© 2016. Published by The Company of Biologists Ltd | Journal of Experimental (2016) 219, 2481-2489 doi:10.1242/jeb.138305

RESEARCH ARTICLE Metabolic allometric scaling model: combining cellular transportation and heat dissipation constraints Yuri K. Shestopaloff*

ABSTRACT body mass M is mathematically usually presented in the following Living organisms need energy to be ‘alive’. Energy is produced form: by the biochemical processing of nutrients, and the rate of B / M b; energy production is called the metabolic rate. Metabolism is ð1Þ very important from evolutionary and ecological perspectives, where the exponent b is the allometric scaling coefficient of and for organismal development and functioning. It depends on metabolic rate (also allometric scaling, allometric exponent). Or, if different parameters, of which organism mass is considered to we rewrite this expression as an equality, it is as follows: be one of the most important. Simple relationships between the b mass of organisms and their metabolic rates were empirically B ¼ aM : ð2Þ discovered by M. Kleiber in 1932. Such dependence is described by a power function, whose exponent is referred to as the Here, a is assumed to be a constant. allometric scaling coefficient. With the increase of mass, the Presently, there are several theories explaining the phenomenon metabolic rate usually increases more slowly; if mass increases of metabolic allometric scaling. The resource transport network by two times, the metabolic rate increases less than two times. (RTN) theory assigns the effect to a single foundational mechanism This fact has far-reaching implications for the organization of life. related to transportation costs for moving nutrients and waste The fundamental biological and biophysical mechanisms products over the internal organismal networks, such as vascular underlying this phenomenon are still not well understood. The systems (Brown and West, 2000; West et al., 1999, 2002; Savage present study shows that one such primary mechanism relates to et al., 2004; Banavar et al., 2010, 2014; Kearney and White, 2012). transportation of substances, such as nutrients and waste, at a Another approach accounts for transportation costs of materials cellular level. Variations in cell size and associated cellular through exchange surfaces, such as skin, lungs, etc. (Rubner, 1883; transportation costs explain the known variance of the allometric Okie, 2013; see also Hirst, et al., 2014; Glazier, 2005, 2014). The exponent. The introduced model also includes heat dissipation similar idea of ‘surface uptake limitation’ was applied to cells constraints. The model agrees with experimental observations (Davison, 1955). Dynamic energy budget theory, whose analysis and reconciles experimental results across different taxa. It and comparison with RTN can be found in Kearney and ties metabolic scaling to organismal and environmental White (2012), uses generalized surface area and volume characteristics, helps to define perspective directions of future relationships. Heat dissipation limit (HDL) theory (Speakman research and allows the prediction of allometric exponents based and Król, 2010) and its discussion (Glazier, 2005, 2014; Hudson, on characteristics of organisms and the environments they et al., 2013) suggests and provides convincing proofs that the live in. ability of organisms to dissipate heat influences their metabolic characteristics. The metabolic-level boundaries hypothesis (Glazier, KEY WORDS: Cellular transportation networks, Living organisms, 2010) generalizes observations that allometric scaling (1) is Metabolic rate, , Nutrients, Plants a variable parameter and (2) its value depends on the level of metabolic activity in a U-shaped manner, with extrema INTRODUCTION corresponding to low (dormant, hibernating) and high (extreme Consumption of energy by living organisms is characterized by a physical exertion) metabolic activities. metabolic rate, which in physical terms is power (amount of energy The study of Darveau et al. (2002) adheres to an idea of multiple per unit time). In the SI system, the unit of measure of energy is the factors contributing to the value of the allometric exponent, joule (J) and the unit of power is the watt (W). The issue of how considering different biomolecular mechanisms participating in metabolic rate scales with the change in an organism’s mass is ATP turnover as the major contributors, together with a ‘layering of special interest for biologists (Schmidt-Nielsen, 1984), as it is tied of function at various levels of organization’, such as different to many important factors influencing life organization – from scale levels. Allometric exponents found separately for each organismal biochemical mechanisms to evolutionary developments contributing mechanism are combined as their weighted sum. to ecological . The relationship of metabolic rate B and the Using energy that is tied to particular biomolecular mechanisms as a common denominator to unite different contributing factors is a significant step. R and D Lab, Segmentsoft, 70 Bowerbank Drive, Toronto M2M 2A1, Canada. Because the metabolic properties of organisms depend on many factors and environmental conditions, the explanation of allometric *Author for correspondence ([email protected]) scaling should instead involve a combination of interacting Y.K.S., 0000-0003-4251-0618 mechanisms. The hierarchy, ‘weight’ and triggering into action of particular mechanisms should be defined by an organism’s

Received 28 January 2016; Accepted 1 June 2016 characteristics and interaction with its environment. At a systemic Journal of Experimental Biology

