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Scaling


Scaling Graph

Life is amazing. Even the smallest bacterium is far more complex in its structure and function than any known physical system. The largest, most complex organisms, large mammals and giant trees, weigh more than 21 orders of magnitude more than the simplest microbes, yet they use basically the same molecular structures and biochemical pathways to sustain and reproduce themselves.

From these incontrovertible observations, two fundamental features of life follow: the first is that biological diversity is largely a matter of size, the second is that in order to achieve such diversity, organisms must adjust their structure and function to compensate for the geometric, physical and biological consequences of being different sizes.

The variety of sizes plays a central role in the ability of organisms to make their living in so many different ways that they have literally covered the earth, exploiting nearly all of its environments.

The principles of mathematics and physics are universal, but their biological consequences depend on the size of the organism.

Slow pulse, long life. The baffling correlation between body size and total metabolic rate may stem from nutrient distribution.

 


     One of the most longstanding and challenging problems in biology appears to be solved: why metabolic rate scales as the 3/4 power of body mass.

     The answer is that organisms effectively live in four spatial dimensions. They have exploited fractal geometry so that critical linear dimensions and surface areas scale as the 1/4 and 3/4 powers of body mass, respectively, rather than the 1/3 and 2/3 powers expected from conventional Euclidean geometry.

     The solution is the result of three years of work by two biologists, myself and Brian Enquist of the National Center for Ecological Analysis and Synthesis and The Santa Fe Institute), and a physicist, Geoffrey West of Los Alamos National Laboratory (Interim President of Santa Fe Insititute, LANL, and UNM Adjunct Professor).

     We have developed a general model which is based on the assumption that biological rates and times are ultimately limited by the rates at which limited energy and materials can be supplied to cells through a hierarchical branching network. It further assumes that the distribution system has three attributes: i) it is space-filling (i.e., it reaches all parts of the organism); ii) it minimizes the energy required for distribution; and iii) it has size-invariant terminal units (e.g., capillaries or terminal xylem).

     From these assumptions we have derived a quantitative model for the geometry and physics of the entire distribution system. The model predicts: i) a fractal-like branching network with scaling laws governing the sizes of the branches; ii) whole-organism metabolic rate scales as M3/4; and iii) many other anatomical and physiological characteristics of mammalian cardiovascular and respiratory systems (West et al. 1997).

     This model, which claims to solve the longstanding problem of quarter-power scaling in biology, has elicited considerable attention, both from biologists who work on related problems and from the press: both scientific publications (commentaries or articles in Science, Nature, The Scientist, BioScience, and Trends in Ecology and Evolution, and an upcoming one in The New Scientist) and the news media (feature articles in the New York Times, Washington Post, and Dallas Morning Star).

     Since publication of this paper in 1997, we have made considerable progress. We have a new manuscript, provisionally accepted by Nature, which describes a similar quantitative model for vascular systems and branching architecture of plants.

     This synthetic, whole-system model incorporates hydrodynamic principles, biomechanical constraints, and branching geometry. It answers some longstanding questions, such as how plants deal with resistance in their microcapillary vessels and what determines the maximum height of trees. It closely predicts many empirical scaling relationships and measured fractal dimensions of roots and shoots in the botanical and forestry literature. The model predicts that metabolic rate or resource use of whole plants scales as M3/4.

     In another paper (Enquist et al. Nature 395:163, 1998), we have used allometric scaling of resource use to model the structure of plant populations and communities. We derived a resource-based thinning law which predicts that average population density scales as M-3/4, giving a -4/3 thinning law rather than the -3/2 one derived on purely geometric grounds. We also predicted that productivity of ecosystems is independent of the size of the dominant plants.

     An important feature of our general model is that it characterizes the integrated structure and function of entire systems. Thus, for example, our models of both mammalian cardiovascular and plant vascular systems predict the scaling exponents (and sometimes the normalization constants) for more than 15 variables, ranging from volume of fluid and dimensions of vessels to resistances, pressure gradients, and rates of flow. Empirical falsification of just one of these predictions could reveal a serious flaw in the model.

     The fact that the measurements which do exist are typically within a few per cent of predicted values suggests that our models capture the essential features of these systems, even though they necessarily make important simplifying assumptions. Furthermore, by making predictions about deliberately simplified systems, our models provide a theoretical basis for interpreting how some real systems deviate from idealized ones.

     Perhaps our biggest breakthrough was the recent discovery of a very general, geometric principle of biological scaling. We proved that the 4 in the denominator of allometric scaling exponents is due to the fact that organisms effectively function in four spatial dimensions.

     The special fractal properties of biological resource distribution networks cause exchange surface areas to scale as M3/4 and internal distances and transport times to scale as M1/4, rather than as M2/3 and M1/3 as expected from conventional Euclidean geometry. This has many implications. For biology, it implies that organisms exhibit "maximum fractality" because they have been selected to be simultaneously maximally powerful and maximally efficient.

     More broadly, it provides a rigorous theoretical foundation for understanding structures and behaviors of networks. Many hierarchical systems, which have been described empirically by measuring their fractal dimensions, still lack a theoretical explanation in terms of basic scientific principles.

     Outside of biology, these include stream networks, human distribution systems (e.g., the Internet and electrical, telephone, water, and road grids), and perhaps even social and economic organizations.


Information contained in this document is © copyright James H. Brown, 2007. All rights reserved.

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