Metabolic allometric scaling model: combining cellular transportation and heat dissipation constraints

Here, a is assumed to be a constant.

Presently, there are several theories explaining the phenomenon of metabolic allometric scaling. The resource transport network (RTN) theory assigns the effect to a single foundational mechanism related to transportation costs for moving nutrients and waste products over the internal organismal networks, such as vascular systems (Brown and West, 2000; West et al., 1999, 2002; Savage et al., 2004; Banavar et al., 2010, 2014; Kearney and White, 2012). Another approach accounts for transportation costs of materials through exchange surfaces, such as skin, lungs, etc. (Rubner, 1883; Okie, 2013; see also Hirst, et al., 2014; Glazier, 2005, 2014). The similar idea of ‘surface uptake limitation’ was applied to cells (Davison, 1955). Dynamic energy budget theory, whose analysis and comparison with RTN can be found in Kearney and White (2012), uses generalized surface area and volume relationships. Heat dissipation limit (HDL) theory (Speakman and Król, 2010) and its discussion (Glazier, 2005, 2014; Hudson, et al., 2013) suggests and provides convincing proofs that the ability of organisms to dissipate heat influences their metabolic characteristics. The metabolic-level boundaries hypothesis (Glazier, 2010) generalizes observations that allometric scaling (1) is a variable parameter and (2) its value depends on the level of metabolic activity in a U-shaped manner, with extrema corresponding to low (dormant, hibernating) and high (extreme physical exertion) metabolic activities.

The study of Darveau et al. (2002) adheres to an idea of multiple factors contributing to the value of the allometric exponent, considering different biomolecular mechanisms participating in ATP turnover as the major contributors, together with a ‘layering of function at various levels of organization’, such as different scale levels. Allometric exponents found separately for each contributing mechanism are combined as their weighted sum. Using energy that is tied to particular biomolecular mechanisms as a common denominator to unite different contributing factors is a significant step.

Because the metabolic properties of organisms depend on many factors and environmental conditions, the explanation of allometric scaling should instead involve a combination of interacting mechanisms. The hierarchy, ‘weight’ and triggering into action of particular mechanisms should be defined by an organism’s characteristics and interaction with its environment. At a systemic level, in the author's view, heat dissipation (HDL) theory is of importance. Temperature is directly tied to foundational-level physiological and biochemical mechanisms, as they are very sensitive to temperature.

The present study supports the multifactor nature of allometric scaling. However, there are the following differences. (1) The overwhelming majority of previous works promote an idea that allometric scaling is due to systemic-level mechanisms. The present study shows that cellular-level transportation mechanisms, although coordinated at a systemic level, present a factor of primary importance for metabolic allometric scaling. (2) Although the cell's size changes significantly less than the interspecific allometric scaling across a wide range of animal sizes, it follows from the present study that the transportation costs, when associated with the cell size, present an important contributing factor into ontogenetic differences in allometric scaling.

The present study introduces a general method for calculating transportation costs. Then, it presents arguments in favor of the cell size factor and associated transportation costs as important mechanisms influencing allometric scaling, and introduces formulas for calculating allometric exponents. Briefly, the HDL theory is considered. Finally, an allometric scaling model incorporating cellular transportation costs and heat dissipation constraints is introduced.

Nutrient distribution networks models were criticized as fairly restrictive in applicability, problematic with regard to some initial assumptions, and that they could not explain the known diversity of metabolic scaling relationships (White et al., 2011, 2009, 2007; Weibel and Hoppeler, 2005; Glazier, 2005; Kozlowski and Konarzewski, 2004). Here, I present a general method for determining the amount of nutrients required for transportation needs, which uses substantially less restrictive assumptions. The model is applicable both to interspecific and intraspecific allometric scaling, although the present study considers only intraspecific scaling.