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As in other works, the present study assumes that the amount of List of symbols and abbreviations nutrients that is required to support transportation networks is – – a constant in the formula B=aMb (kg1 b m2 s 3) proportional to the distance the substances travel along these A constant in the formula for finding the amount of nutrients networks. Validity of this postulate was rather an axiomatic −1 for transportation (m ) proposition in previous works. In fact, several previous studies b allometric exponent (Shestopaloff, 2012a,b, 2014a,b, 2015; Shestopaloff and Sbalzarini, B metabolic rate (W) c cell enlargement 2014) confirm this postulate through calculations of nutrient d dimensionality of cell increase influxes for microorganisms, and for dog and human livers. − f density of nutrients per unit volume (kg m 3) Another indirect proof relates to stability of the growth period of F nutrients required for transportation (kg) large and small organisms. It was shown previously (Shestopaloff, −1 fl density of nutrients in a tube (kg m ) 2012a,b, 2014a,b) that if transportation nutrient expenditures are not h heat dissipation coefficient proportional to the traveled distance, then the growth periods of Hdis dissipated heat (W) HDL heat dissipation limitation (theory) large and small organisms would be substantially different, which is not the case. K1,2,3 scaling coefficients for transportation nutrients; index denotes dimension −1 kt transportation costs as a fraction of transported nutrients (m ) Determining the amount of nutrients required for M mass (kg) transportation r radius for disk and sphere (m) First, we consider organisms that grow or differ in size in one RTN resource transport network S surface area (m2) dimension. We assume that substances are transported along the 3 f ‘ ’ v cell volume (m ) tube. The density of nutrients per unit of length is l ( food ), −1 V organism volume (m3) measured in kg m . Then, the mass of nutrients in a small tube’s β a constant included in the scaling coefficient section with length dl is fl×dl. So, the amount of nutrients dF μ cell mass (kg) required to transport the nutrients’ mass fl×dl by a distance l is:

dF ¼ ktlfl dl; ð3Þ level, in the author’s view, heat dissipation (HDL) theory is of where kt is a constant coefficient that represents the cost of importance. Temperature is directly tied to foundational-level transportation as a fraction of the transported nutrients per unit physiological and biochemical mechanisms, as they are very length (so that the unit of measure is m–1), and distance l is the sensitive to temperature. consequence of the postulate that transportation costs are The present study supports the multifactor nature of allometric proportional to distance. Note that the cost of transportation of scaling. However, there are the following differences. (1) The nutrients F is also valued in nutrients and measured in kg. overwhelming majority of previous works promote an idea that Then, the total amount of nutrients required for transportation to allometric scaling is due to systemic-level mechanisms. The present fill the tube of length L with nutrients is: study shows that cellular-level transportation mechanisms, although coordinated at a systemic level, present a factor of primary ðL 1 2 importance for metabolic allometric scaling. (2) Although the F ¼ ktlfldl ¼ ktftL : ð4Þ ’ 2 cell s size changes significantly less than the interspecific allometric 0 scaling across a wide range of animal sizes, it follows from the present study that the transportation costs, when associated with the In other words, F is proportional to a square of the distance nutrients cell size, present an important contributing factor into ontogenetic are transported to. Assuming that the cross-sectional area of a differences in allometric scaling. ‘feeding pipe’ is S and the length is L, we find the volume as V=SL. The present study introduces a general method for calculating Expressing the length L=V/S and substituting this value into Eqn 4, transportation costs. Then, it presents arguments in favor of the we obtain: cell size factor and associated transportation costs as important 1 1 F ¼ k f L2 ¼ k f V 2=S2: ð5Þ mechanisms influencing allometric scaling, and introduces 2 t l 2 t l formulas for calculating allometric exponents. Briefly, the HDL theory is considered. Finally, an allometric scaling model Similarly, we can find transportation costs for two- and three- incorporating cellular transportation costs and heat dissipation dimensional variations in size. Let us consider a disk increasing constraints is introduced. in two dimensions, with constant height H. The disk has a current radius r and maximum radius R. Then, we can write the following: MATERIALS AND METHODS 2 Nutrient distribution networks models were criticized as fairly dF ¼ ktf 2pr r Hdr ¼ 2pktfHr dr; ð6Þ restrictive in applicability, problematic with regard to some initial assumptions, and that they could not explain the known diversity where f is the density of nutrients per unit of volume (kg m−3) of metabolic scaling relationships (White et al., 2011, 2009, and kt is the cost of transportation, as in Eqn 3. 2007; Weibel and Hoppeler, 2005; Glazier, 2005; Kozlowski and A note about integration limits: in cells, much traffic goes Konarzewski, 2004). Here, I present a general method for determining from the periphery (membrane) and toward the inner regions, in the amount of nutrients required for transportation needs, which uses particular nuclei, and then waste and other substances go back to substantially less restrictive assumptions. The model is applicable both the periphery and leave the cells. This justifies integration in a to interspecific and intraspecific allometric scaling, although the radial direction, from zero to R (this assumption can be significantly present study considers only intraspecific scaling. relaxed, which we proved mathematically, but do not present here). Journal of Experimental Biology

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Doing integration, we find: transportation networks in the larger cell. Of course, a larger cell ðR may also compensate for the higher transportation costs by different 2 2 3 means. F ¼ 2pktfHr dr ¼ pktfHR : ð7Þ 3 Let us generalize this result for multicellular organisms with 0 equal masses M, but different cell sizes. We assume that Certainly, the unit of measure of F remains kg. Taking into account organisms differ proportionally in three dimensions. Densities that the disk volume V=2πHR2, and consequently R3=V3/2/(2πH )3/2, are constant and equal, so that volumes are equal too. Cell we can rewrite Eqn 7 as follows: compositions are accordingly assembled from cells with volume 3=2 v1 and v2, and v1=kv2. If masses of cells are accordingly μ1 and ktfV F ¼ pffiffiffiffiffiffiffiffiffiffi : ð8Þ μ2, it also means that μ1=kμ2. Then, using Eqn 11, we can find 3 2pH the ratio K3 of transportation costs for these compositions of cells In other words, transportation costs in a disk increasing in two as follows: dimensions are proportional to 3/2 power of volume. It is interesting  F Afv4=3N Afv4=3kN 1=3 to note that the amoeba, when it grows, also increases its nutrient K 2 2 2 2 1 1 3 ¼ ¼ 4=3 ¼ 4=3 ¼ consumption in a manner similar to that expressed in Eqn 8 F1 Afv N Af kv N k 1 1 ð 2Þ 1 (Shestopaloff, 2012a, 2015).  