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Metabolic Rate and Kleiber's Law

Metabolic Rate Metabolic Rate, Small Size The first accurate measurements of body mass versus metabolic rate in 1932 shows that the metabolic rate R for all organisms follows exactly the 3/4 power-law of the body mass, i.e., R M3/4. This is known as the Kleiber's Law. It holds good from the smallest bacterium to the largest animal (see Figure 01). The relation remains valid even down to the individual components of a single cell such as the mitochondrion, and the respiratory complexes (a subunit of the mitochondrion) as shown in Figure 02. It works for plants as well. This is one of the few all-encompassing principles in biology.

Figure 01 Metabolic Rate
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Figure 02 Small Size MR[view large image]

But the law's universality is baffling: Why should so many species, with their variety of body plans, follow the same rules?

Fractal in Nature Then in 1997, a couple of physicist and biologists successfully derive the 3/4 power-law using the concept of fractal. The theory considers the fact that the tissues of large organisms have a supply problem. That is what blood systems in animals and vascular plants are all about: transporting materials to and from tissues. Small organisms don't face the problem to the same extent. A very small organism has such a large surface area compared to its volume that it can get all the oxygen it needs through its body wall. Even if it is multicellular, none of its cells are very far from the outside body wall. But a large organism has a transport problem because most of its cells are far away from the supplies they need. Insects literally pipe air into their tissues in a branching network of tubes called tracheae. 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" (Figure 03) and in part from the conjecture that natural selection has tended to maximize metabolic capacity "by maintaining

Figure 03 Fractal in Nature
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networks that occupy a fixed percentage (6 - 7%) of the volume of the body".

Effort has been made to derive the 3/4 power-law for a broader category that includes plants, animals, and even one-celled organisms lacking a vascular system. The latest derivation is based mostly on geometry, particularly the hierarchical nature of circulatory networks. It is argued that an organism's "internal area" -- the total area of its capillary walls -- fills up space so efficiently that it, in effect, adds another dimension (similar to the compactification of extra dimensions in the Superstring Theory). Therefore, the "internal volume" of all the vessels feeding the capillaries acts as an extra dimension, scaling as the fourth power of internal length.
Life Span Brain Mass There is a more general 1/4 power-law applicable to many physicological variables y as shown in Figures 04, 05, and Table 01. The general form of the power-law is y Mb, where b is a multiple of 1/4. It seems that all these physicological variables have something to do with the nutrient distribution networks and the dimensional dependence. For example, the life span is proportional to the linear dimension L, and the heart beat rate is related to the inverse L-1. The other variables simply follow the same 3/4 power-law or L3 as for the case of metabolic rate.

Figure 04 Life Span
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Figure 05 Brain Mass
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It is obvious that the various physicological variables are determined primarily by the dimensional dependence. The corresponding power-laws are the consequence of the relationship L M1/4.

physiological variables (y) Dimension scaling exponent (b)
Heart Beat Rate -1 -1/4
Period of Heart Beat§ 1 1/4
Life Span 1 1/4
Diameter of Tree Trunks 3 3/4
Diameter of Aortas 3 3/4
Brain Mass 3 3/4
Metabolic Rate 3 3/4
Metabolic Rate (latest observations) 4 1

Table 01 The 1/4 Power-Law

Plant Mass A report in the January 26, 2006 issue of Nature indicates that the 3/4 power law is not observed in plants. The new experiment involved 500 individual plants, across 43 species, from varying environments, and covering six orders of magnitude variation in plant mass. It is found that the slope for a plot with respiration (metabolic rate) against plant mass is close to 1 (see Figure 06). The differences in the intercepts between the indoor and outdoor groups disappears if plant nitrogen mass is used instead of the total plant mass. This new result can be explained by the theory if

Figure 06 Plant Mass
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the metabolic rate of plants R is proportional to L4 by including the internal dimension as another metabolic site, so that R M. With the older theory, which gives the 2/3 power law, R L3 would be sufficient to yield the same result.
A 2008 research indicates that most organisms' metabolisms are clustered between 1 and 10 watts per kgm of mass. It seems that an optimum metabolic rate is located within this range. Mathematically, this finding is consistent with the R M law because it implies a constant derivative dR/dM 5x10-3 watts/gm. Whereas the R M3/4 law would yield a mass dependent derivative dR/dM M-1/4.

A 2010 article in Nature proposes an improved theoretical formula, which provides a better fit for the observed data, by adding a quadratic term (in logarithmic mass). A best fit can be obtained by including a temperature-corrected term. The resulting formula can be written in the form:

log10R = B0 + B1log10M + B2(log10M)2 - B3/T
Power Law Deviation where T is body temperature in kelvin, and the B's are constants to be evaluated by best fit to the observed data as shown in Figure 07 from which yields B0=14.0149, B1=0.5371, B2=0.0294, and B3=4799.0. The quadratic model takes care of the small curvatures at both the upper and lower ends. It is suggested that the quadratic correction could be related to the transition of large vessels with pulsating blood flow and small vessels with smooth blood flow. The correction can be visualized mathematically by taking the derivative (neglecting the temperature term):

d(log10R)/d(log10M) = B1 + 2B2log10M
which shows that the 3/4 exponential is replaced by B1+2B2log10M. The 3/4 law applies only to

Figure 07 Power Law Deviation [view large image]

organisms with mass around 1000 gm. For the rest of the organisms, the 3/4 slope is modified by the mass term. It implies that for some unknown reason, the metabolic rate varies slightly for living organisms with different mass "internal volume" notwithstanding.
It is observed that the maximum animal mass is determined by the limit at which the slope equals to 1. Beyond this point, bigger is no longer better. Intriguingly, this point occurs close to 108 gm (100 tonnes, calculated from the values of the B's), which is about the mass of the blue whale - the largest animal that has ever lived.

Another interesting consequence comes from the temperature term B3 = E/R, where R is the gas constant and E is the effective activation energy. It turns out that E = 0.95 ev from the value of B3. This value of E is less than the free energy of the full hydrolysis of ATP to AMP under standard cellular conditions, indicating that the model produces a biologically realistic coefficient.

§Since Period = 1 / Rate L, this relation shows that the pumping cycle takes longer for organism with larger size.
It is known that all organisms expire after 1 - 2 billion heartbeats, i.e., the life span t x (heart beat rate) = constant or
t 1 / (heart beat rate) L. For example, if we assume 2 billion heartbeats in the lifetime of an average human with 60 heart beats per minute, then t = 64 years.