Universal Relation for Life-span Energy Consumption in Living Organisms: Insights for the origin of ageing
UUniversal Relation for Life-span Energy Consumption in LivingOrganisms: Insights for the origin of ageing.
Andr´es Escala
Departamento de Astronom´ıa, Universidad de Chile, Casilla 36-D, Santiago, [email protected]
ABSTRACTMetabolic energy consumption has long been thought to play a ma-jor role in the aging process ( ). Across species, a gram of tissue onaverage expends about the same amount of energy during life-span( ). Energy restriction has also been shown that increases maximumlife-span ( ) and retards age-associated changes ( ). However, thereare significant exceptions to a universal energy consumption duringlife-span, mainly coming from the inter-class comparison (
5, 6 ). Herewe present a unique relation for life-span energy consumption, validfor ∼
300 species representing all classes of living organisms, from uni-cellular ones to the largest mammals. The relation has an averagescatter of only 0.3 dex, with 95% ( − σ ) of the organisms having de-partures less than a factor of π from the relation, despite the ∼ N r ∼ of respiration cyclesper lifetime for all organisms, effectively predetermining the extensionof life by the basic energetics of respiration, being an incentive for fu-ture studies that investigate the relation of such constant N r cycles perlifetime with the production rates of free radicals and oxidants, whichmay give definite constraints on the origin of ageing.1. Introduction Ageing seems to be an inherent characteristic of all living organisms, while some inertobjects can easily subsist on Earth for many centuries, living systems have a much narrow a r X i v : . [ q - b i o . O T ] J un ), compared energy metabolismand lifespans of five domestic animals (guinea pig, cat, dog, cow and horse) and man, findingthat the life-span (total) energy expenditure per gram for the five species is approximatelyconstant, suggesting the total metabolic energy consumption per lifespan is fixed, which laterhas become known as the ‘rate of living’ theory ( ).Decades later, a mechanism was found in which the idea behind a fixed energy consump-tion per lifespan might work in the ‘free-radical damage’ hypothesis of aging (
8, 9 ), in whichmacromolecular components of the cell are under perpetual attack from toxic by-productsof metabolism, such as free radicals and oxidants. In addition, energy restriction has beenexperimentally shown to increase maximum life-span and retard age-associated changes inanimals, such as insects, rats, fish, spiders, water fleas and mice (
3, 4 ).Rubner’s relation was confirmed for around hundred mammals ( ) and extended it tobirds ( ), ectotherms ( ) and even unicellular organisms such as protozoa and bacteria ( ),totalizing almost three hundred different species in a range of 20 orders of magnitude in bodymass. Although the total metabolic energy exhausted per lifespan per body mass of givenorganism appears to be relatively constant parameter, at about the same number determinedby Rubner ( ) of a million Joule per gram of body weight for mammals, variations over anorder of magnitude are found among different animal classes, a result also found by otherauthors (i.e. ) and considered the most persuasive evidence against the ‘rate of living’theory ( ).The origin of such variations in the lifespan energy consumption might come from intrin-sic variations in the quantities used to estimate it: lifespans and especially, the metabolicrate relation which universality is still debated (see and references therein). Recently,the empirical metabolic rate relation was corrected in order to fulfill dimensional homo-geneity ( ), a minimal requirement of any meaningful law of nature ( ), proposing a newmetabolic rate (B) formula: B = (cid:15) (T) η O f H M, where M is the body mass, f H is a char-acteristic (heart) frequency, η O is an specific O absorption factor and (cid:15) (T) = (cid:15) e − E a / kT is a temperature correction inspired in the Arrhenius formula, in which E a is an activationenergy and k is the Boltzmann universal constant. Compared to Kleiber’s original formu-lation ( ), B = B (M / M ) . , this new metabolic rate relation has the heart frequency f H as independent controlling variable (a marker of metabolic rate) and the advantage of beingan unique metabolic rate equation for different classes of animals and different exercisingconditions, valid for both basal and maximal metabolic rates, in agreement with empiricaldata in the literature ( ). 3 –In addition, this new metabolic rate relation can be directly linked to the total energyconsumed in a lifespan ( ), being a promising way to explain the origin of variations inRubner’s relation and unify them into a single formulation. In this paper, we will explorethe implications of this new metabolic relation for the total energy consumed in a lifespanand the ‘rate of living’ theory, being organized as follows. We start reviewing the results ofthe new metabolic rate relation found in ( ), deriving its prediction for the total energyconsumed in a lifespan in §
2. Section 3 continues with testing the empirical support of thepredicted relation for total energy consumed in a lifespan, with satisfactory results. Finallyin §
4, we discuss the results and implications of this work.