m 1=3 The obtained result is valid for other forms increasing 2 ; ¼ m ð12Þ proportionally in two dimensions, such as squares, rectangles and 1 stars, whose height remains constant. Similarly, we can find transportation costs for organisms whose where N denotes the number of cells and the index in K size variation is due to proportional increase in three dimensions. refers to dimensionality. We took into account that N2=kN1, Let us consider a sphere as an example: because the masses of the cells’ compositions are equal 2 3 [indeed, N1=M/V1=M/(kV2), N2=M/V2, from which it follows dF ¼ ktf 4pr r dr ¼ 4pktfr dr: ð9Þ that N2=kN1]. Taking into account that nutrient traffic goes mostly between the Similarly, we can consider compositions of cells that grow in one periphery and the inner regions, we can do integration from zero to and two dimensions. Using Eqns 5 and 7 and performing similar R. Doing this, we find: transformations as in Eqn 12, we obtain the following values for R 1/2 ð ratios of transportation costs: K1=k and K2=k . 3 4 What does Eqn 12 mean? If we have two organisms with the same F ¼ 4pktfr dr ¼ pktfR : ð10Þ mass, but composed of different sizes of cells, then to obtain the 0 same amount of nutrients per unit of mass, the organism with larger Because the sphere volume V=(4/3)πR3, and consequently R3=V/ cells requires more nutrients for transportation. However, in many (4π/3), we can rewrite Eqn 10 as: instances, organisms, regardless of their cell size, receive about the same amount of nutrients per unit of mass because of the specifics 4=3 1=3 4=3 4=3 F ¼ð3=4Þ p ktfV ¼ AfV : ð11Þ of nutrient distribution networks, such as the blood circulation system. In this case, the transportation costs have to be scaled 4/3 –1/3 −1 Here, A=(3/4) π kt is a constant (measured in m ). Note that according to nutrient availability. The question is, by how much? this result (proportionality to V4/3 of transportation costs in a There is no simple answer to this question, because it depends on the sphere-like organism) is valid for other geometrical shapes that organism’s biochemistry and specifics of nutrient distribution increase proportionally in three dimensions. networks, among other factors. However, we can be certain in two The formulas above were obtained under the assumption that things: (1) such scaling affects the total amount of nutrients; and (2) cells are able to increase nutrient consumption to cover a faster the distribution of nutrients between transportation needs and other increase of transportation costs than increase of volume and, biochemical activities (meaning the ratio) will remain, because associated with it, mass. Examples of such unicellular transportation costs represent an intrinsic factor to cells of a organisms (amoeba, fission yeast Schizosaccharomyces pombe particular type and size. In other words, the decrease of nutrients and Saccharomyces cerevisiae) are presented in Shestopaloff used for transportation will proportionally reduce the total amount (2012a,b, 2015). of transported nutrients (of course, including the nutrients used for transportation). The metabolic rate, in turn, is proportional to the Transportation costs in organisms with different cell sizes total amount of consumed nutrients. This last assumption is a but the same mass reasonable one, because we consider the same organisms, whose Let us compare nutrient transportation costs for a single cell metabolic processing efficiency per unit of nutrient mass should not growing proportionally in three dimensions, with a grown volume V, vary much. and two equal cells each with grown volume V/2, so that their total volume is also V. The first organism, according to Eqn 11, requires Influence of cell size on metabolic rate – organisms with 4/3 F1=AfV nutrients for transportation. Two smaller organisms different masses correspondingly need: Let us consider two compositions (1 and 2) of cells. Within each composition, cells have the same size, but cell sizes between 4=3 4=3 1=3 4=3 F2 ¼ Af ðV=2Þ þ Af ðV=2Þ ¼ Af ð2 ÞV : compositions are different. As discussed in the previous section, the metabolic rate B is proportional to the total amount of nutrients T, 1/3 Thus, the ratio F1/F2=2 ≈1.26>1, which means that a larger cell which in turn is proportional to the number of cells, which in our needs more nutrients for transportation than two smaller cells case also means proportionality to mass of a given organism. First, together with the same total mass. This is due to longer we assume that the amount of nutrients used for transportation is Journal of Experimental Biology

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1/3 scaled according to Eqn 12, that is by a factor (μ1/μ2) : complies with numerous experimental observations. Indeed, the position of regression lines often varies vertically for different =  B T m 1 3 M organisms, although the allometric slopes can be the same. 2 2 1 2 : 13 B ¼ T ¼ m M ð Þ According to Eqn 18, the reasons could be different values of 1 1 2 1 scaling coefficients (lnβ) for different organisms, and different The masses of organisms are M1=μ1N1 and M2=μ2N2, where N1 values of lnB1. When the number of cells remains the same (the and N2 represent the number of cells. When N2=N1=N, Eqn 13 cell’s mass increases proportionally to the organism’s mass), the transforms to: fourth term (1/3)ln(N2/N1) also becomes a constant. If an organism’s mass increases as a result of both cell enlargement B B m =m 1=3 M =M 2 ¼ 1ð 1 2Þ ð 2 1Þ and cell number, then this fourth term provides a line with a non- B m =m 1=3 m N= m N zero slope, which, summing with the line defined by the third term ¼ 1ð 1 2Þ ½ 2 ð 1 Þ (2/3)lnB1(M2/M1), adds to the value of the allometric exponent 2/3, B m =m 2=3 B M =M 2=3: ¼ 1ð 2 1Þ ¼ 1ð 2 1Þ ð14Þ thus providing the range from 2/3 to 1. Note that the fraction of nutrients (from the total amount of In other words, the metabolic rate of organisms whose mass nutrients) used for needs other than transportation will be smaller in increases solely because of the increase of cell size scales as a power organisms with larger cells than in organisms with smaller cells. of 2/3 of the masses’ ratio. Thus, fewer nutrients may be available to support other cell When the number of cells is different, then: functions, for instance, DNA repair mechanisms, whose inefficiency may cause transformation to cancerous cells. It is B B m =m 1=3 M =M 2 ¼ 1ð 1 2Þ ð 2 1Þ well known that small cells generally are significantly more resistant 1=3 to cancerous transformations. However, once such a transformation ¼ B1ðm =m Þ ½m N2=ðN1m Þ 2 1 2 1 occurs, they represent the most malicious cancer forms, which are B m =m 2=3 N =N ¼ 1ð 2 1Þ ð 2 1Þ very difficult to cure. Although speculative at this point, this is an = = ¼ B ðM =M Þ2 3ðN =N Þ1 3: ð15Þ observation worth exploring. 