Metabolic cascades of energy

M.A. Fardin1, 2, ∗

1Universite de Paris, CNRS, Institut Jacques Monod, F-75006 Paris, France2The Academy of Bradylogists.

(Dated: December 16, 2021)

Life has a special status, it even has its own science: biology. In many ways, the logic of life seemsto differ from that of atoms, molecules, planets, or any other ‘inanimate object’. However, life isincreasingly measured using quantities shared by all sciences, like mass, force, energy or power.An analysis of the dimensions of these quantities provides powerful ways to infer the relationshipsthey might have with one another. Here we show that a dimensional analysis of the metabolic lawsconnecting the characteristic powers and masses of living organisms offers new ways to understandthe deep connections between the chemistry of microscopic molecules and the physics of macroscopicobjects bound by gravity. This analysis reveals a link between metabolism and the cascades of energyobserved in turbulent flows, opening new perspectives for both fields.

The rate of energy expenditure of living organisms isusually called the ‘metabolic rate’ [1]. As an energy perunit time it is a power P , with dimensions M.L2.T −3,and its standard unit is the Watt. It can be expressedin kg.m2.s−3, but is more often expressed in J/s, orkcal/day, i.e. in some unit of energy over some unit oftime. Beyond any particular choice of units, the dimen-sions remain.

There has been a considerable amount of discussion inthe literature on the relationship between the metabolicrate P and the mass m of the organisms across kingdomsand scales [1–5]. Obviously, an elephant eats more thana mouse, but how much exactly? Such metabolic lawsare particularly interesting because they can be used toinfer other relationships between mass and growth rate,mortality rate, or abundance, to cite just a few exam-ples [2, 5].

There is a mass in the dimensions of any power(M.L2.T −3), and so it is natural to ask how this dimen-sion of mass may be connected to the mass m. Amongthe many types of relationships between P and m thathave been investigated, one continues to have a particu-larly stimulating impact: P ∼ m3/4. This relationship isknown as Kleiber’s law [6]. In some instances, it has beenverified to a large extent [3], in others it has served as apoint of comparison to design alternative metabolic re-lationships [7]. Theoretical monuments have been builtin order to explain or refute it. Our purpose here isnot to review this extensive literature, but to show thata few dimensional arguments can be used to navigatemore easily through it. Once carried out, dimensionalanalysis actually reveals stimulating connections betweenmetabolic laws and turbulence spectra. Consequencesof these connections run wide, as we shall progressivelyshow throughout this article.

∗Corresponding author ; Electronic address: marc-antoine.fardin@ijm.fr

The metabolic law behind turbulence

Let us assume that the relationship between metabolicrate P and mass m is a power law. In doing so, weneglect the ‘curvature’ that data might present in loga-rithmic scale [13]. This amounts to neglecting so-called‘logarithmic corrections’, providing a degree of approxi-mation that we acknowledge, but that will be sufficientfor our purpose. We will write P ∼ mα, where the sign‘∼’ means that if one were to plot log(P) vs log(m) onewould find a straight line of slope α. The intercept log(K)with the log(P)-axis would give rise to the prefactor ofthe power-law, such that we have P = Kmα. We willcall such P (m) relationships ‘metabolic laws’, even whenwe venture outside the realm of living organisms.

Whereas a lot of contributors on this topic have de-voted their efforts to finding the value of α, the prefactorK has not received the attention it deserves. Both Kand α are intimately connected if one analyses the dimen-sions of the equation. As is customary, we will write [K]to denote the dimensions of a quantity K. Since [P ] =M.L2.T −3, and [m] = M, then [K] = M1−α.L2.T −3.This equation is true regardless of the value of α.

One value of α is special by construction. Indeed, α =1 removes the mass dimension of K, [K] =M0.L2.T −3.This would give a metabolic rate proportional to mass:

P = K1m (1)

where we use the subscript 1 to keep in mind that thiscoefficient corresponds to α = 1. In this elementary case,what could be the interpretation of the proportionalityconstant K1? Because the dimensions of K1 only involvespace and time we say that it is a kinematic quantity.For comparison, a speed would have dimensions L.T −1,a coefficient of diffusion would have dimensions L2.T −1,or an acceleration would have dimensions L.T −2. In con-trast, K1 has dimensions L2.T −3. This quantity is notas well known as a speed or a coefficient of diffusion, butit appears prominently in the field of turbulence [14, 15].In this context, one studies the complex flows made upof many swirls as seen for instance in some tumultuous

