Thermodynamics of metabolic energy conversion under muscle load

The metabolic processes complexity is at the heart of energy conversion in living organisms and forms a huge obstacle to develop tractable thermodynamic metabolism models. By raising our analysis to a higher level of abstraction, we develop a compact -- i.e. relying on a reduced set of parameters -- thermodynamic model of metabolism, in order to analyze the chemical-to-mechanical energy conversion under muscle load, and give a thermodynamic ground to Hill's seminal muscular operational response model. Living organisms are viewed as dynamical systems experiencing a feedback loop in the sense that they can be considered as thermodynamic systems subjected to mixed boundary conditions, coupling both potentials and fluxes. Starting from a rigorous derivation of generalized thermoelastic and transport coefficients, leading to the definition of a metabolic figure of merit, we establish the expression of the chemical-mechanical coupling, and specify the nature of the dissipative mechanism and the so called figure of merit. The particular nature of the boundary conditions of such a system reveals the presence of a feedback resistance, representing an active parameter, which is crucial for the proper interpretation of the muscle response under effort in the framework of Hill's model. We also develop an exergy analysis of the so-called maximum power principle, here understood as a particular configuration of an out-of-equilibrium system, with no supplemental extremal principle involved.

The complexity of the metabolic processes at the heart of energy conversion in living organisms is a great obstacle to the development of tractable thermodynamic models of metabolism, which would be based on the definition of a small set of physical parameters. In this article, we construct such a thermodynamic model and show how it can apply in biomechanics by considering the case of muscle load. We assume that living organisms are dynamical systems experiencing a feedback loop, in the sense that they can be considered as thermodynamic systems subjected to mixed (coupled potentials and fluxes) boundary conditions. These feedback effects give rise to homeostatic mechanisms keeping key physiological parameters such as, e.g., body temperature and osmotic pressure, within specific and narrow ranges. The complex nature of such a kind of systems may be dealt with at the cost of raising the abstract analysis to a higher level, but with the advantage of offering a compact approach that relies on a few generic parameters only. To this purpose, we consider a conversion zone as an equivalent thermodynamic working fluid under mixed boundary conditions. The energy conversion process becomes a combination of "one-to-many" (entropy generating) and "many-to-one" (work producing) processes. We introduce and derive the generalized thermoelastic coefficients of the working fluid equivalent, as well as the metabolic transport coefficients from which we define the metabolic figure of merit. Intrinsic muscle friction dissipates the converted energy during muscle mechanical activity, but the chemical-mechanical coupling also causes a dissipative mechanism, which we call the feedback resistance. We discuss the impact of the feedback resistance on the metabolic energy conversion under muscle load. Finally, our approach allows to give a thermodynamic ground to Hill's widely used muscular operational response theory.

