Another predictable Scaling Law?

Leading on from last week’s discussion of universal scaling laws within Nature, brings us to yet another seemingly universal phenomenon; a  recurrent and predictable embedded pattern,  formulated  and traditionally associated with the experimental agriculturalist Max Kleiber. His theory relates to the apparent predictable ¾ power laws where metabolic rate and mass are plotted for a diverse range of animals (see Kleiber, 1932 – Body size and metabolism). Link

Note, that all the images used in graphs within the following charts are my addition to existing graphs and data and are used to simplify and clearly illustrate the concept of scaling.

 scales metabolismFig. 1:  Based upon the metabolic rate to mass/surface area presented by Kleiber’s ¾ scaling law. As mammal mass/surface area increases: so does its metabolic rate increase on a scalable gradient slope.

Applying Kleiber’s laws in a more modern era has proved rather fruitful and more researchers have added on anything from amoebas to antelopes since his original proposal. Everything really would appear to scale in terms of metabolic rate to mass by factors of predictable magnitude. Now there has been hot debate regarding this law and the fact that it seems to apply almost universally and there are always exceptions to the rule. However, as seen in a science paper entitled: Experimental allometry: effect of size manipulation on metabolic rate of colonial ascidians published in more recent years, this issue has been fully addressed by using actual living populations:

The allometric scaling of metabolic rate of organisms, the three-quarters power rule, has led to a questioning of the basis for the relation. We attacked this problem experimentally for the first time by employing the modular organism, the ascidian that forms a single layered flat colony, as a model system. The metabolic rate and colony size followed the three-quarters power relation, which held even after the colony size was experimentally manipulated. Our results established that the three-quarters power relation is a real continuous function, not an imaginary statistical regression.

Nakaya, Saito and Motokawa (2005. abstract),


 The ¾ law is best explained using the information provided by A Van Aken from Brighton and Sussex medical school (slide-share) entitled: Neuroscience made understandable (link) which I have paraphrased as follows: When reading these charts where (body weight) mass is plotted against metabolic rate, the chart is showing a gradient and scalable linear relationship (obeying the ¾ power laws). The numbers that you see above are transformed from the actual weights etc of different animals to a logarithmic scale. Base 10 is used in this case and the reason for this is that if you attempted to plot the actual range of magnitudes such as a mouse of less than 100 grams against elephants weight over 1000kg on same scale, these would be off the charts. Therefore logarithmic scales are used so that we can view them on the same chart. However, it does not take away from the fact that these animals when body weight and metabolism are plotted reflect the seemingly universal scaling law or the ¾ power law as seen on the gradient slope.

All in all, the same predictable properties of scale are detectable across all of Nature according to many in-depth studies since Kleiber’s time. For instance, this seemingly universal pattern of the ¾ power laws is highlighted in an interview with West in the New York Times: Of Mice and Elephants: a Matter of Scale in the following:

This relationship seems to hold across the animal kingdom, from shrew to blue whale, and it has since been extended all the way down to single-celled organisms, and possibly within the cells themselves to the internal structures called mitochondria that turn nutrients into energy.

—Johnston (The New York Times dated to Jan 12th 1999)


 Although, the ¾ power law applies to animals ranging from microbial life to elephants as will become clearly apparent as we proceed, West and Brown encountered a problem which arose when they attempted to apply the same ¾ scaling law to the very different world of plants. This is important to mention, as we are attempting to find universals for evolutionary complexity itself, but I also point to this issue to demonstrate how science should work as exemplified in a study undertaken by West et all and illustrates that when physicists get it wrong, as they occasionally do: they admit it and go back and pose different questions and keep investigating. This is taken from the  same interview article as above: Of Mice and Elephants: a Matter of Scale:

 What emerged closely approximated a so-called fractal network, in which each tiny part is a replica of the whole. Magnify the network of blood vessels in a hand and the image resembles one of an entire circulatory system. And to be as efficient as possible, the network also had to be ”area-preserving.” If a branch split into three daughter branches, their cross-sectional areas had to add up to that of the parent branch. This would insure that blood or sap would continue to move at the same speed throughout the organism. The scientists were delighted to see that the model gave rise to three-quarter-power scaling between metabolic rate and body mass. But the system worked only for plants.