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FIG. 1: Conversion from power spectrum (a) to metabolic law (b) for representative data from the literature on the turbulenceof various fluids in different flow geometries. (a) The wind tunnel data correspond to ρ ' 1.2 kg/m3 [9, 10]. The turbulencewas excited by a grid, and the different trends corresponds to different types of grids or to measurement realized at differentdistances downstream from the grid. The turbulence data of liquid helium correspond to ρ ' 148 kg/m3 [12]. The tidalchannel data correspond to underwater turbulence in the sea, with ρ ' 103 kg/m3 [8]. The swirling flow data were obtained onmixtures of water and glycerol, with ρ ' 998, 1000, 1166 kg/m3 [11]. (b) The same turbulence data are plotted as metaboliclaws, together with the metabolic rates of a vast range of organisms from bacteria to large mammals and trees, reproducedfrom Hatton et al. [5]. More details on the different taxonomic groups will be given in Fig. 2. On both graphs (a) and (b), thediagonal grey lines correspond to different values of transfer rate ε, expressed in m2.s−3.

rivers or in puffs of smokes and clouds.Almost tautologically, one can call K1 the power per

unit mass, since K1 = P/m. In the context of turbulence,a coefficient with the same dimensions than K1 is usu-ally called the ‘rate of energy transfer’ or the ‘dissipationrate’, and the symbol used is ε. This rate of dissipation isthe parameter of the so-called ‘Kolmogorov energy spec-trum’, which is usually expressed as:

E(k) = CKε23 k−

53 (2)

This equation might seem odd to those unfamiliar withit, but we shall see that it is essentially equivalent toP = εm. The term ‘CK ’ is the dimensionless ‘Kol-mogorov constant’, with a value around 1

2 [16]. In thefollowing, we will ignore such dimensionless coefficient ‘oforder 1’ and write E(k) ∝ ε

23 k−

53 , where the sign ‘∝’ is

here to remind us that we neglect these coefficients. Thesymbol ‘k’ refers to the ‘wavenumber’, with dimensions[k] = L−1. The ‘energy spectrum’ E(k) is a kinematicquantity constructed from the fluctuations of the veloc-ity of the flow over a size ` ∝ k−1. Its dimensions are[E ] = L3.T −2. In the context of turbulence, one typi-cally measures how the many-sized vortices in the fluidaffect its flow. The apparent complexity of the scalingE(k) ∼ k−

53 comes from translating simple assumptions

to the particular frame of reference most commonly usedby experiments.

To show how Kolmogorov’s energy spectrum can beexpressed as a metabolic law, we can first translate thevariable k into a mass m. If the spectrum depends onk, then it depends on the size ` ∝ k−1, and on the massm ∝ ρ`3 of a parcel of fluid of size ` and density ρ. Thus,

we have k ∝ (ρ/m)13 . On the left-hand side of Eq. 2, how

can we connect E(k) to P , m and ρ? The answer is givenby dimensional analysis:

E ∝ mβP γρδ (3)

An equation like this is just a formal way of saying thatE depends on the mass m, the power P , and the densityρ. We can translate the equation into an equation on thedimensions:

L3.T −2 =Mβ(M.L2.T −3)γ(M.L−3)δ

M0.L3.T −2 =Mβ+γ+δ.L2γ−3δ.T −3γ (4)

We then have a system of three equations, one for eachdimension: M : β + γ + δ = 0

L : 2γ − 3δ = 3T : −3γ = −2

(5)

This system can easily be solved to get β = − 19 , γ = 2

3

and δ = − 59 . We thus have translation formulas to go

from turbulence spectra to metabolic laws and vice-versa:

k ≡ (ρ/m)13

E ≡ m− 19P

23 ρ−

59⇔ m ≡ ρk−3

P ≡ ρE 32 k−

12

(6)

By translating the energy spectrum and wavenumber inEq. 2 one recovers Eq. 1, with the symbol ε used insteadof K1:

E ∝ ε 23 k−

53 ⇔ m− 1

9P23 ρ−

59 ∝ ε 2

3 (ρ/m)−59 ⇔ P ∝ εm

(7)

3

Both equations encompass the same statement but ex-pressed in two distinct systems of units. The metabolicvariables P and m provide a quite synthetic way to ex-press Kolmogorov’s spectrum.