I. INTRODUCTION
Thermodynamics provides the proper framework to describe and analyse the rich variety of existing sources of energy and the processes allowing its conversion from one form to another. Yet, of all known energy converters, man-made or not, living organisms still represent a formidable challenge in terms of thermodynamic modeling due to their complexity, which far exceeds that of any other artificial or natural system [1][2][3][4][5][6][7][8]. Energy conversion in living bodies is driven by metabolism, which through chemical reactions at the cellular level ensures, along with other vital functions, the provision of the heat necessary to maintain a normal body temperature and to a lesser extent, the energy required for muscular effort and motion. In the present work, we are particularly interested in catabolism, i.e. the energy-releasing process of breaking down complex molecules into simpler ones, and in particular how, from a nonequilibrium thermodynamics viewpoint, the energy yielded by the breaking of the digested food substances is made available to muscles for motion.
Living bodies are open nonequilibrium and dissipative systems as they continuously exchange energy and matter with their environment [1,2]. Unlike classical thermodynamic engines for which equilibrium models may be built using extremal principles, no such a possibility exists for the case of living organisms because of the absence of identifiable genuine equilibrium states. Nevertheless, assuming a global system close to equilibrium, the development of a tractable thermodynamic model of metabolism may rely on notions pertaining to classical equilibrium thermodynamics on the one hand: the working fluid, acting as the conversion medium, and on the other hand the characterization of its thermoelastic properties. The identification of the nonequilibrium processes driving the transformation of the chemical potential of the digested food into a macroscopic form of energy made available for muscle work may be obtained from an effective local linearized approach.
Before we proceed with the thermodynamics of metabolism, it is useful to first clarify what we mean by energy conversion in a standard thermodynamic engine. When energy is transferred to a system, its response manifests itself on the microscopic level through the excitation of its individual degrees of freedom and also on the global level whenever collective excitations are possible. Broadly speaking, energy conversion in a thermodynamic engine submitted to suitable boundary conditions permitting energy transfer (heat) to its working fluid, proceeds from the collective response of the working fluid's microscopic degrees of freedom. Hence part of the energy received by the working fluid may be made available on a global scale to a load for a given purpose as useful work, the rest of it being redistributed (dispersed) on the microscopic level, and dissipated because of internal friction and any other dispersion process imposed by boundary conditions [9]. The conversion efficiency is thus tightly related to the sharing of the energy allocated to the collective modes of the system. Metabolism differs from energy conversion in standard heat engines in the sense that dissipation cannot be seen as a mere waste since, in biological processes, the dispersion of energy is, rather than a production of heat, a production of secondary metabolites [10]. Nevertheless, for the purpose of describing a short duration metabolic effort, the production of dispersed energy can be considered as a waste.
The main objective of the present work is the development of a thermodynamic description of the out-of-equilibrium steady-state biological chemical-tomechanical energy conversion process in the conditions of the production of a mechanical effort of moderate duration. To this aim, we focus on uncovering the essential features of this process considering a basic power converter model system of incoming dispersed power (the chemical energy flux) to aggregated macroscopic (the mechanical power) energy flux. This conversion zone is connected to two reservoirs of chemical energy, respectively denoted source and sink, and acts the main zone, where energy and matter fluxes are coupled and actual "dispersed-to-aggregated" conversion occurs. An important feature of the model is that the connection between the converter and the reservoirs produces a dispersion of the energy due to resistive coupling. A convenient and coherent way to describe all these processes, would ideally borrow from the phenomenological approach to nonequilibrium thermodynamics developed by Onsager [11] though the complex features of biological systems force us to proceed with great care.
The formulation of the basic model we develop is quite abstract, and hence it is devoid of the detailed characteristics of actual biological processes, which for a bird'seye view thermodynamic approach of the problem, are far too complex to account for properly. As a matter of fact, what we call the conversion zone may describe a mitochondria where the Krebs cycle takes place, a whole biological cell, a single organ or even a whole organism. We thus must place ourselves on a sufficiently high level of description, introducing an effective, fictitious, working fluid, which is not an actual material medium, but which transposes Onsager's force-flux formalism to the study of catabolism, with a generalized definition of the transport coefficients. These generalized coefficients are mathematical objects that act as the building blocks of our abstract description of metabolism. Our model thus provides a thermodynamic "meta-description" of biological energy conversion inasmuch as the energy and matter fluxes are defined on the relevant scale and satisfy conservation laws. It is then clear that a large scale description of the converter, say an organ, may be envisaged in terms of a thermodynamic network of subsystems, say unit cells. Consequently, the properties of the effective working fluid must be defined for the considered level of description.
The article is organized as follows. In Sec. II, we recall some of the basic thermodynamic concepts related to the working fluid properties and the close-to-equilibrium force-flux formalism. We discuss the assumption of linearity, defined on the local level, on which we base the model development. We then specify our approach to biological energy conversion. The full development of the thermodynamic model of metabolism is presented in Sec. III. Some of the insights that our thermodynamic model of metabolism can offer on locomotion are discussed in Sec. IV, where we provide a full thermodynamic interpretation of Hill's theory of muscle loading [12]. Concluding remarks and an appendix presenting our exergy analysis of the so-called concept of maximal power principle in biology, in the frame of our approach, end the article.