1 2 1 2 1 The obtained results from the present study agree with and Here, we took into account that M1=μ1N1 and M2=μ2N2. explain known empirical facts. Larger organisms ontogenetically According to Eqns 14 and 15, the size and number of cells are often have larger cells. In Chown et al. (2007), seven of eight ant important parameters defining metabolic scaling ontogenetically. species increased their size primarily through the enlargement of Allometric scaling much depends on relative size of the cells cells, but not through the number of cells. When this is the case, composing the organisms. When body size correlates well with cell then, according to Eqn 14, the allometric exponent is close to 2/3. size, the allometric exponent should be close to 2/3, which Eqn 14 Indeed, in Chown et al. (2007), the authors found for ants that ‘… shows. Eqn 15 presents an intermediate situation. When correlation the intraspecific scaling exponents varied from 0.67 to 1.0. between the organism’s size and size of its cells is weak (cell size Moreover, in the species where metabolic rate scaled as mass1.0, remains the same), we should expect isometric scaling (b=1). cell size did not contribute significantly to models of body size Indeed, assuming μ1=μ2=μ, we obtain from Eqn 15 isometric variation, only cell number was significant. Where the scaling scaling: exponent was <1.0, cell size played an increasingly important role in accounting for size variation.’ Also, the authors confirmed that B B m =m 2=3 N =N B N =N 2 ¼ 1ð 2 1Þ ð 2 1Þ¼ 1ð 2 1Þ when the cell size remains the same, the metabolic allometry is close ¼ B ðmN =mN Þ¼B ðM =M Þ1: ð16Þ to isometric (which is the case in Eqn 16). So, the present results 1 2 1 1 2 1 about the role of cellular transportation costs in metabolic scaling What would happen if scaling of transportation costs is different comply with all experimental observations of Chown et al. (2007). 1/3 from (μ1/μ2) , as defined by Eqn 12? The last value, representing Therefore, the way organisms increase their size during an intrinsic factor, as we said already, will remain the same. We also – that is, either through the increase of cell number or cell size, or a admitted that the total amount of nutrients possibly can be scaled. combination of both – explains the variability of metabolic scaling Let us denote such a scaling as β. Then, the scaling coefficient is from 2/3 to 1, with a value of 2/3 corresponding to an increase by 1/3 β(μ1/μ2) . Substituting this into Eqns 13–16, we obtain the same cell enlargement, and a value of 1 corresponding to mass increase by expressions, multiplied by a coefficient β. However, this will not number of cells. change our results with regard to the allometric exponent. For If we repeat computations similar to Eqns 12–16 for a two- 1/2 instance, in the case of Eqn 15 we obtain: dimensional increase (K2=k ), then the range for the allometric exponent will be 1/2–1, and for a one-dimensional increase, 0–1. In B bB m =m 1=3 M =M 2 ¼ 1ð 1 2Þ ð 2 1Þ the last case, these have to be some exotic, if not hypothetical, 2=3 1=3 organisms composed of tube-like cells of the same width fed from ¼ bB ðM =M Þ ðN =N Þ : ð17Þ 1 2 1 2 1 ends. In allometric studies, such dependencies are usually represented In addition to the described results from Chown et al. (2007), in a logarithmic scale. In this case, Eqn 17 becomes as follows: previous works (Glazier, 2005, 2014) present data from different sources that support the present findings both for the boundary 2 1 ln B2 ¼ ln b þ ln B1 þ lnðM2=M1Þþ lnðN2=N1Þ: ð18Þ values and for transitional growth scenarios. 3 3 First, regarding organisms increasing their size by cell We do not know how much the parameter β can vary. At a first enlargement: (1) nematodes have an allometric exponent b close glance, there are no reasons for it to significantly change depending to 2/3 [four species, b=0.64–0.72; one species, b=0.43; one species on the state of the organism (e.g. torpid or physically active), but the (low food levels), b=0.77; mean value of b for seven nematode issue requires further study. In the case when β≈const, Eqn 18 species is 0.677]; (2) rotifers have b close to 2/3; (3) in the later Journal of Experimental Biology

2484 RESEARCH ARTICLE Journal of Experimental Biology (2016) 219, 2481-2489 doi:10.1242/jeb.138305 stages of growth, increase their size through cell The heat dissipation is done via heat conduction, convection and enlargement, and have a smaller allometric exponent; and (4) in radiation. Dissipated heat is usually accounted for using heat the later development phase, when increase is due mostly to cell transfer equations, applied to certain expanding geometrical forms enlargement, have smaller allometric exponents. modeling bodies of different size. Such is the method introduced in And second, regarding organisms increasing size by cell number: Speakman and Król (2010). It requires evaluation of parameters (1) small metazoans show near-isometric metabolic scaling; (2) related to an insulating layer and its thermal conductivity, squid grow throughout their entire life by increasing cell numbers, accounting for the difference between the core body temperature and have isometric scaling; (3) in the early stages of growth, and the surface temperature. In addition, the complex relationships mammals increase their size through cell number, and have a greater between the size of the body and the size of its limbs have to be allometric exponent; and (4) fishes in the early development phase, taken into account. Despite of all these complexities, the authors when increase is due to increase in cell number, have a greater created a workable model. They found the value of the scaling allometric exponent. exponent to be 0.63, considering it rather as underestimated value. Finally, let us consider a scenario when the cells of a larger Heat dissipation through radiation is usually assumed to be small organism are smaller than in a small organism, that is, when μ2<μ1. compared with heat conduction and convection. For example, in Transforming Eqn 15, we obtain the following: Wolf and Walsberg (2000), the authors estimated that radiation accounts for approximately 5% of heat losses for birds with B B m =m 1=3 M =M B M =M 1þa; 2 ¼ 1ð 1 2Þ ð 2 1Þ¼ 1ð 2 1Þ ð19Þ plumage. The result is not directly transferable to animals without plumage, and for such animals probably more studies are needed. 