In practice, the equivalence between the metabolic lawin Eq. 1 and the energy spectrum in Eq. 2 means thatfor a given density ρ we can use the translation formulasto express a metabolic law into an energy spectrum, ora spectrum into a metabolic law. These are just twoequivalent ways to represent data. For instance, in Fig. 1we compare a few turbulence spectra from the literatureexpressed as E vs k, or as P vs m.

As is apparent, the data do not systematically followthe E ∼ k−

53 scaling, i.e. the data points extend beyond

P ∼ m. Generally, the data points follow P ∼ m onlyin the range m1 < m < m2, i.e. between `1 < k−1 < `2,with m1 ∝ ρ`31 and m2 ∝ ρ`32. The values of these boundsvary from one experiment to another, and so does thevalue of ε characterizing the data between these bounds.For instance, in Fig. 1, the data from the tidal chan-nel correspond to ε ' 5 10−8 m2.s−3 for `1 < k−1,with `1 ' 1 cm [8]. No bound `2 could be identi-fied. In contrast, some of the wind tunnel data provideboth bounds. For instance, the cyan dots correspond toε ' 10−1 m2.s−3 for `1 < k−1 < `2, with `1 ' 2 mm and`2 ' 2 cm [10]. For a particular experiment, the bounds`1 and `2 respectively correspond to the crossovers to-ward what are usually called the ‘dissipation scale’ and‘the integral scale’. We will more neutrally call these the‘microscopic’ and ‘macroscopic’ scales.

What sets the values of the transfer rate ε and of thebounds `1 and `2 in a particular experiment? The sizes`1 and `2 are identified graphically as the points of depar-ture from the scaling E ∼ k−

53 . They are the last points

abiding to the scaling, and so we necessarily have:

ε ∝(E3i`5i

) 12 ∝ Pi

mi(8)

where i = 1, 2. These expressions explain why ε is oftencalled ‘the dissipation rate’. Indeed, if ε is expressed fromthe lower bound `1 then it is associated with the powerP1, which is indeed where dissipation occurs in standardfluids. However, one could very well call ε the ‘injectionrate’ if defined from the integral scale where a power P2

is generated. In practice, the control parameters for agiven experiment are usually connected to P2 and `2.

Universality of the metabolic power density

In turbulent studies, the value of ε typically variesfrom one experiment to another depending both on thefluid properties and on the way energy is fueled into thesystem at large scales. It can be expressed externallyfrom the macroscopic scale, or internally from the micro-scopic scale. This versatility is no surprise if one writesε ∝ P/m ∝ P/ρ`3. We can define the power densityΠ ∝ P/`3, i.e. the power per unit volume, in the same

way that ρ ∝ m/`3 is the mass per unit volume. Withthese definitions, we have:

ε ∝ Π

ρ(9)

Whereas P and m are extensive properties, Π and ρ areintensive, i.e. they are expected to remain constant overa broad range of sizes. For a fluid like water in normalconditions, the density will remain constant over a verylarge range of size, down to atomic scale. However, thepower density Π will be constant over a smaller range, setby `1 and `2. In stark contrast, living organisms seem tobroadly share the same universal metabolic power den-sity, over a very large range of sizes [4, 5].

In Fig. 1b we plot the metabolic rates of a vast range ofliving organisms obtained from Hatton et al. [5]. On thesame P vs m graph, we also reproduce some turbulencedata from Fig. 1a. The metabolic data as well as theturbulence data express how the rate of energy transfer(i.e. the power P ) depends on the mass m (and so size

` ∝ (m/ρ)13 ). In this analogy, each organism is like a

fluid vortex, or each vortex is like an organism.For living organisms, the rate of energy transfer per

unit mass ε typically varies from 0.1 to 10 W/kg [5], asmall range in comparison to turbulence studies. If weconsider a rough average ε0 ' 1 W/kg, and a density al-ways close to that of water ρ0 ' 103 kg/m3, we obtain afairly universal metabolic power density Π0 ' 103 W/m3.How can this value be understood? The insight from tur-bulence studies given in Eq. 8 would suggest that one canobtain equivalent expressions for Π0 (and so ε0) by con-necting it to the bounds of the range of validity of thescaling P ∼ m. A microscopic expression of Π0 can beobtained by considering the smallest living organisms,whereas a macroscopic expression can be obtained byconsidering the largest ones. This is what we shall donow.