As in other works, the present study assumes that the amount of nutrients that is required to support transportation networks is proportional to the distance the substances travel along these networks. Validity of this postulate was rather an axiomatic proposition in previous works. In fact, several previous studies (Shestopaloff, 2012a,b, 2014a,b, 2015; Shestopaloff and Sbalzarini, 2014) confirm this postulate through calculations of nutrient influxes for microorganisms, and for dog and human livers. Another indirect proof relates to stability of the growth period of large and small organisms. It was shown previously (Shestopaloff, 2012a,b, 2014a,b) that if transportation nutrient expenditures are not proportional to the traveled distance, then the growth periods of large and small organisms would be substantially different, which is not the case.

where k_t is a constant coefficient that represents the cost of transportation as a fraction of the transported nutrients per unit length (so that the unit of measure is m^–1), and distance l is the consequence of the postulate that transportation costs are proportional to distance. Note that the cost of transportation of nutrients F is also valued in nutrients and measured in kg.

where f is the density of nutrients per unit of volume (kg m⁻³) and k_t is the cost of transportation, as in Eqn 3.

In other words, transportation costs in a disk increasing in two dimensions are proportional to 3/2 power of volume. It is interesting to note that the amoeba, when it grows, also increases its nutrient consumption in a manner similar to that expressed in Eqn 8 (Shestopaloff, 2012a, 2015).

The obtained result is valid for other forms increasing proportionally in two dimensions, such as squares, rectangles and stars, whose height remains constant.

Here, A=(3/4)^4/3π^–1/3k_t is a constant (measured in m⁻¹). Note that this result (proportionality to V^4/3 of transportation costs in a sphere-like organism) is valid for other geometrical shapes that increase proportionally in three dimensions.

The formulas above were obtained under the assumption that cells are able to increase nutrient consumption to cover a faster increase of transportation costs than increase of volume and, associated with it, mass. Examples of such unicellular organisms (amoeba, fission yeast Schizosaccharomyces pombe and Saccharomyces cerevisiae) are presented in Shestopaloff (2012a,b, 2015).

Thus, the ratio F₁/F₂=2^1/3≈1.26>1, which means that a larger cell needs more nutrients for transportation than two smaller cells together with the same total mass. This is due to longer transportation networks in the larger cell. Of course, a larger cell may also compensate for the higher transportation costs by different means.

Similarly, we can consider compositions of cells that grow in one and two dimensions. Using Eqns 5 and 7 and performing similar transformations as in Eqn 12, we obtain the following values for ratios of transportation costs: K₁=k and K₂=k^1/2.

What does Eqn 12 mean? If we have two organisms with the same mass, but composed of different sizes of cells, then to obtain the same amount of nutrients per unit of mass, the organism with larger cells requires more nutrients for transportation. However, in many instances, organisms, regardless of their cell size, receive about the same amount of nutrients per unit of mass because of the specifics of nutrient distribution networks, such as the blood circulation system. In this case, the transportation costs have to be scaled according to nutrient availability. The question is, by how much? There is no simple answer to this question, because it depends on the organism's biochemistry and specifics of nutrient distribution networks, among other factors. However, we can be certain in two things: (1) such scaling affects the total amount of nutrients; and (2) the distribution of nutrients between transportation needs and other biochemical activities (meaning the ratio) will remain, because transportation costs represent an intrinsic factor to cells of a particular type and size. In other words, the decrease of nutrients used for transportation will proportionally reduce the total amount of transported nutrients (of course, including the nutrients used for transportation). The metabolic rate, in turn, is proportional to the total amount of consumed nutrients. This last assumption is a reasonable one, because we consider the same organisms, whose metabolic processing efficiency per unit of nutrient mass should not vary much.

In other words, the metabolic rate of organisms whose mass increases solely because of the increase of cell size scales as a power of 2/3 of the masses' ratio.

Here, we took into account that M₁=μ₁N₁ and M₂=μ₂N₂.

We do not know how much the parameter β can vary. At a first glance, there are no reasons for it to significantly change depending on the state of the organism (e.g. torpid or physically active), but the issue requires further study. In the case when β≈const, Eqn 18 complies with numerous experimental observations. Indeed, the position of regression lines often varies vertically for different organisms, although the allometric slopes can be the same. According to Eqn 18, the reasons could be different values of scaling coefficients (lnβ) for different organisms, and different values of lnB₁. When the number of cells remains the same (the cell's mass increases proportionally to the organism's mass), the fourth term (1/3)ln(N₂/N₁) also becomes a constant. If an organism's mass increases as a result of both cell enlargement and cell number, then this fourth term provides a line with a non-zero slope, which, summing with the line defined by the third term (2/3)lnB₁(M₂/M₁), adds to the value of the allometric exponent 2/3, thus providing the range from 2/3 to 1.