1/3 α where (μ1/μ2) =(M2/M1) and, because μ2<μ1, α>0, so that the One possible approach to address the issue could be using the allometric exponent is greater than 1. Stefan–Boltzmann equation (or Planck’s formula) for the total energy emitted by an absolutely black body (Shestopaloff, 1993, Representing allometric scaling relationships 2011). ‘Absolute blackness’, that is, when the reflection coefficient Let us compare two forms of representation of metabolic allometric is equal to zero, is a reasonable approximation, because most energy scaling relationships – Eqn 2, and the one we used in our at normal body temperatures is emitted in the infrared spectrum, for derivations: which reflection coefficients of living organisms and natural surfaces are small, usually less than 0.02 (Vollmer and Mollman, b B ¼ B0ðM2=M0Þ ; ð20Þ 2010). Living organisms both emit and absorb energy, so we should find the difference, not the total energy. a B M b which is effectively the same as Eqn 2, if we assume ¼ 0 0 . The can sustain the temperature difference of few Here, M0 is some reference mass with associated metabolic rate B0. degrees Celsius with the surrounding environment for a long time. The allometric exponent b can be variable, as we found above. Let us find the power difference D of emitted and absorbed When mass is measured in absolute values, such as in Eqn 2, the radiation, for a 2°C difference, assuming the outer temperature of a constant a will also be measured in variable units of kg(1–b) m2 s–3 body is 35°C, which approximately corresponds to 308 K: (because the units of measure of the left part in Eqn 2 can be –1 2 –3 a represented as J s =kg m s ). The physical meaning of for such D ¼ sðT 4 T 4 Þ¼6:57108ð3084 3064Þ units of measure is not easy to define. In this regard, the approach in body env : 2: Eqn 20 is more comprehensible, because B0 is always measured in 15 02 W m ð22Þ watts. Mathematically, a reference point (B0,M0) can correspond to any Here, σ is a Stefan–Boltzmann constant. Calculating the total mass. It can be shown as follows. Suppose we want to substitute amount E of required energy for a time period t=24 h for an average b 2 another reference point with index 1. Because B1=B0(M1/M0) ,we human being with the body surface S=1.5 m , we find: can write: E ¼ D t S 15:02 1:5 24 3600 B B M=M b B M =M b M=M b B M=M b; ¼ 0ð 0Þ ¼ 1ð 0 1Þ ð 0Þ ¼ 1ð 1Þ ð21Þ 1:947 106 J: ð23Þ which is the same as Eqn 20. According to Human (2001), an average human requires approximately 9×106 J a day. The obtained value is RESULTS approximately 20% of this amount. In other words, the heat Thermogenesis and heat dissipation – amount of dissipated dissipation through radiation can be of the order of heat losses heat through conduction or convection. Allometric scaling includes two opposite processes existing in Overall, heat dissipation should be considered as an influential inseparable unity: one is energy supply, and the other is energy factor affecting allometric scaling. If the heat dissipation constraints dissipation. Constraints imposed on both sides of this energy balance affect allometric scaling, then this should affect organisms living at are considered as possible causes of allometric scaling. Temperature colder temperatures. Indeed, such higher energy and increased milk variations strongly affect biochemical processes (Hedrick and production at colder temperatures were confirmed for mice Hillman, 2016). Could thermogenesis, heat dissipation constraints (Speakman and Król, 2010). and related changes of body temperatures be the major mechanisms If all organisms had the same geometry, then the allometric responsible for the allometric scaling? Such views were considered scaling due to heat dissipation constraints would be proportional to in previous works (Speakman and Król, 2010; White et al., 2006; surface area, that is to volume2/3. Organisms have different shapes. Hudson, et al., 2013). In the first work, the authors state: ‘We contend For instance, elongation increases the body surface more than that the HDL is a major constraint operating on the expenditure side volume, which can increase the allometric exponent. Besides, HDL of the energy balance equation’. may have different intensity of expression in different organisms. Journal of Experimental Biology

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So, in the real world, the HDL effect should provide a range of energy required for the propulsion of blood by a heart. In an values of allometric exponents. interview, answering a question about the secretory and signal recognition particle pathways, one of the authors, Theg, says: ‘But if Distribution of transportation costs across different scale we do make such a back-of-the-envelope extrapolation, we can levels estimate that the total energy cost to a cell for all its protein transport Organisms transport nutrients and their by-products at different activity may be as high as 15 percent of the total energy scale levels, such as the gastrointestinal tract, the blood supply expenditure’. Another example of high transportation energy costs system, the extracellular matrix, intracellular pathways, etc. in Escherichia coli (but in ATP molecules) is presented in Driessen Different needs are supported by different transportation systems. (1992). Note that cells carry on other transport-related activities Supporting blood supply is different from transportation at the besides protein transport. cellular level. Both require energy, both have certain scaling Of course, more such studies are needed to obtain a mechanisms, and so it is important to know how much energy is comprehensive understanding of the structure of cellular spent for transportation at each scale level. Some works suggest that transportation costs. (Hopefully, this article will initiate interest in the transportation costs in nutrient distribution networks are such studies.) Nonetheless, based on the presented data, we can say responsible for much of the metabolic allometric scaling (West that the total transport expenditures in living organisms at the et al., 2002; Banavar et al., 1999), of which the blood supply system cellular level significantly (tens of times) exceed the amount of is assumed to be a major contributor. Previous work (Banavar et al., energy required for functioning of the blood supply system. 