2. New Metabolic Rate Relation and its Prediction for ‘Rate of Living’Theory
In the proper mathematical formulation of natural laws, a minimum requirement is to beexpressed in a general form that remain true when the size of units is changed, simply becausenature cannot fundamentally depend on a human construct such as the definition of units.In mathematical terms, this implies that meaningful laws of nature must be homogeneousequations in their various units of measurement ( ). However, most relations in allometrydo not satisfy this basic requirement, including Kleiber’s ‘3/4 Law’ ( ), that relates the basalmetabolic rate and body mass as B = B (M / M ) / = C M / , being this ‘3/4 Law’ a typicalexample ( ) of a relation where the proportionality factor C has a fractal dimensionalityand its value depends on the units chosen for the variables (B , M), therefore, do not fulfillthe minimum requirement for being a natural law. Mathematically speaking, to qualify forbeing a natural law the metabolic rate relation must be first rewritten such the constantswith dimensions are universal and restricted to a minimum number, which in no case couldexceed the total number of fundamental units of the problem.To solve this issue, ( ) proposed a new unique homogeneous equation for the metabolicrates that includes the heart frequency f H as independent controlling variable, proposinga the metabolic rate (B) formula: B = (cid:15) (T) η O f H M, where M is the body mass, f H isa characteristic (heart) frequency, η O is an specific O absorption factor and (cid:15) (T) is atemperature-dependent normalization. The new metabolic rate relation is, in addition tobe in agreement with the empirical data ( ), valid for different classes of animals and forboth resting and exercising conditions. Using this formula, it can be shown ( ) that mostof the differences found in the allometric exponents are due to compare incommensurablequantities, because the variations in the dependence of the metabolic rates on body mass aresecondary, coming from variations in the allometric dependence of the heart frequencies f H H can be seen as a new independent physiological variablethat controls metabolic rates, in addition to the body mass.One of the advantages of having a metabolic rate relation with the heart frequency f H asindependent controlling variable, is that it can be straightforwardly linked ( ) to the totalenergy consumed in a lifespan, by the relation of total number N b of heartbeats in a lifetime,N b = f H t life , which is empirically determined to be constant for mammals and equals to7 . × heartbeats ( ). Therefore, for a constant total number of heartbeats in a lifetime,t life = N b / f H , the metabolic rate relation ( ) can be rewritten as B t life = (cid:15) (T) η O N b M,giving a straightforward prediction for life-span energy consumption and that can be seenas a test for the ‘rate of living’ theory ( ).However, since the total energy consumed in a lifespan relation is valid even in unicel-lular organisms without heart ( ), in order to find an unique relation for all living organismsthe concept of characteristic (heart) frequency must be first generalized. A natural candidateis the respiration frequency, f resp , since this frequency is observed in animals to be strictlyproportional to the heart one, f H = a f resp ( ), and is a still meaningful frequency for organ-isms without heart. Under this proportionally between frequencies, the empirical relationwith lifetime can be rewritten to be also valid for a total number N r (= N b / a) of ‘respira-tion cycles’: t life = N b / f H = N b / af resp = N r / f resp . This total number of ‘respiration cycles’,N r = f resp t life , will be assumed from now to be the same number for all living organisms andin §
4, we will explore the implications of this conjecture for the origin of aging.Under the condition of a constant total number N r of respiration cycles in a lifetimet life , multiplying the new metabolic rate relation ( ) by t life / a = N r / af resp = N r / f H isnow equivalent to B t life / a = (cid:15) (T) η O N r M. The factor (cid:15) (T) η O = (cid:15) η O e − E a / kT can berewritten as E e ( − ) Eak , where T a in a normalizing ‘ambient’ temperature and E =10 − . mlO g − ≈ − Jg − (converting 1 ltr O =20.1 kJ; ) a constant that comes fromthe best fitted value for the corrected metabolic relation ( ). Therefore, for the metabolicrate relation given in ( ) and under the condition of fixed respiration cycles in a lifetime,the following relation is predicted to be valid:exp (cid:16) E a kT (cid:17) B t life a = E exp (cid:16) E a kT a (cid:17) N r M . (1)Eq. 1 is a prediction from the mathematically-corrected metabolic relation ( ) underthe assumption of constant N r respiration cycles in a lifetime, t life = N r / f resp , which is inprinciple valid for all living organisms. It is important to emphasize that this relation forlifespan energy consumption is not assumed or hypothesized to be fixed like in the ‘rate ofliving’ theory, instead is derived directly from the metabolic relation under the conjectured 5 –invariant N r . Compared to allometric relations, characterized mathematically for havingas many dimensional constants (B , M , in Kleiber’s case) as there are variables (B , M),the relation given by Eq. 1 has (in principle) 7 variables (t life , B , a , M , E a , T a & T), onedimensionless number (N r ) and only two constants with units: one universal one (Boltzmannconstant k) and another one coming from best fitting the energetics of respiration (E ),which possible universality will be discussed in §