Note that the fraction of nutrients (from the total amount of nutrients) used for needs other than transportation will be smaller in organisms with larger cells than in organisms with smaller cells. Thus, fewer nutrients may be available to support other cell functions, for instance, DNA repair mechanisms, whose inefficiency may cause transformation to cancerous cells. It is well known that small cells generally are significantly more resistant to cancerous transformations. However, once such a transformation occurs, they represent the most malicious cancer forms, which are very difficult to cure. Although speculative at this point, this is an observation worth exploring.

The obtained results from the present study agree with and explain known empirical facts. Larger organisms ontogenetically often have larger cells. In Chown et al. (2007), seven of eight ant species increased their size primarily through the enlargement of cells, but not through the number of cells. When this is the case, then, according to Eqn 14, the allometric exponent is close to 2/3. Indeed, in Chown et al. (2007), the authors found for ants that ‘… the intraspecific scaling exponents varied from 0.67 to 1.0. Moreover, in the species where metabolic rate scaled as mass^1.0, cell size did not contribute significantly to models of body size variation, only cell number was significant. Where the scaling exponent was <1.0, cell size played an increasingly important role in accounting for size variation.’ Also, the authors confirmed that when the cell size remains the same, the metabolic allometry is close to isometric (which is the case in Eqn 16). So, the present results about the role of cellular transportation costs in metabolic scaling comply with all experimental observations of Chown et al. (2007).

Therefore, the way organisms increase their size during ontogeny – that is, either through the increase of cell number or cell size, or a combination of both – explains the variability of metabolic scaling from 2/3 to 1, with a value of 2/3 corresponding to an increase by cell enlargement, and a value of 1 corresponding to mass increase by number of cells.

If we repeat computations similar to Eqns 12–16 for a two-dimensional increase (K₂=k^1/2), then the range for the allometric exponent will be 1/2–1, and for a one-dimensional increase, 0–1. In the last case, these have to be some exotic, if not hypothetical, organisms composed of tube-like cells of the same width fed from ends.

In addition to the described results from Chown et al. (2007), previous works (Glazier, 2005, 2014) present data from different sources that support the present findings both for the boundary values and for transitional growth scenarios.

First, regarding organisms increasing their size by cell enlargement: (1) nematodes have an allometric exponent b close to 2/3 [four species, b=0.64–0.72; one species, b=0.43; one species (low food levels), b=0.77; mean value of b for seven nematode species is 0.677]; (2) rotifers have b close to 2/3; (3) in the later stages of growth, mammals increase their size through cell enlargement, and have a smaller allometric exponent; and (4) in the later development phase, when increase is due mostly to cell enlargement, fishes have smaller allometric exponents.

And second, regarding organisms increasing size by cell number: (1) small metazoans show near-isometric metabolic scaling; (2) squid grow throughout their entire life by increasing cell numbers, and have isometric scaling; (3) in the early stages of growth, mammals increase their size through cell number, and have a greater allometric exponent; and (4) fishes in the early development phase, when increase is due to increase in cell number, have a greater allometric exponent.

where (μ₁/μ₂)^1/3=(M₂/M₁)^α and, because μ₂<μ₁, α>0, so that the allometric exponent is greater than 1.

which is effectively the same as Eqn 2, if we assume ⁠. Here, M₀ is some reference mass with associated metabolic rate B₀.

The allometric exponent b can be variable, as we found above. When mass is measured in absolute values, such as in Eqn 2, the constant a will also be measured in variable units of kg^(1–b) m² s^–3 (because the units of measure of the left part in Eqn 2 can be represented as J s^–1=kg m² s^–3). The physical meaning of a for such units of measure is not easy to define. In this regard, the approach in Eqn 20 is more comprehensible, because B₀ is always measured in watts.

