Kleiber's law

From Wikipedia, the free encyclopedia

Kleiber's law, named after Max Kleiber's biological work in the early 1930s, is the observation that, for the vast majority of animals, an animal's metabolic rate scales to the 3/4 power of the animal's mass. Thus a cat, having a mass 100 times that of a mouse, will have a metabolism roughly 31 times greater than that of a mouse.

Kleiber's law, as many other biological allometric laws, is a consequence of the physics and geometry of animal circulatory systems according to some authors. Young (small) organisms respire more per unit of weight than old (large) ones of the same species because of the overhead costs of growth, but small adults of one species respire more per unit of weight than large adults of another species because a larger fraction of their body mass consists of structure rather than reserve; structural mass involves maintenance costs, reserve mass does not.

The exponent for Kleiber's law, what is called a power law, was a matter of dispute for many decades. It is still contested by a diminishing number as being 2/3 rather than the more widely accepted 3/4. Because the law concerned the capture, use, and loss of energy by a biological system, the system's metabolic rate was at first taken to be 2/3 because energy was thought of mostly in terms of heat energy. Metabolic rate was expressed in calories/sec. 2/3 expressed the relation of the square of the radius to the cube of the radius of a sphere, with the volume of the sphere increasing faster than the surface area, with increases in radius. This was purportedly the reason large creatures lived longer than small ones - that is, as they got bigger they lost less energy per unit volume through the surface, as radiated heat.

The problem with 2/3 as an exponent was that it did not agree with a lot of the data. There were many exceptions, and the concept of metabolic rate itself was poorly defined and measurable. It seemed to concern more than rate of heat generation and loss. Since what was being considered was not Euclidean geometry, the appropriateness of 2/3 as an exponent was questioned. Kleiber himself came to favor 3/4, and that is the number favored today by the foremost proponents of the law. Their reasons for favoring this exponent suggest, however, that their understanding of metabolism is still a problem. Geoffrey West, Brian Enquist, James Brown, and their school favor 3/4 claiming that it accounts for what West calls the 'extra dimensionality' of fractal branching of the capillaries that deliver blood-borne oxygen in biological systems with vascularity. Their understanding of an organism's metabolic/respiratory chain is based entirely on blood-flow considerations. Consequently, despite claiming the relevance of Kleiber's scaling law to biology over 27 powers of magnitude, they limit its relevance to only animal phyla, and, even then, mostly to mammals. Metabolism applies to far more than respiration and oxygen delivery at the basal level. It includes the derivation of energy from food sources.

Their position has been criticized for two key reasons. First is that the size-invariance of capillaries, the same from leaves to mammalian blood flow, dooms attempts to account for motor activity as part of metabolism. West et al. claim that Kleiber's law refers to the basal metabolic rate (BMR) of an organism's cells, and biologists point out that BMR cannot possibly account for motor activity. The second reason is that fractal branching does not account for any special increase in dimensionality - it is still Euclidean, and does not result in the creation of new space or advanced efficiency of delivery.

Since calories/sec is a rate, and is directly translatable to watts, it is safe to conclude that this alleged extra-dimensionality is actually one of time and not of space. Furthermore, the translation to watts suggests that the form of energy to be considered is electrical or chemical energy. This is in keeping with the idea that thermogenesis is not part of metabolism, that metabolism is all about reduction reactions like those behind the creation of cellular ATP. Although these arguments might favor 3/4 as an exponent of the Kleiber power law, there are still a large number of exceptions and anomalies to the law, and the law is still not able to account for motor activity.

The number of exceptions is reduced significantly if the law is taken to apply not just to BMR, but also to field metabolic rate (FMR). FMR includes, but is not limited to BMR, and accounts for motor behavior. The reason BMR is the preferred explanation for biologists is that it is mathematically easy to arrive at independently of organism mass. This was called mass-specific metabolic rate, and the story was it allowed the BMR of whales to be calculated as if they were yeast. But it wasn't really a rate at all. Instead it was division of both sides of the equation by organism mass such that the exponent of body mass became -1/4, and metabolic rate became calories/sec/gram, which is not a rate, however interesting mathematically it might be. The problem with mass-specific MR is that it completely dismisses the efficiency of the organism's mass at capturing energy through its motor behavior, something yeast does not have. As a result the values for BMR, given the nature of the -1/4 exponent, were much lower for the phyla animalia because it included their motor behavior as well as basal functioning. This re-enforced the idea that the lower values of BMR for larger creatures was due to a lower pulse rate, and that smaller creatures with higher pulse rates and less mass lived shorter lives because of higher metabolic rates tied to those pulse rates. In essence mass specific metabolic rates are entirely useless for comparing yeast and whales, or for comparing whales to mice, or primates, or birds, etc.