Microscopic scale and viscosity

Using the power density, Kolmogorov’s scaling canbe written as E ∝ (Π/ρ)

23 k−

53 . Using the power and

mass variables, this is simply P ∝ (Π/ρ)m, i.e. itis just expressing the extensive-to-intensive relationshipP/m ∝ Π/ρ. Part of Kolmogorov’s insight was to supple-ment the intensive quantities Π and ρ by a third quantityη (M.L−1.T −1), which is the viscosity of the fluid [14].With this last quantity, one can build a complete set ofunits, equivalent to M, L and T . Using dimensionalanalysis one can show that Π, ρ and η can be combinedin unique ways to construct a time scale τK , a length `Kand a mass mK ∝ ρ`3K :

`K ∝( η3

ρ2Π

) 14

(10)

τK ∝( η

Π

) 12

(11)

4

Note that one can recast the first equation as `K ∝(ν3/ε)

14 [15], where ν ∝ η/ρ is the kinematic viscos-

ity (momentum diffusivity). One can also express Kol-

mogorov’s length as `K ∝ (ντK)12 or `K ∝ (ε/τK)

12 . For

the turbulence data in Fig. 1, the Kolmogorov’s lengthsranged from a few microns to a few millimeters [8–11].

For the metabolic scaling of organisms, we can use thereference values Π0 and ρ0 to infer `K ' 5 10−3η

34 . In

other words, the Kolmogorov length associated with theaverage parameters of the metabolic scaling is directlyconnected to the value of the viscosity η. The smallestbacteria in Fig. 1b correspond to a mass m ' 10−17 kgand size ` ' 10−7 m. The average metabolic law P ∼ mextends at least down to this scale, so it is the highestpossible value for Kolmogorov’s size, which would thencorrespond to a viscosity η ' 5 10−7 Pa.s. Lower choicesof `K , would yield even smaller values of viscosity. Forcomparison, the viscosity of water is two thousand timeslarger than the inferred viscosity from the size of bacteria.Blindly applying Kolmogorov’s units leads to aberrantvalues of viscosity, far from the material properties ofliving organisms. We need a different interpretation ofthe microscopic scale.

Microscopic scale and thermal energy

In turbulence studies the microscopic length scale isvariable, whereas for the metabolic rate its value can beset to the size of the smallest living organisms. Let us call`T this microscopic length, associated with a mass mT =ρ`3T . Instead of using Kolmogorov’s units to describe themicroscopic scale, one may use ‘thermal units’, based onthe density ρ, the thermal energy E = kBT , and thelength `T . Using dimensional analysis one can show thatthe associated time scale is:

τT ∝(ρ`5TE

) 12

(12)

In this system of thermal units, one can obtain a powerdensity:

ΠT ∝ mT `−1T τ−3

T ∝( E3

ρ`11T

) 12

(13)

If we assume the reference density ρ ' ρ0, room tempera-ture such that E ' 2 10−21 J, and a microscopic size `T '400 nm, the thermal power density is ΠT ' 103 W/m3,which is similar to the average Π0 for all living organ-isms. As is apparent in Eq. 13, the microscopic powerdensity depends most sensitively on `T . Changes of tem-perature or density have comparatively small effects. InFig. 2a, we give the values of power density predictedby the thermal units for ρ ' ρ0 and room temperature,for `T ' 200 nm or `T ' 1 µm, which respectively giveΠT ' 5.7 104 W/m3 and ΠT ' 8 W/m3, a range thatencompasses most of the metabolic rates of organisms.

One may wonder what sets the value of `T . This ques-tion is more easily answered by looking first at the otherend of the spectrum.

Macroscopic scale and gravity

At the microscopic scale, no clear departure fromP ∼ m could be identified, so we used `T to establishan expression of the power density based on the thermalenergy. At the macroscopic scale, the metabolic data ac-tually show significant departure from P ∼ m, in partic-ular for mammals, as shown in Fig. 2b. These data setsare better represented by Kleiber’s law, where P ∼ m

34 .

To understand this scaling, one may return to an oldquestion: why giants don’t exist? This question has stim-ulated the development of dimensional analysis since itsinception, and was famously addressed by Galileo [17].

