Home Page |
Overview |
Site Map |
Index |
Appendix |
Illustration |
About |
Contact |
Update |
FAQ |

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 M^{3/4}. 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 |
## 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? |

- An explanation for this kind of relationship was proposed further back in 1883:
- Suppose the organism has a size of L, then the surface area A L
^{2}, while the volume V L^{3}assuming that it is in the shape of a sphere. - If the density in the organism M / L
^{3}is constant, then L M^{1/3}, where M is the total mass of the organism. - Since the heat dissipation from an organism is proportional to its surface area, the total metabolic rate R L
^{2}M^{2/3}, which is close but not quite the same as the 3/4 power-law.

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 |
networks that occupy a fixed percentage (6 - 7%) of the volume of the body". |

- A simplified derivation that takes the fractals out of the picture, goes like this:
- It is assumed that a network for distributing nutrients has circulation length L, which in three dimensions serves roughly L
^{3}sites where nutrients are delivered by capillaries. - The transport networks essentially create an extra (internal) dimension L so that the effective density
_{eff}M / L^{4}. If_{eff}is a constant, then M L^{4}or L M^{1/4}. - The total metabolic rate is proportional to the number of sites served, i.e., R L
^{3}M^{3/4}.

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 M^{b}, 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 L^{3} as for the case of metabolic rate.
| ||

## Figure 04 Life Span |
## Figure 05 Brain Mass |
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 M^{1/4}. |

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 |

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 |
the metabolic rate of plants R is proportional to L^{4} 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 L^{3} would be sufficient to yield the same result. |

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:

log

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 B_{0}=14.0149, B_{1}=0.5371, B_{2}=0.0294, and B_{3}=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(log _{10}R)/d(log_{10}M) = B_{1} + 2B_{2}log_{10}Mwhich shows that the 3/4 exponential is replaced by B _{1}+2B_{2}log_{10}M. 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. |

Another interesting consequence comes from the temperature term B

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.