A. The linear approximation
The validity of the linear approach for the description of living systems has been recently considered [13], and the study shows that the linear response approach is well-suited to the modeling of living organisms as the non-linear signatures are subjected to strong feedback effects, which maintain the organism to steady states. This permits analyses based on the linearization of the biological system's responses, considering sufficiently high metabolic levels. As recently reported, this conclusion extends beyond the case of biological systems [14]. Besides the question of linearity, it must be noted that processes on the local level are neither "linear" nor "nonlinear", but discrete and that modeling is considering the continuous limit. Recent progress in mesoscopic physics and biology is based on discretized modeling, Brownian motors being one of the main modeling tools [4,5]. The motors derive directly from the Feynman ratchet, modeled as an engine connected to two sources [15,16]. Our present work is quite in line with this approach. The constitutive relations of a Feynman ratchet model can be linearized on the condition that the mechanical energy scales are smaller than the thermal energy scales of the reservoirs [15].
Living bodies are complex systems which produce entropy but the ranges of variations of life-sustaining physiological parameters that present a thermodynamic character (such as body temperature, osmotic pressure, mechanical pressure, etc.) are indeed limited. The study of a living body under normal conditions is thus performed in the vicinity of a working point from which thermodynamic quantities only weakly deviate. Hence, our model follows Onsager's approach which is locally linear but does not preclude the study of a whole range of complex macroscopic signatures of microscopic processes on the condition that the local description is completed by an integration over a finite volume, under specific boundary conditions. In the present work, we show that the overall energy balance is obtained after integration of the local energy budget. This integration yields a quadratic term characterizing the energy dissipation at the volume level, so that the conversion and dissipation processes are properly described, and on the same footing, which is not possible with the sole use of Onsager's linear formalism.
B. The many-to-one and one-to-many picture The phase-space representation of the time-evolution of a thermodynamic system is constrained by its equation of state and yields to global increase of the explored phase-space volume. Considering more closely heat and work transfers between a system and its environment, heat-to-work conversion may be viewed as a process that reduces the explored phase space volume when work is produced, while as the system receives heat, the phase space volume increases. In the limiting case of a pure Hamiltonian dynamics the phase space volume remains constant, in accordance with Liouville's theorem. Now, recalling our remarks on energy conversion stated in the Introduction, a collective response naturally constrains the individual dynamics of the working fluid degrees of freedom, in such a way that the volume of the phase space that can be explored is reduced. On the other hand, the individual excitation of each of the many degrees of freedom (dispersion of energy) of the working fluid can only augment the portion of the phase space that it explores. We may be even more specific: the production of work from heat-to-work conversion, is a "many-to-one" (MO) process: the energy is gathered from the individual degrees of freedom to produce a collective effect; conversely energy dissipation implies a "one-to-many" (OM) process, where part of the energy transferred to the working fluid is dispersed among its individual degrees of freedom. In other words, all the thermodynamic processes that partake in energy conversion can be either seen as of the OM type if they essentially generate entropy, or as of the MO type if there is production of work or a structuration of, e.g., a living system through growth.
Hence, operating a thermodynamic engine in a generator mode thus amounts to engaging the MO processes in the limits imposed by the second law of thermodynamics. The same statement holds for a living body, which needs to produce mechanical work and/or grow. More specifically, production of work from chemical reactions is a global MO process, which is necessarily constrained or limited by one or other OM processes (second law), with a strict global increase of the total phase space volume. In contrast, a pure Hamiltonian evolution may be seen as a one-to-one process. More extensive discussions on phase space analysis of thermodynamic systems and many-to-one and one-to-many processes may be found in, e.g., Refs. [17,18].

C. Thermoelastic coefficients
In terms of thermodynamic variables for a standard thermodynamic engine, entropy S and temperature T are associated with the energy dispersion processes because of their direct link to heat, while pressure P and volume V (or other variables like voltage, electric charge, chemical potential, particle number) may be associated with the energy aggregation processes because of their direct link to work. In a more general framework, we may define a set of coupled variables for dispersive processes (Π m , m), and aggregative processes (Π M , M ). The local formulation of the Gibbs relation reads in standard notations dU = T dS − P dV , with U being the internal energy, can be generalized as follows: where m and M are extensive variables, respectively standing for the dispersive and aggregative processes, and the thermodynamic potentials Π m and Π M are their intensive conjugate variables. The Π m dm term is the generic mathematical expression for the one-to-many transformation, while the Π M dM term represents the many-to-one transformation. Although Eq. (1) merely represents a higher level of description of the thermodynamic system in the sense that the variables m and M are assigned a dispersive and agregative microscopic and a macroscopic character, it provides a quite convenient starting point for the study of the fictitious working fluid. For instance, in a heat engine that involves the coupled transport of energy and matter, M would classically correspond to matter as it characterizes a collective modality in the energy conversion process, and m would characterize the dispersed energy and the related entropy variation. The only physical constraint in the definition of the variables m and M is that their products with their intensive conjugate variables, mΠ m and M Π M , have the dimension of an energy. This aspect is completely in line with Carathéodory's axiomatic formulation of thermodynamics based on the properties of Pfaff's differential forms [19]. In a metabolic description, the thermoelastic coefficients define the conversion ratios and energy capacities for the dispersed energy as reported in Table I. The thermodynamic characterization of the coupling between the energy conversion zone and the muscles, which are the engine equivalent, is of importance to understand how muscular effort is achieved under different situations. We now introduce the coupling coefficient α = −(∂Π M /∂Π m ) M of the thermodynamic potentials Π m and Π M to provide a quantitative means to evaluate the energy conversion efficiency. Note that α is reminiscent of the so-called entropy per particle introduced by Callen in the context of thermoelectricity [22], and by extension, it may be seen as a direct measure of the entropy per unit of working fluid. This coupling coefficient is related to the thermoelastic coefficients (see Table I) and can be explicitly derived provided the working fluid equation of state f (U, Π M , Π m , M, m) = 0, which relates the system's internal energy U to its thermodynamic variables, is given. Indeed, the heat capacity ratio γ = C Π M /C M yields the analogue of the classical isentropic expansion factor, a measure of the "quality" of energy conversion in the sense that the system tends to minimize dissipation; it is given by: In Eq. (2), we made use of the extended Maxwell relations to link the thermoelastic coefficient β to the coupling coefficient α as was done in Refs. [20,21]: so that β = αχ Πm .
The equivalent working fluid has, of course, no material existence per se, unlike, e.g., water in a steam engine, but the meta-description it allows is very helpful to understand the different contributions to the metabolic energy conversion "from local to global". The dimensionless quantity: ZΠ m = β 2 χΠ m C M Π m , which we obtain from the definitions above, characterizes the working fluid as it provides a direct measure of its dispersed-to-aggregated energy conversion efficiency, just like the heat capacity ratio does for a usual gas in a heat engine.