Allometric scaling includes two opposite processes existing in inseparable unity: one is energy supply, and the other is energy dissipation. Constraints imposed on both sides of this energy balance are considered as possible causes of allometric scaling. Temperature variations strongly affect biochemical processes (Hedrick and Hillman, 2016). Could thermogenesis, heat dissipation constraints and related changes of body temperatures be the major mechanisms responsible for the allometric scaling? Such views were considered in previous works (Speakman and Król, 2010; White et al., 2006; Hudson, et al., 2013). In the first work, the authors state: ‘We contend that the HDL is a major constraint operating on the expenditure side of the energy balance equation’.

The heat dissipation is done via heat conduction, convection and radiation. Dissipated heat is usually accounted for using heat transfer equations, applied to certain expanding geometrical forms modeling bodies of different size. Such is the method introduced in Speakman and Król (2010). It requires evaluation of parameters related to an insulating layer and its thermal conductivity, accounting for the difference between the core body temperature and the surface temperature. In addition, the complex relationships between the size of the body and the size of its limbs have to be taken into account. Despite of all these complexities, the authors created a workable model. They found the value of the scaling exponent to be 0.63, considering it rather as underestimated value.

Heat dissipation through radiation is usually assumed to be small compared with heat conduction and convection. For example, in Wolf and Walsberg (2000), the authors estimated that radiation accounts for approximately 5% of heat losses for birds with plumage. The result is not directly transferable to animals without plumage, and for such animals probably more studies are needed. One possible approach to address the issue could be using the Stefan–Boltzmann equation (or Planck's formula) for the total energy emitted by an absolutely black body (Shestopaloff, 1993, 2011). ‘Absolute blackness’, that is, when the reflection coefficient is equal to zero, is a reasonable approximation, because most energy at normal body temperatures is emitted in the infrared spectrum, for which reflection coefficients of living organisms and natural surfaces are small, usually less than 0.02 (Vollmer and Mollman, 2010). Living organisms both emit and absorb energy, so we should find the difference, not the total energy.

According to Human (2001), an average human requires approximately 9×10⁶ J a day. The obtained value is approximately 20% of this amount. In other words, the heat dissipation through radiation can be of the order of heat losses through conduction or convection.

Overall, heat dissipation should be considered as an influential factor affecting allometric scaling. If the heat dissipation constraints affect allometric scaling, then this should affect organisms living at colder temperatures. Indeed, such higher energy and increased milk production at colder temperatures were confirmed for mice (Speakman and Król, 2010).

If all organisms had the same geometry, then the allometric scaling due to heat dissipation constraints would be proportional to surface area, that is to volume^2/3. Organisms have different shapes. For instance, elongation increases the body surface more than volume, which can increase the allometric exponent. Besides, HDL may have different intensity of expression in different organisms. So, in the real world, the HDL effect should provide a range of values of allometric exponents.

Organisms transport nutrients and their by-products at different scale levels, such as the gastrointestinal tract, the blood supply system, the extracellular matrix, intracellular pathways, etc. Different needs are supported by different transportation systems. Supporting blood supply is different from transportation at the cellular level. Both require energy, both have certain scaling mechanisms, and so it is important to know how much energy is spent for transportation at each scale level. Some works suggest that the transportation costs in nutrient distribution networks are responsible for much of the metabolic allometric scaling (West et al., 2002; Banavar et al., 1999), of which the blood supply system is assumed to be a major contributor. Previous work (Banavar et al., 2010) also considers vascular networks as major contributors to allometric scaling. According to the authors, ‘an exponent of 3/4 emerges naturally … in two simple models of vascular networks’. In contrast, some studies, in particular Weibel and Hoppeler (2005), do not support this idea. However, no estimation of energy consumption by different systems was done. Let us do this.

Here, g is the free-fall acceleration equal to 9.8 m s⁻², H is height (m) and M is mass (kg). For a blood volume from 1 to 18 l min⁻¹, this gives a range from 5645 to 101,606 J.