This was amended by alterations to the exponent incorporating the term for metabolic efficiency, that is, the ability to capture energy from ambient sources. These ambient sources were food sources, and what the term successfully reflected was the efficiency of the motor behavior of that organism in capturing food sources. BMR cannot reflect this. Only FMR can. The equation was changed to FMR = άW^(4μ-1)/4μ) where μ became the term for metabolic efficiency. μ was the ratio of the rate of reduction reactions associated with the development, maintenance, and functioning of the organism, to the rate of flow of electrons from food and all oxidative energy sources, whether basal or digestive. This went along with the treatment of metabolism as a chemical process involving the delivery and expenditure of energy, with metabolic rate expressed in watts rather than calories/sec. μ became a statement of the efficiency of redox coupling by the organism of mass W. This amendment expanded the range of relevance of Kleiber's law, from bacteria to whales to social organizations. For otherwise, with just a 3/4 exponent (μ = 100%), the calculations for MR from whales to bacteria were so divergent that they offered no clue to the relation of MR and longevity, with bacteria living less than one second, and whales living thousands of years.

The suggestion is made by West et al. that Kleiber's power equation of the metabolism of life might hold the secret to the aging process in view of the fact that larger creatures live longer. The suggestion is also annihilated by those same proponents who also argue that the equation concerns only BMR, and that metabolic rate is only a function of body mass without consideration of μ. For example West et al. claim that a rat and a pigeon of the same mass differ in life spans by a power of 10 despite having the same metabolic rate. They also note that primates live far longer than they should, given their difference in mass from elephants and cows.

These difficulties are easily accounted for by appeal to the term μ for metabolic efficiency. A 100 gram rat with a μ of 20% has an FMR one tenth that of a pigeon with an μ of 30%. A 75 kg. human being with a μ of 32% lives just over two times as long as a 1000 kg. elephant with a 28% μ. FMR, not being entirely mass dependent, is not the same for the rat and the pigeon. Difference in eating habits and activity levels is introduced by the term μ. This appeal to μ to explain what would otherwise be insurmountable anomalies in attempts to relate metabolic rate to potential life span, begs for clarification as to what bears upon the value of the term μ. Since FMR includes motor activity driven by the nervous system, the numerator of μ includes the neuronal synthesis of ATP to drive that activity in the quest for food. Since the denominator includes energy from food sources, what is implied is a neuro-gastric coupling. The more neurons taking part in this coupling, in other words, the greater the organism's encephalization, the higher the rate of capture of energy from food sources, the numerator of μ. This is what is behind the evolution of longevity in mammals, why the primate lives longer than quadrupeds far more massive.

If values for Kleiber's law ranging from -13e grams to +13e grams, and from 0% μ to 100% μ are graphed and placed in a table, the evolution of life on earth in metabolic terms is clearly depicted, from mitochondria and bacteria, to single celled animals, multi-cellular creatures, and all vertebrates. The equation is numinous in its electrochemical application to all of life, and, when properly regarded, does not have any exceptions at all, clearly depicting the seams along which life evolved. One seam is at the one gram level, the other is at the 25% μ level. All of life can be found between 15 and 70% μ. At masses on the order of -13e to -5e grams high MRs are more indicative of the rate at which energy can be throughput, in watts or calories/sec., then potential life span, though the reputed immortality of some cells fits the mathematics. And at values for μ less than 15%, for small things MR skyrockets, representing destruction by the overwhelming availability of capturable energy, while for large things (over one gram) MR plummets as μ drops below 15%. In the case of cells, the skyrocketing of MR is the condition of cellular suicide. The mathematization of biology in this way brings to it the same analyticity that Galileo brought to the study of the solar system, and that Newton brought to mechanics. In both cases the math was based upon ideal situations not found in the everyday world of the layman, e.g., Galileo discounted air resistance in his treatment of the acceleration of gravity. Mathematical idealization like that in Kleiber's law at first was riddled with exceptions that prompted biologists to scoff at his scheme. But improvements in that math, discussed above, have eliminated all those exceptions. Many academic biologists still deride the equation as not pertinent to biology, even seeing its relevance as a threat to cherished ideas that life is something special and too complicated to fit such a simple formula.