2010) also considers vascular networks as major contributors to Consequently, we are left with the only possible inference that the allometric scaling. According to the authors, ‘an exponent of 3/4 overwhelming fraction of transportation costs is due to cellular emerges naturally … in two simple models of vascular networks’.In biomolecular activity. Thus, if the transport networks are the factor contrast, some studies, in particular Weibel and Hoppeler (2005), do that significantly contributes to the phenomenon of metabolic not support this idea. However, no estimation of energy scaling, then it should be the cells’ transportation costs that consumption by different systems was done. Let us do this. contribute the most. The heart of an adult human pumps approximately 1 to 18 liters of blood per minute, depending on the individual and the level of DISCUSSION − physical exertion, with an average value of 3–4 l min 1. At rest or Cellular transportation costs versus the cellular surface during moderate physical activity, the blood pressure changes from hypothesis diastolic to systolic in the range of 7 to 13 cm (mercury). So, we can In Davison (1955), the author introduced an idea that the size and assume that the heart’s ‘pump’ produces work equivalent to lifting a number of cells are limiting the metabolic rate by restricting nutrient certain volume of mercury to the height of 6 cm. Taking into and waste fluxes through the cells’ surface. He obtained the range of account that the specific weight of mercury is approximately metabolic scaling of 2/3–1, where a value of 2/3 corresponds to an 13.5 g cm−3, it will be equivalent to lifting the same weight of blood increase in organism size that is due to cell enlargement, and a value to the height of 13.5×6=81 cm=0.81 m (as the blood’s specific of 1 corresponds to growth that is entirely due to an increase in cell weight is ∼1gcm−3). Because the systolic blood pressure is number. The idea about the relationship between the allometric maintained for only a fraction of the whole cycle (let us assume, half exponent and cell size and number was also explored in Kozlowski of the cycle), then we should accordingly reduce the height to et al. (2003). The authors argue that the allometric scaling of approximately 0.4 m. Then, the work w done by the heart’s ‘pump’ metabolic rate is a by-product of evolutionary diversification of during the day (24 h) can be found as: genome size, which ‘underlines the participation of cell size and cell number in body size optimization’. The range of the exponent’s w ¼ MgH: ð24Þ values and relationships obtained in the cited article are similar to Davison’s results. Numerically, the same values were obtained in Here, g is the free-fall acceleration equal to 9.8 m s−2, H is height the present study. However, here the similarities end. In essence, the (m) and M is mass (kg). For a blood volume from 1 to 18 l min−1, ‘cell surface’ approach and that of the present study (considering this gives a range from 5645 to 101,606 J. cellular transportation networks) are very different. On the positive According to a previous report (Human, 2001), the daily energy side, this numerical similarity means that all empirical observations requirements for an adult of 17–18 years old with a weight of 68 kg supporting the main idea in Davison (1955) and Kozlowski et al. is 14.3 MJ. Thus, the average energy (and accordingly food) (2003) support the results of the present study as well. required for the heart is approximately 0.14% of the overall average Regarding Davison’s assumption about the restrictiveness of cell energy consumption, for 100% heart efficiency. In reality, surfaces to transfer nutrients and waste products, it was shown efficiency is less, so that the actual energy expenditures could be experimentally in Maaloe and Kjeldgaard (1966) that few times higher, according to Schmidt-Nielsen (1984). Blood microorganisms can increase nutrient flux through the membrane pressure during physical exertion can elevate as well, by a few by several folds instantly, once nutrients become available. Also, it centimeters of mercury, but neither of these adjustments would was found (Shestopaloff, 2014b, 2015) that during normal growth in change our results substantially. In any case, the value is very small a stable nutrient environment, the amoeba increases nutrient compared with the total energy production by a human. consumption per unit of surface by approximately 68%, while However, for our purposes, it is important to know how these fission yeast increases consumption by 340%. In other words, the energy costs relate to the total transportation costs, but not to the cell membrane capacity of these organisms has several-folds reserve total energy production. Shi and Theg (2013) state, ‘We estimate to accommodate the sharp increase of nutrient and waste fluxes that protein import across the plastid envelope membranes when in need. It does not mean that the surface cannot be a factor consumes ∼0.6% of the total light-saturated energy output of the affecting allometric scaling. Rather, it means that the surface area is organelle’. Note that this is only one of many transport pathways unlikely to be the universal factor for this phenomenon we are working in cells. Nonetheless, its relative value is more than the looking for. Journal of Experimental Biology

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For completeness, we should mention the surface area present study discovered that nutrient acquisition and delivery form hypothesis, considered in Hirst et al. (2014) and Glazier et al. the primary level in the hierarchy of different factors contributing to (2015). The authors related surface area to allometric exponents allometric scaling. Other factors can modify, to a certain extent, the using pelagic invertebrates (they change their body shape during allometric exponent defined by cellular transportation mechanisms, growth, which gives a wider range of geometrical forms for testing but not override it. One of the reasons is that nutrient acquisition and the surface area hypothesis). The authors showed that the RTN transportation, considered as a single process, form the foundation of method produces invalid results for these animals, and so they any metabolic activity in any living organism. concluded that the RTN model does not explain the metabolic Such modifications, in