”We worked through the model and made clear predictions about mammals,” Dr. Brown said, ”every single one of which was wrong.”

In making the model as simple as possible, the scientists had hoped they could ignore the fact that blood is pumped by the heart in pulses and treat mammals as though they were trees. After studying hydrodynamics, they realized they needed a way to slow the pulsing blood as the vessels got tinier and tinier. These finer parts of the network would not be area-preserving but area-increasing: the cross sections of the daughter branches would add up to a sum greater than the parent branch, spreading the blood over a larger area. After adding these and other complications, they found that the model also predicted three-quarter-power scaling in mammals. Other quarter-power scaling laws also emerged naturally from the equations. Evolution, it seemed, has overcome the natural limitations of simple geometric scaling by developing these very efficient fractal-like webs.

—Johnston (The New York Times dated to Jan 12th 1999)


At least these scientists were willing to ask the right question using their multidisciplinary tools of enquiry. Anyway, they figured it out in the end and this gives us a much better understanding of how the same laws apply across all species, but that Nature has sometimes made adaptations to these systems to work at their optimum level according to resources and it would appear to do it using fractal networks. The nature of fractal-like patterning and the rules of growth and development reflect the scaling concepts of D’Arcy Thompson’s research (Link to Article) discussed a few weeks ago on this site. There is seemingly, a common underlying (universal) principle that guides growth and development, no matter what the species and these fractal-like networks relating to the efficiency of the biological systems (in energy exchange with their environment) seems to be the cause that gives rise to predictable scales of complexity and results in our measurements consistently showing the ¾ power law.

For instance, these fractal networks operating within actual organisms (species) are outlined in functional terms and in terms of the same fundamental system operating within species as diverse as mammals, fish and plants in the following in the Universal Review:

Mammals have richly branched air tubes, but they are confined to special organs, the lungs. Fish do a similar thing with gills. Trees use their richly dividing branches to supply their leaves with water and pump sugars back from the leaves to the trunk. The 3/4-power law is derived in part from the assumption that mammalian distribution networks are “fractal like”…


 Recall that D’Arcy Thompson’s observations  led to the identification of the common underlying principles of scale and the fact that everything grows in near perfect proportion to itself via repeating patterns on every scale (reiterations) (Link to Article). For instance, there may also be a reason why our five fingers don’t sprout five smaller fingers at the tips of our existing one. This may be because everything is a whole system within a larger whole system, just at different (repeating patterns) scales of complexity.

Again, it would appear to be all about the efficiency of the whole system in relation to everything else. This is perhaps why we don’t have five digit extremities and not many continuous branching and ever decreasing multiple miniature digits, apart from the fact that they would look ridiculous and you would fall over a lot, it simply would be a very efficient way of getting blood circulating and everything else that you need around the whole body.


Fig. 2: ‘freaky-fractal-fingers’ from article in Live Science

However, a tree would put on continuous branches reflecting the ratio of distance of the previous branches and the overall shape of the tree itself and the veins of some leaves can be seen to form this same branching pattern, on ever decreasing scales. But, even the tree will eventually stop branching and just consolidate what it has, growing thicker and maturing (stabilising phase in the universal growth curve – see last week’s article) according to its natural life cycle and the resources available within the context of the whole forest.

To emphasis this further, the following explanation for these principles of scale according to efficiency of the whole biological system is given in The Theory of allometric scaling laws in biology from genomes to ecosystems: towards a quantitative unifying theory of biological structure and organization, by Geoffrey B. West and James H. Brown.


… all organisms share a common structural and functional basis of metabolism at the molecular level. The basic enzymes and reactions are universal, at least across the aerobic eukaryotes. Additional general rules based on first principles determine how this molecular-level metabolism is supplied and regulated at higher levels of organization: from organelles, to cells, to organisms, to ecosystems. The most important of these rules are those relating to the size of the systems, including the body size of the individual organisms, and the temperature at which they operate. Our theory of quarter-power scaling offers a unified conceptual explanation, based on first principles of geometry, biology, physics and chemistry for the size-dependence of the metabolic process. The theory is based on generic properties of the metabolic distribution networks in simplified, idealized organisms.