D. Metabolic forces and fluxes
To develop an out-of-equilibrium description of the metabolic process, we transpose the phenomenological linear force-flux formalism approach and Onsager's reciprocal relations. We consider a thermodynamic unit cell (in the general sense), open to exchange of energy and matter with its environment, and where we can assume local equilibrium. Further, we assume a quasistatic transformation to describe the coupled transport processes schematically depicted on Fig. 1. As a matter of fact, both the fluxes J Em and J M (defined below) are conserved quantities but, due to their coupling the potentials Π m and Π M , are modified across the cell. To account for the fluxes and the forces which derive from the thermodynamic potentials, we extend the Gibbs relation (1) assuming quasi-static working conditions: where J U represents the total energy flux with Π m J m ≡ J Em being the dispersed microscopic energy flux, and with Π M J M being the aggregated energy flux, which is proportional to the power produced on the macroscopic level, such as, e.g, mechanical power. The flux J M may be seen as the metabolic intensity necessary to maintain a given metabolic state for the living system. For our purpose we focus on the coupled fluxes J Em and J M , which satisfy: where the off-diagonal kinetic matrix coefficients satisfy Onsager's reciprocity relations L 12 = L 21 [11]. For a living system Π m is the chemical potential of the digested food, which we denote µ according to standard notations, so that J m = µJ S ; and Π M is the macroscopic potential from which derives the muscular force. For simplicity and with no loss of generality we consider a 1D description and we rewrite Eq. (5) as: where we introduced the macroscopic driving force F M = −dΠ M /dx.

Isochemical potential conductivity
In this configuration the gradient of the chemical energy distribution inside the body is zero, so µ is constant and J M = L11 µ F M . This result shows that the metabolic intensity J M is driven by the force F M = −dΠ M /dx which is nothing but the muscular force on the macroscopic level. The metabolic conductivity is then given by The quantity σ is the isochemical potential conductivity, which embodies dissipation when the muscles produce a mechanical activity, including locomotion but also any motionless efforts. So, considering a segment of muscle of length l and section A we obtain the expression of the dissipative resistance R = l/Aσ.

Basal metabolic conductivity
Considering the situation when the animal is at rest with no mechanical activity, the metabolic intensity is zero J M = 0; we denote F M0 the macroscopic force under this specific condition. Then the metabolic power flux J * Em under basal conditions, J M = 0, is where κ J M ≡ κ(J M ), and is the metabolism conductivity. Since J * Em is a measure of the consumed power density when the animal is at rest, κ J M can be considered as a metabolic conductivity under zero load, from which we get the basal metabolic impedance: R E = l/κ J M A, considering a system of length l and section A.

Exhaustion metabolic conductivity
Let us now consider the opposite configuration when the animal experiences an exhausting effort. In this case the animal dissipates all the mechanical power without producing any contribution to the motion. This occurs for metabolic intensities far above the location of the maximal power. In this configuration the organism is no longer able to sustain the required effort. In this case, the acceleration vanishes as well as the net mechanical power transferred to the environment and the net associated mechanical force, so F M = 0. All the metabolic power produced by the muscles is consumed inside the animal. In other words, the animal metabolism is in the shortcircuit configuration, fully loaded, and the metabolic intensity density J M reaches a critical value J X at the exhaustion stage. Then, from Eq. (6), the associated energy flux, J † Em , under the condition ∇(Π M ) = 0 is: where is the exhaustion conductivity.