3. Empirical Support for the Corrected Relation of Total Energy Consumedin a Lifespan
In this section we test the validity and accuracy of the derived relation for total energyconsumed in a lifespan (Eq. 1), predicted from the new formulation of the metabolic raterelation ( ) and the conjectured constant total number N r of respiration cycles per lifespanfor all living organisms. For that purpose, we use data from 277 species of all classes of livingorganisms, from unicellular organisms and other ectotherms species, to mammals and birds,listed in Table 1 of ( ) with their body mass M, total metabolic energy per lifespan B t life and body temperature T.Fig 1 shows the relation predicted by Eq. 1 for the 277 living organisms listed in ( ),where the activation energy E a was chosen to the average value of 0.63 eV, independentlydetermined to temperature-normalize the metabolic rates of unicells and poikilotherms toendotherms (
18, 19 ). The parameter a was chosen to the empirically determined values of4.5 for mammals and 9 for birds ( ), estimated to be 3 for ectotherms with heart (from therelative size of their hearts; ) and assumed unity for ectotherms without heart, such asunicellular organisms. The total number N r = N b / a = 1 . × of respiration cycles in alifetime, was determined from the best fitted values for mammals: N b = 7 . × heartbeatsin a lifetime ( ) and a=4.5 ( ).The data displayed in Fig 1 strongly supports the unique relation predicted by Eq. 1,in a dynamical range of 20 orders of magnitude, for all classes of organisms from Bacteria to Elephas Maximum . The solid blue curve displays the best fitted value of slope 0.997 andnormalization of 1.05, almost undistinguishable from the identity predicted by Eq 1 (dashedred curve). The only free parameter (not predetermined by an independent measurement)in Eq 1 is the ‘ambient’ temperature T a = 30 o C, which was chosen only to match thenormalization in the best fitted relation (solid blue) to the identity (dashed red), but is anatural choice for normalization, since ectotherms are typically around 20 o C and endothermsclose to 40 o C in Table 1 of ( ). Moreover, the slope close to unity (0.997) is independent ofthe T a choice and for example, if we instead decide to preset T a to the value of mammals 6 – E e E a / kT a N r M [kJ]10 e E a / k T B T li f e / a [ k J ] Fig. 1.— Figure shows the relation predicted by Eq. 1 for 277 living organisms, listedin ( ). The dashed red curve displays the identity given by Eq 1. The solid blue curvedisplays the best fitted value of slope 0.997 and normalization of 1.05 for ambient temperatureT a = 30 o C. The two curves are almost undistinguishable in the dynamical range of 20 ordersof magnitude.(37 o C), it only changes the best fitted normalization value to 1.81. Therefore, the relationbetween 5 physiological variables (t life , B, a, M & T) given by Eq 1 is confirmed without thechoice of any free parameter.Fig 2 shows the residuals from the relation predicted by Eq 1, as a function of theorganism’s body mass. The relation has only an average scatter of 0.339 dex around thepredicted value (E N r = E N b /a; dashed line in Fig 2), which impressively smalltaking into account the 20 orders of magnitude variations in mass and that the values ofE = 10 − Jg − ( ), N b = 7 . × heartbeats in a lifetime ( ) and a=4.5 ( ), comesfrom three completely independent measurements in mammals. Moreover, around 95% ofthe points (2 σ ) has departures from the relation less than a factor of π ( ≈ ± -17 -15 -13 -11 -9 -7 -5 -3 -1 M [kg]3.03.54.04.55.05.56.06.57.0 L o g [ e E a k ( T − T a ) B T li f e / M a ] [ k J ] Fig. 2.— Residuals from the relation predicted by Eq 1 as a function of the organism’sbody mass for 277 living organisms, listed in ( ). The relation has only an average scatterof 0.339 dex around the predicted value (E N r ) denoted by the dashed line. The coloredregion between the solid curves denotes residuals less than a geometrical factor of π fromthe relation and 95% of the points (2 − σ ) fulfill such criterion.dimensionless numerical factor of order unity ( π ). If that is indeed the case, this is probablythe first relation in life sciences including all the relevant controlling parameters, reachingan accuracy comparable to the ones in exact sciences. Moreover, Fig 2 implies no cleartrend (larger than an e-fold ≈
2, 5 ).These exceptions were erased in Fig 2 due to predicted secondary parameters that are notincluded in the original formulation (mainly the parameter a = f H / f resp for the particular caseof birds and bats to mammals). Moreover, Eq 1 implies a dependence of t life ∝ MB exp (cid:16) E a kT a (cid:17) that it is in agreement with the three regimens experimentally known to extend life-span( ): lowered ambient temperature T a in poikilotherms, decrease of physical activity inpoikilotherms (lower B) and caloric restriction (lower B). 8 –
4. Discussion
We showed that the new metabolic rate relation ( ) can be directly linked to the totalenergy consumed in a lifespan, if it is conjectured a constant number N r of respiration cyclesper lifespan, finding a corrected relation for the total energy consumed in a lifespan (Eq 1)that can explain the origin of variations in the ‘rate of living’ theory (