According to a previous report (Human, 2001), the daily energy requirements for an adult of 17–18 years old with a weight of 68 kg is 14.3 MJ. Thus, the average energy (and accordingly food) required for the heart is approximately 0.14% of the overall average energy consumption, for 100% heart efficiency. In reality, efficiency is less, so that the actual energy expenditures could be few times higher, according to Schmidt-Nielsen (1984). Blood pressure during physical exertion can elevate as well, by a few centimeters of mercury, but neither of these adjustments would change our results substantially. In any case, the value is very small compared with the total energy production by a human.

However, for our purposes, it is important to know how these energy costs relate to the total transportation costs, but not to the total energy production. Shi and Theg (2013) state, ‘We estimate that protein import across the plastid envelope membranes consumes ∼0.6% of the total light-saturated energy output of the organelle’. Note that this is only one of many transport pathways working in cells. Nonetheless, its relative value is more than the energy required for the propulsion of blood by a heart. In an interview, answering a question about the secretory and signal recognition particle pathways, one of the authors, Theg, says: ‘But if we do make such a back-of-the-envelope extrapolation, we can estimate that the total energy cost to a cell for all its protein transport activity may be as high as 15 percent of the total energy expenditure’. Another example of high transportation energy costs in Escherichia coli (but in ATP molecules) is presented in Driessen (1992). Note that cells carry on other transport-related activities besides protein transport.

Of course, more such studies are needed to obtain a comprehensive understanding of the structure of cellular transportation costs. (Hopefully, this article will initiate interest in such studies.) Nonetheless, based on the presented data, we can say that the total transport expenditures in living organisms at the cellular level significantly (tens of times) exceed the amount of energy required for functioning of the blood supply system. Consequently, we are left with the only possible inference that the overwhelming fraction of transportation costs is due to cellular biomolecular activity. Thus, if the transport networks are the factor that significantly contributes to the phenomenon of metabolic scaling, then it should be the cells' transportation costs that contribute the most.

In Davison (1955), the author introduced an idea that the size and number of cells are limiting the metabolic rate by restricting nutrient and waste fluxes through the cells' surface. He obtained the range of metabolic scaling of 2/3–1, where a value of 2/3 corresponds to an increase in organism size that is due to cell enlargement, and a value of 1 corresponds to growth that is entirely due to an increase in cell number. The idea about the relationship between the allometric exponent and cell size and number was also explored in Kozlowski et al. (2003). The authors argue that the allometric scaling of metabolic rate is a by-product of evolutionary diversification of genome size, which ‘underlines the participation of cell size and cell number in body size optimization’. The range of the exponent's values and relationships obtained in the cited article are similar to Davison's results. Numerically, the same values were obtained in the present study. However, here the similarities end. In essence, the ‘cell surface’ approach and that of the present study (considering cellular transportation networks) are very different. On the positive side, this numerical similarity means that all empirical observations supporting the main idea in Davison (1955) and Kozlowski et al. (2003) support the results of the present study as well.

Regarding Davison's assumption about the restrictiveness of cell surfaces to transfer nutrients and waste products, it was shown experimentally in Maaloe and Kjeldgaard (1966) that microorganisms can increase nutrient flux through the membrane by several folds instantly, once nutrients become available. Also, it was found (Shestopaloff, 2014b, 2015) that during normal growth in a stable nutrient environment, the amoeba increases nutrient consumption per unit of surface by approximately 68%, while fission yeast increases consumption by 340%. In other words, the cell membrane capacity of these organisms has several-folds reserve to accommodate the sharp increase of nutrient and waste fluxes when in need. It does not mean that the surface cannot be a factor affecting allometric scaling. Rather, it means that the surface area is unlikely to be the universal factor for this phenomenon we are looking for.

For completeness, we should mention the surface area hypothesis, considered in Hirst et al. (2014) and Glazier et al. (2015). The authors related surface area to allometric exponents using pelagic invertebrates (they change their body shape during growth, which gives a wider range of geometrical forms for testing the surface area hypothesis). The authors showed that the RTN method produces invalid results for these animals, and so they concluded that the RTN model does not explain the metabolic scaling phenomenon. However, this statement is true only for global transportation networks, such as blood supply systems, whose energy consumption, as we found, is small. However, if we consider cellular transportation costs, then our results agree with the presented experimental data, provided cell size in pelagic invertebrates does not change much. Change of cell size during ontogeny should be known in order to definitively answer the question.