Kleiber's law clearly models the relation between food sources and reproductive strategies, where such strategies are seen as alteration of mass (through division or parturition) prompted by equilibration of MR following changes in μ. Changes in μ strictly reflect the fluctuations in available energy for all biological organisms, from food sources to sunlight to hydrogen sulfide escaping at submarine volcanic vents. The equation accounts for why more massive rats live shorter lives than less massive rats. It also explains the effects of caloric restriction on longevity, and why this is not a good strategy for increased longevity of more encephalized creatures, that is, creatures with a higher value for μ. This equation explains why exercise results in longer life, why over-eating results in shorter life, why stem cells proliferate, and why they stop as they differentiate. This equation explains why cancer cells proliferate, and how to get them to destroy themselves with skyrocketing BMRs. This equation models how the evolution of mammals was made possible by fat cells, and why dieting is a bad idea for fat cell loss of mass. This equation tells how to build muscle using direct current, how to reverse the degeneration of aging, and how to increase one's potential life span by from 30 to 100%. There is a reason this equation is called the equation of life. Metabolism necessarily preceded and was a requirement for the evolution of RNA and DNA. It is truly unfortunate that metabolism is not well understood by the foremost proponents of the law who restrict it to respiration and hemodynamics. But their misunderstanding is to be expected in view of their dedication to the perpetuation of a biology whose mathematical foundations, with regards to the electricity of the cell, are still rooted in the 19th century thermodynamics of the Nernst equation. This misunderstanding prevents realizing the potential of Kleiber's law for intercession into the human equation to promote life and health using electochemistry.

West et al. make the claim that Kleiber's law applies over twenty-seven orders of magnitude. At first blush this seems rather exaggerated. It seems 23 or 24 orders of magnitude would cover life from mitochondria to pods of whales. Consideration of the mathematics reveals how the effects of fluctuations in available energy, the denominator of μ, are equilibrated through the organism, appearing either as increases in mass, in activity of that mass, and in division of that mass, as in proliferation or procreation. Consequently the equation is applicable to social organisms, that is, ant colonies, pods of whales, and human society.

In this latter regard, if we consider W as the population of a human society, with the basal unit being a couple, and if we consider the increase or collapse of this society as due to fluctuations in available wealth generation from whatever circumstance, and if we consider money or exchange of value as the unit of energy in this society based upon the division of labor and human activity, then we must consider wages in the denominator of the basal couple's μ with mathematics applicable for organisms less than one gram. And we must consider wages in the numerator of the employer's μ as a cost of doing business, while in the denominator of the employer's μ is the ingenuity and industriousness of his worker's who can generate wealth by increasing productivity. And in the case of the employer, or the state, the appropriate mathematics are for organisms over 1 gram.

Metabolic rate becomes then market activity.

The mathematics clearly show that, as in the evolution of organisms from bacteria, so with social organisms, the best way to insure the adaptability and survival of the organism is to distribute wealth through increased wages at the basal level, and to draw upon accumulations of wealth during the lean times. The suggestion then is that accumulations of wealth as capital that are socially worthless, such as standing armies, imperil survival of the organism and/or many of its basal couples, during lean times. The numbers show that one way to increase the employer's/state's μ, and consequently market activity, is to raise wages. This effect is equilibrated at the basal level by increased spending.

This is one aspect of Kleiber's law that should be critically examined since it takes the order of magnitude for that law to the 30th power of ten since human society numbers in the billions, a thing not possible without human activity and food, the division of labor and money.