Schematically, the size of large living organisms canbe understood from a balance between the weight, whichtends to compress the organism, and the internal ‘pres-sure’ maintaining the shape of the organism. The weightper unit volume is Ψ = ρg0 (M.L−2.T−2), where g0 '10 m.s−2 is the acceleration of gravity. The ‘pressure’Σ is the elastic modulus of the organism (M.L−1.T−2).The size is then given as `? ∝ Σ/Ψ. Taking ρ ' ρ0, and amodulus Σ ' 105 Pa, intermediate between muscles andbones, one gets `? ' 10 m. Of course, slightly differentdensities or elastic moduli would change this value, as isthe case for trees.

In the framework of metabolic laws, Galileo’s argu-ments can be adapted by looking for a relationship be-tween power on one side, and mass m, force density Ψand power density Π on the other. Dimensional analysiswould lead to:

P ∝ Π52 Ψ− 9

4m34 (14)

This metabolic law is an example of Kleiber’s law, whereP ∼ m

34 . Assuming Π ' 6Π0 and Ψ0 = ρ0g0 '

104 kg.m−2.s−2, this metabolic scaling is shown in Fig. 2bto fit well with the metabolic rates of mammals. Slightlydifferent values of Π can be used to fit different taxonomicgroups. Note that in this regime, Π 6= ερ.

Dimensionally, the ratio between a power density anda force density is a speed, c ∝ Π/Ψ. In the case of mam-mals, this speed is c ∝ 6Π0/ρ0g0 ' 1 m/s. Using thisdefinition, one may rewrite Eq. 14 as:

P ∝ Π14 c

94m

34 ∝ Π

ρm(mG

m

) 14

(15)

Such law can also be obtained directly from dimensionalanalysis, since it is the only metabolic scaling dependingsolely on a power density Π as well as on a characteristicspeed c. In the right-most equation, this metabolic lawis expressed as a correction to the linear scaling, wherethe macroscopic mass mG is defined as the intersectionbetween this Kleiber’s law and the linear scaling, suchthat mG ∝ ρ4c9/Π3 ∝ ρ`3G, with `G ∝ c3/ε.

5

FIG. 2: Illustrations of the microscopic (a) and macroscopic (b) bounds of the metabolic rates of living organisms. (a) Twodifferent choices of microscopic length `T lead to different values of the power density Π. Red dashed lines correspond to`T = 1 µm, whereas brown dotted-dashed lines correspond to `T = 200 nm. The blue lines reproduce some of the turbulencedata from Fig. 1. (b) Close-up on the metabolic rates of larger organisms. The highlighted symbols correspond to mammals,

illustrating the crossover from P ∝ (Π/ρ)m (continuous line) to Kleiber’s law, where P ∝ Π52 Ψ− 9

4m34 (dashed line), assuming

ρ ' ρ0, Π ' 6Π0 and Ψ0 = ρ0g0 ' 104 kg.m−2.s−2. In (a) and (b) the different shades of green correspond to differenttaxonomic groups; see Hatton et al. for details [5].

In analogy with Galileo’s approach, one can assumethat the crossover to Kleiber’s law can be expressed asa ratio of elasticity and force density, `G ∝ `? ∝ Σ/Ψ.If such assumption is combined with the definition `G ∝ρc3/Π, then the speed c can be expressed as a ‘soundspeed’:

ρc3

Π=

Σ

Ψ⇒ c =

(Σ

ρ

) 12

(16)

For mammals, the value c ' 1 m/s inferred empiricallywould give an elastic modulus Σ ' 1 kPa, which corre-spond to soft tissues. Invoking such speed of mechani-cal waves was central to some theoretical approaches toKleiber’s law [3].

Overall, whereas the microscopic scale was well char-acterized by thermal units, the macroscopic scale is bet-ter described by what we might call ‘Galileo’s units’, de-pending on Ψ, ρ and Σ. In addition to `G and mG, thefollowing time scale complements the system of units:

τG ∝(Σρ)

12

Ψ(17)

With these units, the power density can be definedmacroscopically as:

Π ∝ mG`−1G τ−3

G ∝ Ψ(Σ

ρ

) 12

(18)

