


Universal Scaling Laws

Contents of this page


Introduction 

Scaling principles 

Universal scaling laws 

Towards multivariate allometry 

Conclusion 


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Introduction


Physics is the oldest and most mature of the natural sciences. Since Aritstotle physicists have strived for the discovery of natural laws that govern physical phenomena and have concentrated their efforts on the mathematical formulation of these general laws later in history (History of physics, Wikipedia). 

Unlike, biologists while always facing the vast diversity and complexity of living organisms had to confine their work in terms of the number of taxa or scales under investigation. Presumably due to these constraints the retrieval for general laws was of rather little importance in life sciences. In accordance, the belief that general laws do not exist in nature is still widespread among life scientists. 

Brown, West & Enquist, joined by other colleagues, have ultimately challenged this view on the biological world by dozens of papers on "Universal Scaling Laws" published in Nature, Science and elsewhere. 


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Scaling principles


Scaling has a long tradition in biology (see Peters 1983). Scaling deals with sizedependent variation of an organism's form or function (Niklas 1994). The most commonly used scaling equation is the power function: 





Scaling relationships are usually fitted with Reduced Major Axis regression instead of simple Least Squares regression, since Y2 underlies also variation and measurement error. For the same reason, the usage of an independent variable commonly denoted as X is avoided in the formulation (compare Niklas 1994). 

Often, in particular when the variation of Y2 at the high end of the data range is rather high or when the data contains outliers, the regression analysis is better carried out on a linear equation derived by taking logarithms on both sides of the equation: 







Solely estimating the parameters of the linear form is usually sufficient in empirical scaling analyses. However, if we wish to confirm that the estimated &alpha is identical to one that follows from first principles, we need additionally the 95% confidence interval in order to test whether the estimation overlaps with the "theoretical" &alpha. This kind of scaling analysis is referred to as analytical.


In principle, an &alpha of 1 indicates that form or function is similar independent of size. This is called isometry, whereas the opposite is referred to as anisometry.



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Universal scaling laws


In their Science paper "A General Model for the Origin of Allometric Scaling Laws in Biology" West, Brown & Enquist (1997) derived the three quarter power law that for many organisms relates metabolic rate to mass. In their derivation the universal power law originates from traits of the fractallike branching networks that efficiently transport resources to their final locations of cellular supply. Further, the authors give evidence that the model can explain major attributes of vertebrate cardiovascular, plantvascular and several respiratory systems. 

In a number of followup publications large databases were used to authenticate the universal nature of power laws in scaling. 

In other papers the authors and colleagues focused more on the ecological consequences of universal scaling laws which operate at the individual level (e.g. Enquist, Brown & West 1998). Brown et al. (2004) even developed concepts toward a metabolic theory of ecology. 

However, in the series there were several articles that dealt with ontogenetic growth and showed that a negative quarter power relationship between growth rate and mass holds universally (Enquist et al. 1999, West, Brown & Enquist 2001). Those were the most interesting to me. 

In the first phase of UIBM development it was more feasible to use a simple scaling law to drive the growth of the plants. Even later, when the model received a functional basis, the scaling law is convenient for driving initial seed growth until the first structure appears. 


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Towards multivariate allometry


Brown, West & Enquist's research focuses mainly on simple bivariate relationships between form or function and the sizerelated variable that matters most to them, i.e. body mass. 

While those researchers, in principle, advocate analytical scaling analysis (see above), concomitantly empirical scaling analysis has also gained in popularity. 

For instance, in one of the most widely cited botanical papers of recent years Wright et al. (2004) presented a matrix of scaling relationships for the whole set of variables contained in the GlopNet database. However, all those examples have the focus on bivariate relationships in common. 

In contrast, Shipley (2004) "doubts that the many researchers who study allometry truly believe that organismal attributes interact only two at a time". "The tightly integrated attributes of form and function should change together with complicated direct and indirect relationships between them", he stresses. To Shipley allometry is intrinsically multivariate, though. Shipley uses the term allometry in the sense of scaling but his definition of allometry is more general than Niklas' scaling definition. He refers to allometry as the study of how the attributes of organisms change with respect to one another. 

Shipley further argues that traditional allometry per se lacks statistical methods to study multivariate relationships and introduces causal models (Shipley 2002, Shipley 2004) towards this end. 

In his article he uses a set of measured variables as an example. From basic biological knowledge he derives several hypothetical models of causeeffect relations among the variables, presents those models as directed graphs and statistically tests the conditional dependencies of individual variable pairs as well as the whole model. Certainly application of causal models to plant trait databases is not feasible until a sufficiently large data set consisting of the original data measured on individual plants is availabe in the databases. Unortunately, this is not the case at present. 

However, Shipley further uses serial biological reasoning of the kind "if variable A precedes variable B, A may cause B, but not the other way around" to exclude a set of statistically equivalent, but biologically unrealistic models. The application of serial biological reasoning appeared to be applicable to plant trait databases, as well. 


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Conclusion


During UIBM development we will adopt multivariate allometry assisted by serial biological reasoning as a general methodology to empirically scale plant traits to one another. 

As long as UIBM is lacking a functional basis a universal negative quarter power law will be used to drive plant growth. Later on, the same universal scaling law will simulate the growth of a seed until the first structure in the form of cotyledon(s) is built. 


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Copyright © Dec. 2009 Dr. U. Grueters



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