particular, can relate to dynamics of scaling phenomenon. However, this statement is true only for global specific biomolecular energy mechanisms considered in Darveau transportation networks, such as blood supply systems, whose et al. (2002). On a practical note, Eqn 17 can accommodate such energy consumption, as we found, is small. However, if we consider modifications, including the ones that are due to different energetic cellular transportation costs, then our results agree with the regimes, through the parameter B1 (and possibly β, but that depends presented experimental data, provided cell size in pelagic on the particular organism). If that is not enough, still, the model has invertebrates does not change much. Change of cell size during lots of flexibility to adapt inputs from other contributing factors. So, ontogeny should be known in order to definitively answer the here we also see that cellular transportation mechanisms and their question. formal description allow integrating inputs from different contributing factors related to energy production and utilization. Cellular transportation and contributing energy pathways Previously, we briefly discussed the approach proposed in Darveau Integrated model of allometric scaling et al. (2002). It considers different supply and demand energy Here, we propose the model of allometric scaling, which includes pathways, whose allometric exponents are then combined in order to the following interacting mechanisms: Energy consumption by calculate the allometric exponent for the entire organism. Such cellular transportation networks, organismal maintenance and factorization on the basis of real biochemical mechanisms is an synthesis metabolism, and heat dissipation (described by important step forward. However, the approach still does not give an HDL theory). All of them present real influential physiological and answer to what is the fundamental cause of allometric scaling as such. biochemical mechanisms. In contrast, cellular transportation represents a general mechanism There are many experimental examples that the metabolic activity across all living species, which defines the amount of nutrients of unicellular organisms and cells in tissues strongly depends on delivered to and consumed by organisms. Nutrient influxes for all temperature. For example, the average growth time of fission yeast individual biochemical contributors, including the ones participating S. pombe at 25°C is 269 min, while at 28°C it is 167 min in energy production and utilization, represent parts of the overall (Shestopaloff, 2015). In Glazier (2010), the author summarizes nutrient influx. We can think of them as branched influxes distributed results obtained in Ivleva (1980) for crustaceans as follows: ‘the from the main node, which is the total influx. The total influx, as we scaling exponent … decreases significantly with temperature found, depends on how much of the nutrients can be used for (b=0.781±0.016, 0.725±0.086, 0.664±0.026 at temperatures of 20, transportation relative to other biomechanical activities. Indeed, no 25 and 29°C)’. delivery equals no nutrients. In this regard, cellular transportation Lower body temperatures correlate with low metabolic activity. represents the primary mechanism, which defines the amount of So, in this case, heat dissipation does not impose meaningful nutrients available for other biochemical activities. constraints, and, if cell size is approximately the same for small and What the present study found is that the specifics of cellular larger animals, the allometric exponent should be close to 1. Indeed, transportation mechanisms, which is faster than the linear increase of deeply hibernating animals have an allometric exponent close to 1 transportation costs when cells enlarge, caps the total amount of (Glazier, 2010; Hudson et al., 2013). However, when food supply is acquired nutrients (Eqns 16 and 17) regardless of nutrient usage. It is not limited, then energy consumption is high, which apparently like when a host offers a plate of cookies to their guests, and then the triggers heat dissipation limits, reducing the allometric exponent guests distribute the cookies between themselves according to their to 2/3. needs and adopted social practices. However, together, the guests Ectothermic animals produce heat as a by-product of muscle and cannot consume more cookies than the host offers. Fewer cookies other biochemical activity, while endothermic animals have a means a lower average number of cookies per guest. The same separate temperature regulating system, whose input is added on top. situation occurs with nutrients that are brought by and under the This additive effect apparently increases the body temperature of management of nutrient transportation mechanisms, which in this endothermic animals more than in ectothermic animals, so that HDL analogy represent the host. So, the idea of combining different factors trigger earlier for them. This, accordingly, reduces the allometric contributing to allometric scaling through energy production very exponent of endothermic animals compared with ectothermic well matches the cellular transportation mechanism affecting the animals – an effect that many experimental observations confirm total amount of consumed nutrients and consequently organism's (Glazier, 2005, 2014). metabolism. In fact, these two approaches complement each other. High levels of metabolism (for instance, during strenuous The concept of combining different contributing factors also physical activity) increase body temperature, which in turn addresses the variability of the allometric exponent that is due to accelerates metabolic reactions and triggers more energy- variations in the energy regime, such as the difference between the producing mechanisms, such as anaerobic mechanisms. The need basal and maximal metabolic rates. The model, which will be to dissipate more heat is met by triggering different mechanisms, presented in the next section, actually provides variability also such as sweating, and by auxiliary means, such as incoming air or through the heat dissipation constraints, in many instances linked to water flow during movement. The raising of body temperature biochemical mechanisms through the temperature changes. However, during physical activity increases the temperature contrast between the main connection between contributing factors and cellular the body and the surrounding environment, which increases heat transportation costs with regard to variability is the following. The dissipation. In addition, organisms have some tolerance to elevated Journal of Experimental Biology