Metabolic figure of merit
The relationship between κ F M and κ J M follows from the definitions of the transport coefficients: where α is the coupling coefficient introduced in Sec. II. C.; it shows that the most efficient metabolic conditions are those when the basal metabolism is as low as possible while the exhausting metabolism conditions are delayed as long as possible. By most efficient, we mean minimal entropy production, hence maximal possible conversion of the food chemical energy into useful muscular power. The rejected fraction of matter and energy into the sink is here considered as a waste. It is clear that, contrary to the waste heat of steam engine, most of this biological waste should be considered as secondary metabolites for the waste matter, and warming body contribution for the waste energy. All these secondary contributions could be included in a more complex thermodynamic network whose generic building block would be the present Onsager unit cell. The complete treatment is out of the scope of the present article. Equation (12) also indicates that the ratio κ F M /κ J M provides a direct measure of the efficiency of the equivalent working fluid as it contains a term which can be viewed as the metabolic figure of merit, i.e. an extension to biology of the figure of merit in thermoelectricity [23]: In the case of a complete system of length l and section A, see Fig. 1b, the expression becomes Zµ = α 2 R E µ/R.

Degree of coupling and energy conversion efficiency
In their seminal article Kedem and Caplan derived the following expression of the coupling parameter between the two fluxes involved in the conversion process [24]: , involving the kinetic coefficients. So, the figure of merit and the coupling factor q are perfectly equivalent in terms of system performance: the higher they are, in absolute value, the better the energy conversion system is. This is can be evidenced by the derivation of the local maximal efficiency of the conversion process as, The figure of merit Zµ is therefore a direct quantitative indicator of the performance of the metabolic system it characterizes.
6. Many-to-one coupling coefficient As J M = 0 under the basal metabolism configuration, the expression of the "local to global" coefficient α may be rewritten as: so only three transport parameters are needed to study the metabolic machine: two conductivities, respectively for the aggregated energy flux, σ, and dispersed energy flux, κ J M , and a coupling parameter between these two fluxes, α.
Note that in order to study how a metabolic machine operates in a realistic configuration, hence considering the whole biological system, the following points must be addressed i/ expressing the local energy budget; ii/ integrating the local expressions; iii/ taking into account the boundary conditions. While thermodynamic models of biological systems have been extensively developed [1][2][3][4][5][6][7][8][26][27][28], including the thermodynamic network approach and bond-graph methods [29,30], a proper account of the three above points, is sometimes missing, leading to an incomplete thermodynamic description of the biological energy conversion.

III. THERMODYNAMICS OF METABOLISM
A. The metabolic machine

Local energy budget
Replacing the kinetic coefficients L ij in Eq. (5) by the transport coefficients, σ, κ J M and α, yields the following expressions for the fluxes J M and J Em : and assuming constant parameters, the gradient of J Em reads As energy conservation imposes a constant flux J U in Eq.
(4), we get and combining Eqs. (16) and (17), we finally obtain the local energy budget: 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 2 M /σ. In other words, ∇J U = 0 is a local finite-volume condition, and the local working conditions of the metabolic device are totally defined by the local budget and the three transport coefficients σ, κ J M and α. When considering a fraction of an animal of characteristic length l and section A, the conductivities defined on the local level appear in R = l/σA and the metabolic resistance Let us now extend the analysis to the management of this equivalent working fluid inside an organism or part of it.

Metabolic flux
Assuming that the metabolic processes take place in the volume Al of the organism, the complete energy budget is obtained after integration of Eq. (18) The energy flux Φ = AJ Em can hence be expressed as where I M = AJ M is the metabolic intensity so that the metabolic flux may read from Eq. (17): The constant is determined by considering the case when the system does no produce power, hence when I M = 0. Now, in order to complete the description of the system and the model, we turn to the definition of the boundary conditions. B. Metabolic system