Previously, we briefly discussed the approach proposed in Darveau et al. (2002). It considers different supply and demand energy pathways, whose allometric exponents are then combined in order to calculate the allometric exponent for the entire organism. Such factorization on the basis of real biochemical mechanisms is an important step forward. However, the approach still does not give an answer to what is the fundamental cause of allometric scaling as such.

In contrast, cellular transportation represents a general mechanism across all living species, which defines the amount of nutrients delivered to and consumed by organisms. Nutrient influxes for all individual biochemical contributors, including the ones participating in energy production and utilization, represent parts of the overall nutrient influx. We can think of them as branched influxes distributed from the main node, which is the total influx. The total influx, as we found, depends on how much of the nutrients can be used for transportation relative to other biomechanical activities. Indeed, no delivery equals no nutrients. In this regard, cellular transportation represents the primary mechanism, which defines the amount of nutrients available for other biochemical activities.

What the present study found is that the specifics of cellular transportation mechanisms, which is faster than the linear increase of transportation costs when cells enlarge, caps the total amount of acquired nutrients (Eqns 16 and 17) regardless of nutrient usage. It is like when a host offers a plate of cookies to their guests, and then the guests distribute the cookies between themselves according to their needs and adopted social practices. However, together, the guests cannot consume more cookies than the host offers. Fewer cookies means a lower average number of cookies per guest. The same situation occurs with nutrients that are brought by and under the management of nutrient transportation mechanisms, which in this analogy represent the host. So, the idea of combining different factors contributing to allometric scaling through energy production very well matches the cellular transportation mechanism affecting the total amount of consumed nutrients and consequently organism's metabolism. In fact, these two approaches complement each other.

The concept of combining different contributing factors also addresses the variability of the allometric exponent that is due to variations in the energy regime, such as the difference between the basal and maximal metabolic rates. The model, which will be presented in the next section, actually provides variability also through the heat dissipation constraints, in many instances linked to biochemical mechanisms through the temperature changes. However, the main connection between contributing factors and cellular transportation costs with regard to variability is the following. The present study discovered that nutrient acquisition and delivery form the primary level in the hierarchy of different factors contributing to allometric scaling. Other factors can modify, to a certain extent, the allometric exponent defined by cellular transportation mechanisms, but not override it. One of the reasons is that nutrient acquisition and transportation, considered as a single process, form the foundation of any metabolic activity in any living organism.

Such modifications, in particular, can relate to dynamics of specific biomolecular energy mechanisms considered in Darveau et al. (2002). On a practical note, Eqn 17 can accommodate such modifications, including the ones that are due to different energetic regimes, through the parameter B₁ (and possibly β, but that depends on the particular organism). If that is not enough, still, the model has lots of flexibility to adapt inputs from other contributing factors. So, here we also see that cellular transportation mechanisms and their formal description allow integrating inputs from different contributing factors related to energy production and utilization.

Here, we propose the model of allometric scaling, which includes the following interacting mechanisms: Energy consumption by cellular transportation networks, organismal maintenance and biomass synthesis metabolism, and heat dissipation (described by HDL theory). All of them present real influential physiological and biochemical mechanisms.

There are many experimental examples that the metabolic activity of unicellular organisms and cells in tissues strongly depends on temperature. For example, the average growth time of fission yeast S. pombe at 25°C is 269 min, while at 28°C it is 167 min (Shestopaloff, 2015). In Glazier (2010), the author summarizes results obtained in Ivleva (1980) for crustaceans as follows: ‘the scaling exponent … decreases significantly with habitat temperature (b=0.781±0.016, 0.725±0.086, 0.664±0.026 at temperatures of 20, 25 and 29°C)’.