West and Brown(2005, Introduction)


In other words, this is a shared generic system across seemingly most species that have been studied, albeit in an idealised form (physicists tend to see the bigger patterns – the invariants and don’t get bogged down in the details), but it does produce predictable results which are always worth paying attention to in science, they do conform to first principles established in several fields.  Although cells look and behave very differently to microbes and mice are very different to men, there is an underlying commonly shared system that relates to energy exchange between the organism and its environment.

There are other predictable properties that emerge when studied from the fractal perspective such as the heart beats on average throughout the life cycle of an average mouse will scale in the same ratio to that of an elephant and the smaller animal has faster beats, but the scale according to their mass heart rates in mammals showing the ¾ power law of scaling as pointed out in Allometric Relations and Scaling Laws for the Cardiovascular System of Mammals by Thomas H. Dawson Link

As indicated above, West and others have established these scaling laws for a wide range of biological phenomenon, however, their research also shows that they apply to other systems such as the growth of cities to the fractal-like nature of the communication networks and the growth of corporations; just about anything and everything systems (See last week’s article on this site). This is interesting as the Sigmoidal growth curve applies similarly to these ‘manmade’ systems as discussed in last week’s article on the topic. According to these researchers, it is all about distribution of resources which by their very nature forms fractal-like (space-filling) networks, whether the system is directly biological or run by biological entities (See Life’s Universal Scaling Laws, by West and Brown) Link and for more information on these broader scaling laws follow this link to view a video [You Tube] where Geoffrey West gives a very good lecture on the topic, see Scaling Laws In Biology And Other Complex Systems (2012) Link

There is definitely something universal going on if, as these scaling factors apply to anything from small business to big corporate entities and from the rail network to the whole infrastructure that supports big cities that started out as a one horse town, there is a universal growth/evolutionary pattern emerging.  And of course biological systems are not that different as they are all about distribution of resources as well. Below are some examples of related biological phenomenon that scale predictably and follow the ¾ power law, but with a slight twist which gives support to D’Arcy Thompson’s alternative evolutionary scenario of fundamental types and diversification from these generalist types that create all the variations that we call a species.

For instance, another predictable phenomenon identified within biological systems is the concept that small organisms reproduce (multiply/replicate) much faster than larger organisms. Basically, as most of know from common knowledge, a small species like a mouse reproduce very fast compared to larger mammals such as elephants. This is referred to as intrinsic rate and it means that there is an intrinsic difference between the reproductive capacities over time according to size of the organism. Mammals for instance with a self-regulating metabolic system- homeothermic, (warm-blooded) is rather different to a virus, but viruses predictably scale in relation to mammals in terms of their intrinsic rate being 100,000 times greater than a mammals! Essentially, as we know the rate and mass/size of one organism, we can predict according to size, the proportions or intrinsic rate in this case of any organism.  See chart below:

scales metabolism intrinsic rate

Fig.  3:  Chart based upon rate of increase in relation to mass/weight of organism. Note the discrete groupings of fundamental forms/types with only a slight overlap between the more complex animal groups (Diagram based upon figure 11.21 Body size and intrinsic rate of increase – data from Fenchel 1974, Link)

Again, the scaling is very clear where all organisms tend to fall within the predictable slope (gradient) according to their mass/size (fundamental metabolic type) and rate of increase in this case.  These are discrete groups of organism where the bacteria are quite distinct from virus organisms and protozoa (small animals that we can’t see for the most part) are a whole metabolically distinct group that in turn only overlap slightly with cold-blooded animals such as fish, amphibians, reptiles and these are a discrete, and only slightly overlapping group/type with the warm-blooded mammals and birds.  Another simplified chart which plots metabolic rate to mass is illustrated below. Here the discrete groupings of very different organisms are clear as well as the fundamental leaps (discontinuous lines) of metabolic complexity.