Assuming Ψ ' Ψ0 and ρ ' ρ0 gives Π ' 3 102Σ12 . For

instance, if Π ' Π0, this gives Σ0 ' 10 Pa.From this macroscopic perspective, the power density

actually depends on the size of the host planet of the or-ganisms, since Ψ ∝ ρg ∝ ρGM/R2, where G is the gravi-tational constant, and M and R are the mass and radius

of the Earth. At the scale of the Earth, one can definean elasticity ΣE ∝ GM2/R4 ' 1012 Pa. This formulais obtained by applying Galileo’s argument to the Earth,R ∝ ΣE/Ψ, which is known as the ‘hydrostatic equilib-rium’ in astrophysics [18]. One can also define a time

scale τE ∝ (Gρ)−12 [18]. This time scale is sometimes

called the ‘free-fall time’ and interpreted as the time anobject would take to collapse under its own gravity. Notethat this time scale depends on the density rather thanon the absolute size of the object, for a bacteria or thewhole Earth it is around one hour. With these gravita-tional units, the macroscopic formula for the metabolicrate given in Eq. 18 can be written from a simple geo-metric mean of the elasticities of the Earth and of thetissues of the organisms:

Π ∝ (ΣΣE)12

τE(19)

A bridge between the molecule and the planet

The average linear relationship between power andmass found for living organisms is a testimony of the in-tensive nature of the power density over a large range ofsizes. In the microscopic realm of bacteria we have seenthat the power density can be expressed using thermalunits. In the macroscopic realm we have seen that linear-ity is broken due to the more dominant presence of grav-ity. If the power density is indeed intensive, the macro-scopic and microscopic expressions ought to be equiva-

6

lent:

Π ∝( E3

ρ`11T

) 12 ∝ Ψ

(Σ

ρ

) 12

(20)

⇒ `T ∝ (`90`2G)

111 (21)

By equating the microscopic and macroscopic expressionsof the power density, one can obtain a formula for the sizeof the smallest living organisms `T as a weighted geo-metric mean between the macroscopic length of Galileo’sunits, and the following length:

`0 ∝(E

Σ

) 13

(22)

The form of this length scale can be understood if one re-calls that an elasticity Σ has the dimensions of an energydensity. The length `0 is then the scale at which the ‘elas-tic energy’ Σ`30 is equal to the thermal energy. By defini-tion this length is smaller than that of the smallest livingorganism, i.e. `0 < `T . Assuming room temperature andΣ ' Σ0, this gives `0 ' 60 nm. This nanometric scaleis at the crossroad of many physical phenomena combin-ing thermodynamics, elasticity and electrostatics [19]. Itmay be connected to the size of the ‘terminal units’ usedin some models of metabolism [3].

The metabolic outlook on turbulence

So far, we have used the framework of turbulenceto gain new insight on the metabolic relationships un-derlying the vast diversity of living organisms. In thislast section, we would like to suggest how the metabolicframework can be used to study turbulence beyond Kol-mogorov’s ‘inertial regime’.

In fluid turbulence, Kolmogorov’s scaling is usuallycalled ‘inertial’, to underline the fact that the viscos-ity does not appear explicitly in the parameters of thescaling, which are Π and ρ. How can living organismsdisplay Kolmogorov’s scaling? Living organisms do notabide to the Navier-Stokes equations underlying Kol-mogorov’s framework [20]. Nevertheless, they display asimilar ‘inertia-like’ spectrum because this scaling is moregeneral than its particular instance in hydrodynamics.This generality is manifested most evidently by trans-lating the power spectrum into a metabolic law, wherethe inertial spectrum is revealed as the expression of theintensive-to-extensive relationship, P/m ∝ Π/ρ. ‘Non-inertial turbulence’ is then a generic term for types ofturbulence where this intensive-to-extensive relationshipis broken, i.e. where either the density or power densitycan substantially vary. In these cases, the power spec-tra E(k) may be better expressed from quantities beyondρ and Π, ones that would be sufficiently constant to beused as parameters. Since Kolmogorov’s studies in the1940s, the phenomenology of turbulence has been ap-plied to a very vast array of systems, which would beimpossible to review here. To cite just a few examples,

‘non-inertial’ regimes of turbulence have been describedfor compressible fluids [21] and magnetic fluids [22, 23],including the interstellar medium [24, 25], for viscoelasticfluids [26, 27] and active fluids [28–30], for waves [31, 32]or in two-dimensions [33]. For some of these systems,an understanding of the scaling of power spectra is stilllacking.