Lower body temperatures correlate with low metabolic activity. So, in this case, heat dissipation does not impose meaningful constraints, and, if cell size is approximately the same for small and larger animals, the allometric exponent should be close to 1. Indeed, deeply hibernating animals have an allometric exponent close to 1 (Glazier, 2010; Hudson et al., 2013). However, when food supply is not limited, then energy consumption is high, which apparently triggers heat dissipation limits, reducing the allometric exponent to 2/3.

Ectothermic animals produce heat as a by-product of muscle and other biochemical activity, while endothermic animals have a separate temperature regulating system, whose input is added on top. This additive effect apparently increases the body temperature of endothermic animals more than in ectothermic animals, so that HDL trigger earlier for them. This, accordingly, reduces the allometric exponent of endothermic animals compared with ectothermic animals – an effect that many experimental observations confirm (Glazier, 2005, 2014).

High levels of metabolism (for instance, during strenuous physical activity) increase body temperature, which in turn accelerates metabolic reactions and triggers more energy-producing mechanisms, such as anaerobic mechanisms. The need to dissipate more heat is met by triggering different mechanisms, such as sweating, and by auxiliary means, such as incoming air or water flow during movement. The raising of body temperature during physical activity increases the temperature contrast between the body and the surrounding environment, which increases heat dissipation. In addition, organisms have some tolerance to elevated body temperatures. Taken together, all of these adaptation mechanisms support the ability of animals to dissipate heat at maximum metabolic activity, which results in greater values of allometric exponents. This inference is well supported by experimental results from Weibel and Hoppeler (2005), in which the exponent value of 0.872 was obtained.

Quantitatively, the model is as follows. Let us introduce a dimensionless parameter h, the ‘heat dissipation coefficient’, changing in the range of 0–1 and defined as h=H_dis/H_disMax. Here, H_dis is the dissipated heat and H_disMax is a maximum possible dissipated heat under given conditions, both in watts. The words ‘given conditions’ are of importance. At high levels of physical exertion, additional air convection as a result of movement accounts for 80% of all heat losses in birds (Speakman and Król, 2010), so that even when the dissipated energy is high, its fraction from the maximum possible dissipated heat can be relatively small because of the ‘triggering’ of additional cooling mechanisms. The water environment in which aquatic animals live potentially allows the dissipation of much larger quantities of heat than such animals produce, and accordingly a higher value of the allometric exponent.

Evaluation of the heat dissipation coefficient presents a certain challenge, although there are no limitations, in principle, for doing so. The parameter c can be determined from histological studies. In the case of complex organisms, its evaluation will involve a variety of cells of different types, and the weighted average is one of the approaches that can be used.

The presented graph and formula embrace known metabolic 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. 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 through cell enlargement (which produces the value of 2/3, according to the present study) and increase in cell number (which produces an isometric scaling, when the allometric exponent is equal to 1). A combination of these two mechanisms would result in allometric exponents close to a value of 3/4.

Here, d changes from 1 to 3, and u_x, u_y and u_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 as major axes of an ellipsoid). This third parameter d can be easily adopted both into Eqn 26 and Fig. 1, if needed.

The purpose of the present study was to propose an allometric scaling model based on real physiological and biochemical mechanisms, which could explain known facts and predict the allometric exponent. Descriptive models do have value, but to create a quantitative predictive model, the real physical mechanisms have to be considered. In the present work, the role of cellular transportation was studied. It was found that cellular transportation represents a substantial part of all energy expenses. The model balances intrinsic and systemic factors, and incorporates two different mechanisms, thus reflecting on the multifactorial nature of allometric scaling. In the author's view, the model potentially has high predictive power. Of course, much work still has to be done to make it an efficient workable instrument. Nonetheless, in its present form, it allows us to define directions of future studies, such as inclusion of cell size and possibly cell geometry, evaluation of the heat dissipation regime, and determination of the biochemistry and energy requirements of cellular transportation networks.

The author thanks the Editor, Professor H. Hoppeler, whose help in editorial matters went beyond professional duties; Professor C. White, for valuable comments and support of this study; the reviewers, for helpful and important comments and suggestions, and Dr A. Y. Shestopaloff, for editing and discussions.