scales metabolism animals

Fig.  4:  Chart based upon metabolic rate to mass for diverse organisms ranging from unicellular to cold-blooded invertebrates and vertebrates and from warm-blooded vertebrates. Again note the discrete groupings of fundamental forms/types seemingly nested in scales of complexity.(Chart based upon data from  Link and Tatsuo Motokawa, “Elephant’s Time, Rat’s Time” Link

The charts for intrinsic rate for groups of major life-forms from singled-celled  organisms, to cold-blooded and warm-blooded species, clearly cluster into discrete metabolic groups and the leaps of metabolic complexity between each group is worth drawing attention to. In other words, the neat continuous grade seen exclusively for mammals is not the case when we plot these ratios for organisms with distinct metabolic systems, beyond warm-blooded mammalian forms. This might not seem that amazing at this stage, but it does give us a big clue to evolutionary development, which will become much clearer as these articles on alternative evolution continue. It begins to support D’Arcy Thompson’s fundamental types or archetype forms from which divergences can occur.

For instance, by employing the power laws of metabolism to mass, intrinsic rate to mass and other things such as heart rate to mass, we wouldn’t be top of this evolutionary tree metabolically speaking – elephants and whales are because they are bigger mammals than ourselves. Their mass and the fractal-like networks (space-filling needs) would be scaled up proportionately to make the metabolic system (the heat/energy exchange between the organism and its environment) work at optimum level. But as mammals, we out-rank in complexity of our warm-blooded metabolic system, the cold-blooded (simpler) vertebrates such as lizards and amphibians and in turn they out rank fish and fish outrank insects and so on and so forth, if you want to talk in competitive terms.

But, Nature would not appear to be this linearly hierarchical – it is fractal and each part is related to the whole on self-similar scales of complexity and therefore the relationship between the different species may be less literal and not so much about direct common ancestral descent, or as D’Arcy Thompson proposed, not through each other progressively as the Darwinian model would follow, but  from a common underlying patterning with many divergent and distinct forms which we would refer to as species. They share the same systems for patterning, growth and form and now it seems metabolic to mass ratio fractal patterning as well.

Just to reiterate the importance of the discrete fundamental types of species are trying to tell us about evolution, I do find it interesting that a mouse-sized lizard cannot be plotted within the higher scale of metabolism with a mammalian mouse of the same size, but remains fundamentally at the level of metabolic complexity seen in present-day lizard species. Does this mean an invariant? Is this an indication of a fundamental archetype form from which all divergent patterns on the same theme emerge into species as proposed by D’Arcy Thompson? In other words, the species may become fundamentally fixed (stabilised) once they reach the optimum level of their innate complexity and are metabolically in balance with their environment. The book entitled: Evolution, if it wasn’t by Darwinian means, how did it happen? goes into much more detail on this where is much scientific evidence to support this concept (See  main menu of this site under ‘BOOKS’ for more information).

These fundamental forms and their discrete groupings based upon metabolism, another chart, again based upon ratios of mass to metabolism is given below. It is shows the scaling exponents of fish, amphibians, reptiles, birds and mammals (which I have represented literally to illustrate the point with images) and is based upon the actual results of a study by White et al (Abstract 2006). Interestingly, this study disputes the scaling laws as they sometimes show non-universals and it is these differences that are actually quite revealing from the perspective taken here.

scales metabolism fundamental types

Fig.5:  Chart of the fundamental ‘types’ of species that scale according to radically distinct metabolism according to their mass within discrete groups. (chart is based on figure 1 from White et al 2006,

Note that West and others have adequately addressed such criticisms raised by White et al who have proposed heterogeneous scales or scales that don’t exactly conform to the ¾ power rule.  Now, as the chart clearly shows, birds and mammals are discretely grouped and basically fall within the gradient of their own slope and are therefore distinct from cold-blooded animals as they are more akin based upon warm-blooded metabolism. However, although all the groups follow a scaled slope according to mass and rate of metabolism, each group or archetype: fish, amphibian, reptile, bird and mammal, can be seen as discrete groups and only overlap slightly with each plotted gradient within their overall group .