2, 5 ) and unify theminto a single formulation. We test the validity and accuracy of the predicted relation (Eq 1)for the total energy consumed in a lifespan on ∼
300 species representing all classes of livingorganisms, finding that the relation has an average scatter of only 0.3 dex, with 95% of theorganisms having departures less than a factor of π from the relation, despite the ∼
20 ordersof magnitude difference in body mass. This reduces any possible inter-class variation in therelation to only a geometrical factor and strongly supports the conjectured invariant numberof N r ∼ of respiration cycles per lifespan in all living organismsInvariant quantities in physics traditionally reflects fundamental underlying constraints,something also applied recently to life sciences such as Ecology (
21, 22 ). Fig 2 displays thefact that, for a given temperature, the total life-span energy consumption per gram per‘generalized beat’ (N Gb ≡ aN r = a 1 . × ) is remarkably constant on around E ,supporting that the overall energetics during lifespan is the same for all living organisms,being predetermined by the basic energetics of respiration. Therefore, Rubner’s originalpicture it is shown to valid without systematic exceptions, but in this more general form.Moreover, since the value determined from Fig 2 is remarkably similar to E , it can beconsidered an independent determination for E , suggesting that E is a candidate forbeing an universal constant and not just a fitting parameter coming from the correctedmetabolic relation ( ).In addition, we showed here that the invariant total life-span energy consumption pergram per ‘generalized beat’ comes directly from the existence of another invariant: theapproximately constant total number N r ∼ of respiration cycles per lifetime, effectivelyconverting the ‘generalized beat’ into the characteristic clock during lifespan. Thus, the exactphysical relation between (oxidative) free radical damage and the origin of aging, is mostprobably related to the striking existence such of constant total number of respiration cyclesN r in the lifetime of all organisms, which predetermines the extension of life. Therefore,since all organisms seems to live the same in units of respiration cycles, future theoreticaland experimental studies that investigate the exact link between the constant N r ∼ respiration cycles per lifespan and the production rates of free radicals (and other byproductsof metabolism), should shed light on the origin of ageing and the physical cause of naturalmortality.It have been also suggested that an analogous invariant is originated at the molecular 9 –level ( ), the number of ATP turnovers in a lifetime of the molecular respiratory complexesper cell, which from an energy conservation model that extends metabolism to intracellularlevels is estimated to be ∼ . × ( ). Similar number can be determine taking intoaccount that human cells requires synthase approximately 100 moles of ATP daily, equivalentto 7 × molecules per second. For ∼ × cells in human boby and for a respirationrate of 15 breaths per minute, this gives ∼ × ATP molecules synthesized per cell perbreath, which for the invariant total number N r of respiration cycles per lifetime found inthis work, arises to the same number of ∼ . × ATP turnovers in a lifetime per cell,showing the equivalence between both invariants, linking N r to the energetics of respiratorycomplexes at cellular level.The excellent agreement between the predicted relation (Eq 1) and the data acrossall types of living organisms emphasizes the fact that lifespan indeed depends on multiplefactors (B, a, M, T & T a ) and strongly supports the methodology presented in this work ofmultifactorial testing, as done in Fig 1, since quantities in life sciences generally suffers from acofounding variable problem. An example of this problem, illustrated on individually testingeach of the relevant factors is in ( ), which for a large (and noisy) sample test for t life ∝ / B,finding no clear correlation. From Eq 1, it is clear that in an uncontrolled experiment thedependence on the rest of the parameters (M, a, T, & T a ) might erase the dependence on themetabolic rate B (in fact, for the same reason Rubner’s work ( ) focused on the mass-specificmetabolic rate B/M instead of B). This work ( ) only finds a residual inverse dependenceof t life on the ambient temperature T a for ectotherms, something expected according to Eq1 (cid:16) t life ∝ exp (cid:16) E a kT a (cid:17)(cid:17) .Finally, the empirical support in favor of Eq 1 allow us to compute how much will varythe energy consumption in the biomass doing aerobic respiration, as increases the earth’stemperature, relevant in the current context of possible global warming. This is given by thefactor exp (cid:104) E a k (cid:16) − (cid:17)(cid:105) which for an activation energy E a = 0 . o C, implies an increase 8.3% in energy consumption per 1 degree increase on the averageEarth temperature. This result can be straightforwardly applied in ectotherm since theirbody temperatures adapt to the environmental one (T = T a ), but is less clear its implicationsfor the case of endotherms organisms. Acknowledgments
I acknowledge partial support from the Center of Excellence in Astrophysics and Asso-ciated Technologies (AFB-170002) and FONDECYT Regular Grant 1181663. 10 –
References and Notes
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