To the best of our knowledge, no review of the differentobserved energy spectra exist in the literature. This stateof affair is unfortunate since some of these regimes are di-rect consequences of dimensional analysis and could bederived systematically. For instance, Kolmogorov’s scal-ing E ∼ k−

53 is the scaling obtained by considering the

parameters Π and ρ. We have seen here that Kleiber’slaw corresponds to Π and Ψ. If expressed as a powerspectrum, Kleiber’s law becomes:

E ∝ ε 16 c

32 k−

76 (23)

We do not know if this power spectrum has been observedin the context of turbulence. We hope that readers willbe able to answer this question. This power spectrumdepends on Π and ρ through ε, but also on Ψ with thespeed c ∝ Π/Ψ. If one attempts to derive this powerspectrum directly, as E ∝ εαcβkγ , no unique solution canbe found. However, if one starts with a metabolic lawP ∝ mαΨβΠγ , a unique solution is found, and it canthen be translated to a spectrum using Eq. 6. The massdimension provides the additional constraint. As longas the density is constant, it can reappear in the powerspectrum even if it was not used in the metabolic law,just by virtue of the fact that k ∝ (ρ/m)

13 . Under this

condition, we can outline a general scheme, where onewould pick two mass-carrying quantities Q1 and Q2, then

seek the unique associated metabolic law P ∝ mαQβ1Qγ2 ,

and translate it to a power spectrum.To give one last example of this metabolic approach

to turbulence, one can consider the metabolic law basedon a pressure/elasticity Σ and a density ρ. Dimen-

sional analysis would lead to P = m23 Σ

32 ρ−

76 . Such 2

3scaling has been discussed extensively in the metabolicliterature, based on arguments of surface-to-volume ra-tios [1]. We here provide a possible scaling for the prefac-tor. If expressed as a power spectrum, this law is simplyE ∝ c2k−1, with c = (Σ/ρ)

12 the sound speed.

Conclusion

Kolmogorov’s spectrum is often described as formaliz-ing the idea of the ‘energy cascade’. This view of tur-bulence as an energy cascade was famously expressed byRichardson [34]: “Big whirls have little whirls that feedon their velocity, and little whirls have lesser whirls andso on to viscosity”. In this picture, the energy is under-stood as cascading from the large scale where it is fed, allthe way down to the viscous scale where it is dissipatedas heat. In fact, Kolmogorov’s spectrum in itself does

7

not provide any direction in which the flow of energy ishappening, just that the power density is constant over awide range. The direction of the flow can only be inferredby higher order metrics [35]. For simple fluids like wateror air, the energy does cascade down. However, othertypes of turbulence can also be associated with inversecascades, where the energy flows toward large scales [33].

For living organisms, we have seen that the metabolicrate can be understood in analogy with turbulence. Thepower density is constant over a wide range of scales, justas in Kolmogorov’s scaling. One may ask in which direc-tion the energy is flowing? In a way, the energy does seemto be ‘fed’ at large scale, quite literally as ingurgitatedcalories. The bits and pieces of food are chewed downand broken up into their constituent molecules. Con-versely, the metabolic chemical reactions involving thesemolecular nutriments generate an upward cascade, wherethermal energy is progressively converted into work bymolecular motors, cells, tissues, muscles, all the way upto the hand bringing the food to the mouth. Through-out living organisms, energy seems to be able to flow bothways.

Looking back at Fig. 1b or Fig. 2a, the difference be-tween the passive turbulence of fluids and the metabolic

law of organisms seems to reside in what we may call the‘pit of viscosity’. For passive fluids, the rate of energytransfer falls down before reaching the thermal scale wedefined with bacteria. As the mass decreases, the powercascades down the pit of viscosity. In contrast, living or-ganisms seem to throw a bridge over that pit to connectthe molecular scale associated with `0 and the astronom-ical scale associated with R. Life displays a roughly con-stant metabolic power density because it provides thesmooth transition between the characteristic sizes andpowers of the molecules and of the whole Earth. We hopethat this new found connection will generate future stud-ies on the relations between power and mass for objectsbeyond life, in order to refine the broad scaling relationswe drew here, and to understand how life avoids the pitof viscosity.

Acknowledgments: The ideas of this paper were fos-tered by a lecture from Thomas Lecuit at College deFrance, by stimulating discussions with Mathieu Haute-feuille and Vivek Sharma, and by the joyful atmosphereof the Ladoux-Mege lab. Joseph d’Alessandro is thankedfor critically reviewing the manuscript. Ernesto Horne isthanked for his endorsement.

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