In combination, these charts and the scaling principles in general, strongly suggest that the name of the evolutionary game is growth and innovation according to resources. It is seemingly about fractal-like (networks) space-filling and efficiency at the end of the day.  This common metabolic system to mass in exchange with its environment would allow growth on every scale of complexity, according to available resources. In a sense, species could develop and become increasingly complex according to their intrinsic metabolic potential and expand to eventually fill every niche on the planet as long as the resources became available, which looks very likely given the type of evidence is presented  throughout the book noted above (Evolution, if it wasn’t by  Darwinian means…?).  In other words, a form of co-evolution between the species and ecological system itself seems to be occurring.

This understanding and predictable properties of these systems therefore have real potential to project back in time and try to begin understanding the growth patterns that evolution itself might have followed according to these scaling laws. D’Arcy Thompson’s framework which was based upon the universal patterns that he found within all the great diversity of life which were underlain by simple fractal-like and predictable scaling laws, he proposed from these findings that fundamental forms could diverge into many different varieties leading to what we would call a species, may actually be the case.

All in all, what D’Arcy Thompson offers by way of this alternative evolutionary scenario of species going from fundamental unspecified types (generalists) to multitudes of divergent and differentiated forms (specialised species) according to natural laws of scale and form which can be described mathematically, as outlined in previous articles, is beginning to find much support as you will see as we proceed. I will just finish up this article with a quote from D’Arcy Thompson’s book On growth and form and you will find it quite pertinent to the next article (coming shortly on this site) focussing on the lesser known contribution to evolution theory from Alan Turing who is more commonly known in association with early computers and mathematics and his significant contribution to cracking the Enigma Code of WWII. Alan Turing also helped decipher another code  in biology and its wasn’t DNA, but rather the chemical switches that turn genes on or off and in particular ways during development – in his theory of Morphogenesis. Below is D’Arcy Thompson’s pertinent quote:

And while I have sought to shew the naturalist how a few mathematical concepts and dynamical principles may help and guide him, I have tried to shew the mathematician a field for his labour- a field which few have entered and no man has explored…For the harmony of the world is made manifest in Form and Number, and the heart and soul and all the poetry of Natural Philosophy are embodied in the concept of mathematical beauty.

Wentworth-Thompson (1917 Epilogue)



2 thoughts on “Another predictable Scaling Law?

  1. Very interesting content but: (i) terrible web design, probably intended to cause dizziness. The twists ad turns on the argument also seem to have the same objective. If single cell colonies conform to the 3/4 law, how then can you claim to explain the law on the basis of “networks”? There are no networks in single celled organisms are there?


    1. Thank you Jason for your interesting comments about the web design and very astute question regarding 3/4 law scaling and the single-celled organisms.
      First, I apolgise for late delay in responding – I have been really busy working on stuff and secondly for the dizzyness of the site and the twists and turns. I really prefer to write books – not blogs, but I do try and put up my research as I go along, as much as for myself as others interested in the topic of evolution.It is simply a means to an end – hence it is a bit crazy.
      Thirdly, the question about the organisms of only a single cell being independent critters and therefore not forming networks.
      I probably confused you and everyone else – but at least you are paying attention. The networks are fractal-like according to Geoffrey West et al, and are probably fractal because of distribution resources etc. Obviously, a colony of cells, single or otherwise, can operate as a loose type organism (symbiotically) and this hasn’t been assessed as yet (as far as I know).On the other hand, the 3/4 scaling law is not referring directly to these fractal-like networks, but to the individual species types (including single-celled organisms) that according to the scaling charts and Tatsuo Motokawa, “Elephant’s Time, Rat’s Time” ref in link
      a single-celled organism can be plotted (following 3/4 power laws) according to its mass to metabolism, or intrinsic rate etc just as other more complex animals. Fractal networks are predictable and scalable, but the 3/4 scaling law is not applicable to networks as such. I hope this helps.
      Do stay in touch. I have significantly updated and clarified the articles on scaling laws for a book out next month and will upload the first few chapters and Table of Contents in PDF format to download soon. Cheers MariaBrigit


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