Powers
As a thermodynamic device, the metabolic conversion zone defined by the element of lenght 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 condition of the metabolic device. The complete thermodynamic system is represented on Fig. 2, which describes a quite general situation. Due to the presence of dissipative couplings between the conversion zone and the reservoirs, the reservoirs chemical potentials µ + and µ − are modified and become µ +M and µ −M as shown on Fig. 2. FIG. 2. Thermodynamic system: a) general configuration. b) simplified configuration for low-duration, steady-state efforts. A dissipative coupling modifies the chemical potential difference across the conversion zone.
The metabolic converter is connected to a reservoir and a sink by two dissipative elements of resistances, R + and R − . These resistances, together with R E constitute a bridge of resistances which allows to consider all the possible boundary conditions between their two limiting cases, namely i/ the Neumann conditions, where the currents are imposed, considering high, diverging values of R + and R − ; ii/ the Dirichlet conditions, where the potentials are imposed, considering vanishing values of R + and R − .
From the general expression Eq. (21), we get the expressions of the incoming Φ + = Φ(0) and outgoing Φ − = Φ(l) fluxes: where the constant term ∆µ R E comes from the equality Φ + = Φ − in the configuration I M = 0. And, accounting for the connections to the reservoirs, we also get two additional expressions for the fluxes which yield the following simple expression for the metabolic converter output power: As mentioned above, the constant appearing in Eq. (21) is obtained from the condition P = 0, hence Φ + = Φ + , which yields a constant equal to: ∆μ/R E = (µ +M − µ −M )/R E . The expressions (22), (23), (24) and (25) now give a complete set of equations for a proper description and analysis of a given metabolic situation. The general solution may be found without difficulty but it is cumbersome. So, as an illustration, we consider the situation of an effort of reasonable duration, neglecting the drawback effects of the waste fractions. In this case, we may consider R − ≈ 0 so µ −M ≈ µ − , and obtain where I T = R E +R+ αR+R E , is a threshold intensity above which the metabolic force collapses; and the metabolic power delivered during a physical effort is now expressed as: is the available force when the organism produces a moderate effort; it can be interpreted as the driving force when considering vanishing metabolic intensities, which reduces to a constant basal force F B = α∆µ R E R++R E , also called isometric force [31], for I = 0. As expected this basal force is proportional to the converted fraction of the chemical energy difference ∆µ modulated by the resistance bridge value: R E R++R E . The output power contains the power spent for 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 animal motion, the physical effort simply corresponds to the production of the mechanical power necessary for moving. In this case, the metabolic intensity can be expressed directly from the mechanical velocity of the system as I M = v. The metabolic intensity is also non-zero for animals supporting heavy loads, while standing still. So, the concept of metabolic intensity accounts for various types of efforts that a purely mechanical description cannot.
The power P has two zeros, which correspond to two roots: I M = 0 in the absence of effort, and I X in a situation of exhaustion. According to Eq. (29), the maximum power that can be produced lies between the two zeros.
In the most favorable case where R = 0, the exhaustion intensity reaches its maximum value: I X = ∆µ αµ−R+ . The value of the residual friction resistance, induced by the feedback due to the coupling conditions, can be deduced: So, the expression of R fb clearly shows the feedback origin of this dissipative resistance. In short, R fb is an additional dissipation term, due to the presence of mixed boundary conditions. The figure of merit can then be rewritten as This result shows that maximum efficiency is obtained by maximizing the resistance ratio R fb /R; in other words, improving the efficiency of the conversion of the energy from its chemical input to its mechanical output requires the minimization of the mechanical frictions compared to the dissipative contribution of the chemical flux. The characteristic intensity I T is the threshold intensity below which the force does not depend on the metabolic intensity. This particular phenomenon has been reported in the literature, where it is observed a speed-dependence of the basal power for flying hummingbirds [32,33]. One may notice that the output power P = F M I M is completely independent of the detailed nature of the effort. In particular, any effort which does not involve the motion of the organism can also be described using our approach. FIG. 3. a) Generic power plots: external mechanical power P (straight line), input Φ+ (dash-dotted) and output Φ− (dash) metabolic fluxes. The shaded area corresponds to the region where the organism degrades more mechanical power than it uses for the effort. b) Efficiency versus power response. As the metabolic intensity IM increases, the efficiency η reaches its maximum before the mechanical power reaches its own maximum.

Energy currents
Using the set of equations (27), (28), and (29), we can now plot the generic response of an organism to an effort as shown on Fig. 3a where the input, Φ + , output, Φ − and external mechanical power P are reported. We also report on Fig. 3b the efficiency versus power plot: the maximal efficiency is obtained for a metabolic intensity slightly smaller than that needed for the production of maximal mechanical power, which is reached at the cost of efficiency. Depending of the metabolic parameters, every specific organism may undergo a trade-off between its maximal efficiency and its maximal power, or minimum production of waste.

IV. METABOLIC ENERGY CONVERSION UNDER MUSCLE LOAD
A. Physical efforts Figure 4 summarizes the principles of our thermodynamic model of metabolic power generation from chemical power transferred into a mechanical power. The complete system is composed of two parts. The first one is the biochemical converter which delivers an output power P and produces waste as well as and secondary metabolites, denoted Φ − [see Eq. (28)]. The produced power is given by the product of a mechanical force F M acting at a given metabolic intensity I M which corresponds to a muscle bundle velocity v. It should be noticed that this mechanical power is partly dissipated inside the organism through the internal resistance R, and that the output power is The available metabolic power now assumes two possible expressions: P = F M v and P = F M I M . Depending on the particular muscular effort required, which, e.g., would correspond to a given gait of a moving animal, the number N of elementary bundles of muscle fibers involved in the production of the mechanical effort varies, which is in line with Henneman's size principle of recruitment of motor units [34]. It is worth noticing that the quadratic dissipation RI 2 M term is then strongly reduced with an increase of number of acting bundles. For the same reasons the basal metabolism is proportional to N . For a given organism withstanding an effort, there is a trade-off between the number of bundles in action and the intensity of the effort. It follows that v = N I M measured from different gaits provides the relative number of bundles involved in the metabolic response of and organism.