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1.000 so. The parameter c can be determined from histological studies. In 0 Heat dissipation coefficient h the case of complex organisms, its evaluation will involve a variety of cells of different types, and the weighted average is one of the 0.933 0.2 approaches that can be used. b The presented graph and formula embrace known metabolic 0.867 0.4 states and experimental observations, covering a wide range of animals, whose cell sizes change in three dimensions, from torpid and hibernating animals to the highest levels of metabolic activity. 0.800 0.6 There is no explicit incorporation of the famous value of 3/4, but it is included in the range. In the author’s opinion, such a value could be the result of a compromise, when an organisms’ mass increases both Allometric exponent 0.8 0.734 through cell enlargement (which produces the value of 2/3, according to the present study) and increase in cell number 1.0 (which produces an isometric scaling, when the allometric exponent 0.667 0 0.2 0.4 0.6 0.8 1 is equal to 1). A combination of these two mechanisms would result Cell increase c in allometric exponents close to a value of 3/4. The dimensionality of cell increase (in how many dimensions Fig. 1. Dependence of an organism’s allometric exponent on how cells cells increase and by how much) can be accounted for by an contribute to the organism’s mass increase, and on the organism’sheat additional ‘dimensionality’ parameter d, defined as follows: dissipation regime. Parameter c defines to what extent the organism's mass increases as a result of cell enlargement, and to what extent the mass grows as d ¼ðux þ uy þ uzÞ= maxfux; uy; uzg: ð27Þ a result of increase in cell number. Dependence on mass increase of three- dimensional organisms (through cell enlargement or/and increase of cell d u u u number), and on the organism’s heat dissipation regime. Here, changes from 1 to 3, and x, y and z are relative length increases along the major body axes (‘major’ in a mathematical sense, which means axes of canonical representation of a body, such body temperatures. Taken together, all of these as major axes of an ellipsoid). This third parameter d can be easily mechanisms support the ability of animals to dissipate heat at adopted both into Eqn 26 and Fig. 1, if needed. maximum metabolic activity, which results in greater values of allometric exponents. This inference is well supported by Conclusions experimental results from Weibel and Hoppeler (2005), in which The purpose of the present study was to propose an allometric the exponent value of 0.872 was obtained. scaling model based on real physiological and biochemical Quantitatively, the model is as follows. Let us introduce a mechanisms, which could explain known facts and predict the dimensionless parameter h, the ‘heat dissipation coefficient’, allometric exponent. Descriptive models do have value, but to changing in the range of 0–1 and defined as h=Hdis/HdisMax. Here, create a quantitative predictive model, the real physical mechanisms Hdis is the dissipated heat and HdisMax is a maximum possible have to be considered. In the present work, the role of cellular dissipated heat under given conditions, both in watts. The words transportation was studied. It was found that cellular transportation ‘given conditions’ are of importance. At high levels of physical represents a substantial part of all energy expenses. The model exertion, additional air convection as a result of movement accounts balances intrinsic and systemic factors, and incorporates two for 80% of all heat losses in birds (Speakman and Król, 2010), so different mechanisms, thus reflecting on the multifactorial nature that even when the dissipated energy is high, its fraction from the of allometric scaling. In the author’s view, the model potentially has maximum possible dissipated heat can be relatively small because high predictive power. Of course, much work still has to be done to of the ‘triggering’ of additional cooling mechanisms. The water make it an efficient workable instrument. Nonetheless, in its present environment in which aquatic animals live potentially allows the form, it allows us to define directions of future studies, such as dissipation of much larger quantities of heat than such animals inclusion of cell size and possibly cell geometry, evaluation of the produce, and accordingly a higher value of the allometric exponent. heat dissipation regime, and determination of the biochemistry and The next parameter, c, is the cell enlargement parameter, i.e. the energy requirements of cellular transportation networks. fraction of body mass increase that is due to cell enlargement, which changes from 0 to 1 (0 corresponds to mass increase that is due Acknowledgements The author thanks the Editor, Professor H. Hoppeler, whose help in editorial matters solely to increase in cell number, and 1 corresponds to mass went beyond professional duties; Professor C. White, for valuable comments and increases exclusively through cell enlargement). It is defined as the support of this study; the reviewers, for helpful and important comments and ratio of the relative increment of cell mass, (μ/μ0)/μ0=(μ/μ0–1), to the suggestions, and Dr A. Y. Shestopaloff, for editing and discussions. relative increment of the whole organism’s mass, (M/M0–1): Competing interests c m=m = M=M : ¼ð 0 1Þ ð 0 1Þ ð25Þ The author declares no competing or financial interests.

Fig. 1 presents the model in graphical form. The analytical Funding formula describing the lines on the graph, and which can be used for This research received no specific grant from any funding agency in the public, predicting the value of the allometric exponent, is as follows: commercial or not-for-profit sectors. 2 1 b ¼ þ ð1 hÞð1 cÞ: ð26Þ References 3 3 Banavar, J. R., Maritan, A. and Rinaldo, A. (1999). Size and form in efficient transportation networks. Nature 399, 130-132. Banavar, J. R., Moses, M. E., Brown, J. H., Damuth, J., Rinaldo, A., Sibly, R. M. Evaluation of the heat dissipation coefficient presents a certain and Maritan, A. (2010). A general basis for quarter-power scaling in animals. challenge, although there are no limitations, in principle, for doing Proc. Natl. Acad. Sci. USA 107, 15816-15820. Journal of Experimental Biology

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