit, which is unlikely in living systems. Although linear, this response cannot be confused with the response of fast muscles, which are characterized by high values of I_T. In term of Hill parameters, by noting that ${F}_{\mathrm{iso}}=c/b-{a}_{0}$ where ${a}_{0}={R}_{\mathrm{fb}}{I}_{T}$ it comes that ${R}_{H}({I}_{M})=c/b(b+{I}_{M})$, and hence

$$\begin{eqnarray}&&P=\left[\displaystyle \frac{c}{{I}_{M}+b}-a\right]{I}_{M}\end{eqnarray} \tag{ 39 }$$

so the maximal intensity is given by, ${I}_{X}=c/a-b$, which, assuming a ≈ a₀, reduces to

$$\begin{eqnarray}&&{I}_{X}\approx \displaystyle \frac{{F}_{\mathrm{iso}}}{{R}_{\mathrm{fb}}}.\end{eqnarray} \tag{ 40 }$$

We observe here that, although Hill's hypothesis, H = av is tantamount to totally neglecting mechanical viscosity, i.e. R_M in his description of the muscle (which is a daring hypothesis), this has not had any significant consequence because the dynamics of the system is limited by R_fb which dominates and defines the breaking point I_X.

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