B. Hill's muscle load model
Our model of the response of metabolic conversion of chemical to mechanical power must match with the most classical description of the muscle, F (v) = c (v+b) − a, which is that proposed by Hill in 1938 [12]. With no additional hypothesis a straightforward derivation yields the thermodynamic correspondence of the three constants from (29) The term a is usually called the "coefficient of shortening heat" since it drives the amount of dissipation. It is usually found to be constant, from which we conclude that αµ − RI M which is in close agreement with the maximization of the figure of merit of the system discussed above. We report in Fig. 5 the force-velocity response for three different ratios of R/R fb . The response of a traditional system is typically characterized by R/R fb ≈ 1, where the intrinsic resistance R of the system dominates. As the resistance R decreases, the feedback term contribution increases and finally drives the overall response with R/R fb ≈ 0. We conclude that the dissipation appears to be controlled by the boundary condition factor R+ R++R E and the figure of merit Zµ. As for a thermodynamic engine the limitation of the performance of the muscle is not limited by the mechanical viscosity but is rather driven by the feedback term R fb , which defines the exhaustion limitation F = 0 as Under exhausting conditions, the organism cannot provide output mechanical power any longer as all the power is dissipated inside the body. Figure 5 summarizes the behavior of thermodynamic machines according to a parametrization given by the figure of merit. This behavior lies between the two extremes, i) the purely dissipative machine, which has a figure of merit equal to zero and for which the response is strictly linear, and ii) the machine, which has a very large figure of merit and for which the characteristic is strongly curved. In other words, the "muscle machine" is presented as a highly optimized thermodynamic machine, in which the dissipation reaches its minimum value, imposed by the conditions at the limits of the coupling. The force-velocity response of a varying number of bundles leads to the final response of a given organism. Assuming a bundle of fibres to be equivalent to a thermodynamics engine, we easily transpose to the situation where N bundles are working in parallel. A straightforward result is the amplification of the basal force and of the exhausting intensity by a factor N , and the decrease of the total resistance R fb + R by the same factor.

V. CONCLUDING REMARKS
Owing to a long history of studies, metabolism is by now certainly well understood in many of its fundamental aspects, including the universality of the metabolic chemical reactions [35], but its study remains highly topical, notably in the field of biochemistry where questions related to the origins of the well-known Krebs cycle and the early physicochemical conditions on Earth permitting the emergence of metabolic-like reactions are discussed [36,37]. In the present article, we studied metabolism from the viewpoint of thermodynamics, developing a model of energy conversion under muscle load and of the response of a biological system under effort. The sequences of metabolic reactions yielding the synthesis of all molecules needed by the cells (lipids, nucleic acids, and amino acids) and the energy in the form of heat or work for motion, entail highly complex processes making a few-parameter model development quite challenging. Our approach was to build a theoretical framework based on a generalized meta-formulation of linear nonequilibrium thermodynamics to characterize the production of metabolic energy and its use under effort, assuming that the feedback effects in biological systems permit an analysis based on an effective linearized model [13,14].
To adapt Onsager's force-flux formalism to the study of metabolism we resorted to the introduction of an effective metabolic equivalent of a thermodynamic working fluid, which is not a physical substance per se but provides a theoretical ground to compute generalized thermoelastic coefficients, effective transport parameters and a metabolic figure of merit. This figure of merit characterizes the ability of a given organism to produce useful mechanical power from the chemical potential energy, in order to sustain a muscular effort. It depends both on the intrinsic performance of the pseudo-working fluid and also on the boundary conditions that constrain the organism. The use of the one-to-many and many-to-one picture and the accompanying definition of the generic thermodynamic extensive variables m and M (for energy dispersive and aggregative processes), and their conjugate thermodynamic potentials Π m and Π M allowed to develop a compact framework within which we may no longer focus on the concept of heat but rather on dispersed energy, which brings a necessary generalization to study metabolism: the effects of the OM processes do not simply boil down to heat production, but characterize various forms of energy dispersed such as secondary metabolites. Interestingly, the metabolic intensity term shows that the reference level is not equilibrium but the basal point at which a living system is in a steady rest state. The next challenge is to investigate the power budget of locomotion and the related oxygen consumption. In this appendix, we show that the so-called maximum power principle, frequently invoked in biology, can be understood as a specific configuration of an out-ofequilibrium system, and should not be considered as a fundamental principle. For this purpose, it is convenient to move from the concept of work to that of exergy, that is the maximum amount of work that can be recovered for a given heat-to-work conversion, and which decreases as the system gets closer to equilibrium. Stated simply, exergy is a measure of the useful share (or "quality") of the energy involved in a thermodynamic transformation. At equilibrium, exergy is necessarily zero.
The capacity of out-of-equilibrium systems to absorb energy was discussed by Lotka [45][46][47] before a proper framework for nonequilibrium thermodynamics was put on firm grounds. As recently mentioned by Sciubba [38], Lotka's assumption referred to the capacity of a given system to maximize the capture of exergy, which is the free energy fraction of incoming energy. As we show below, Lotka's description can in fact be reconsidered in terms of impedance matching. However, some misunderstanding of Lotka's analyses gave rise to the so-called maximal power principle (MPP), which is sometimes considered as the fourth principle of thermodynamics. But, strictly speaking, up to now, the MPP never received final proof [39][40][41][42][43][44]. On the contrary, there is now an increasing consensus to describe out-of-equilibrium systems not in terms of MPP, but simply using the classical principles of thermodynamics accounting for specific boundary conditions [9].
We now reconsider this question in light of the results of our model, using the generic picture of Fig. 2. Following Lotka we consider the exergy circulation. A straightforward analysis of the system reveals that the exergy is destructed in three different processes: i/ the incoming energy quality is degraded by the presence of R + ; ii/ the basal metabolism is governed by R E + R + which reduces the available exergy for other activities; iii/ the internal dissipation term R reduces the maximal accessible power. To reduce the effects of these dissipation sources, we may first consider a configuration where R + = 0 (although it is obviously unrealistic). The equation (29) becomes P = αI M (µ + − µ − ) − RI 2 M which means that the animal activity would be only limited by its internal dissipation RI 2 M . In addition, the basal metabolism term b M = µ+−µ− R E +R+ increases. In other words the gain obtained by the easiest access to the resources, i.e. R + = 0, is counterbalanced by an increase of the basal metabolism. One can see here that a simple extremal exergy analysis is far to be sufficient to explain the optimal physiological point, if any.
As the condition R + = 0 is unrealistic, we turn to the other two cases with R + = 0. According to process ii/ above, a reduction of the basal metabolism would improve the exergy budget. Then, in the configuration R + ≈ 0, and R E /R + 1, Eq. (29) now gives which means that the available power is vanishingly small, which is a totally useless configuration. It then becomes obvious that the reduction of the basal metabolism under a certain threshold is not desirable unless it is imposed by some external constraints, such as, and mainly, food resources. In short, there is a necessary trade-off between the value of R E and R + in order to reach an optimum. Clearly, the implicit variational principle for a maximum of the exergy is necessary but not sufficient, and the feedback induced by the boundary conditions leads to a distribution of the exergy degradation locations [14]. From an evolutionary perspective, the final distribution depends on the degrees of freedom of each system component, and suboptimal configurations may exist due to absence of any degree of freedom on the local level. From a statistical point of view it is clear that a complex system, composed of a network of various engines, may present various optimal configurations. In other words, there is a matching under constraint of the global exergy impedance of the system, allowing the optimal, but not necessary maximal, exergy flow. Let us finally discuss briefly the question of the extremal principle, deriving an impedance considering the setup depicted in Fig. 6. The energy flux is approximately Φ ≈ µ+−µ− R E +RΣ , where R Σ = R + + R − , and the chemical potential difference ∆µ = µ + − µ − directly governs the output power. Note that strictly speaking ∆µ = 0 correspond to the dead body condition. Using the argument of the minimization of the degradation of the exergy we expect a minimization of R Σ . In addition, if the ratio R E /R Σ → 0 we see that ∆µ M → 0 and both the output power and the efficiency vanish. Now, if we consider the R E /R Σ → ∞ configuration, the efficiency becomes maximal, but the power vanishes again. In both cases we experiment a minimal power principle, with maximal or minimal entropy production. If we now consider R Σ = 0 then the value of R E does not matter any longer, and the output power is maximal, possibly diverging if R decreases. This configuration seems similar to the so-called MPP configuration, and it actually is, but, as said above, the boundary condition R Σ = 0 is not a realistic one. If we finally consider R Σ ≈ 0 then the non vanishing ∆µ condition imposes a non-zero value for R E . Using elementary algebra, the optimal configuration is R E = R Σ where the output power is now maximal. If R E > R Σ the process favors the energy conversion efficiency at the expense of power; conversely, if R E < R Σ the process favors the output power at the expense of the efficiency. Therefore, there is no extremal principle for such out-of-equilibrium systems, but there exists a perfectly determinate configuration, defined by the boundary conditions imposed on the system.