Modeling at the Cutting Edge Universal Scaling Laws in Biology

Modeling at the Cutting Edge: Universal Scaling Laws in Biology September 6, 2003 1 Introduction Why is there a study of scaling in Biology? From the smallest of microbes to the largest creatures on Earth, we all work on the same basic principles. In fact, there is a 21-order of magnitude difference between the smallest of animals to the largest. The question we ask is how does the biology “scale up” when the animal scales up? This is not a trivial question- for an engineering example, consider the problem of spanning a stream versus spanning Puget Sound. As structures get larger, the support for that structure cannot scale linearly to support the weight. When structures pass certain critical points, it is more advantageous to change the structure. We might expect the same from biological organisms. The point is, that while all animals share the same molecular structures and biological pathways to sustain and reproduce themselves, how the structure is realized is a function of the size. We might say therefore that the variety of sizes we find in the biological world is responsible for the wide biological diversity. There is a long history of studying scaling in biology going back about 100 years. When biologists speak of scaling, it is not linear (where two quantities are linearly related): Q1 = kQ2 but rather the form is nonlinear, or biologists use the term allometric. They study scaling as: y = axb To assist in studying this relationship, note what happens when we take the logarithm of both sides: ln(y) = ln(a) + b ln(x) (1) so that the relationship between ln(y) and ln(x) is now a linear one. So when you see graphs in biological texts, you will see plots of ln(y) versus ln(x) (these are called logarithmic plots). Consider the plot of metabolic rate versus body mass given in Figure 1. 1 Figure 1: A logarithmic plot of metabolic rate versus body mass. Even though body mass undergoes a 21 order of magnitude difference, we see that the metabolic rate scales quite nicely with mass. (Reproduced from “Scaling in Biology”, edited by Brown and West) 2 In our discussion of scaling, one might guess that the scaling for the metabolic rate might be analyzed as follows. Let’s simplify things by considering a sphere. Let V be the volume of the sphere, and we see that: S.A. = 4πr2 , V = 4 3 πr 3 so that: r = k1 V 1/3 S.A. = kV 2/3 These equations should apply to any shape, if biological organisms maintain similar structures as they grow. In this event, we would expect the scaling from Equation 1, to be b = 23 . This is not the case, however. The slope of the lines in Figure 1 is 34 . It might be surprising to you that many biological scaling laws give the exponent b as a multiple of 14 . For example, the aortas of mammals and the 3 diameter of tree trunks both scale as M 8 (M is the mass). Heart and respiratory 1 rates in mammals scale as M − 4 . Amazingly enough, it seems that longevity 1 scales as M 4 so that the number of heartbeats in a lifetime is the same, about 1.5 × 109 , for all mammals, regardless of size. To quote West, Brown and Enquist: Allometric scaling laws are special because they express a systematic and universal simplicity in the most complex of all complex systems. They provide rare examples of universal quantitative laws in biology. Their origin presents a major challenge because their very existence, not to mention their common quarter-powers, suggests the operation of a set of principles that are fundamental to all life. Recently a model of universal scaling in biology has been proposed by James H. Brown (University of New Mexico), Geoffrey West (Los Alamos), and their colleagues. Understanding the full model would require some understanding of fractal geometry and partial differential equations, so we won’t go into that here. One of the keys however to understanding the construction of the model is that biological organisms must use networks to obtain sustenance and remove waste. Examples include vertebrate circulatory systems, vertebrate respiratory systems and plant vascular systems. While the function of each system is quite unique, each of these systems exhibit the same set of scaling laws. We will only state the assumptions of the model below, and conclude with some features. 2 Model Assumptions The model assumptions are the following: 1. For the network to supply the entire volume of the organism, a spacefilling hierarchical branching patter is required. This is the portion of the model that uses fractal geometry, or more specifically, self-similarity. 3 2. The final branch of the network (e.e., the capillary of the circulatory system or the petiole of a plant) is a size-invariant unit. 3. The energy required to transport materials through the network is minimized. Some predictions of the model: 1. Blood pressure p and average velocity of blood at the aorta u are independent of mass. For example, the aorta of a whale has a radius of 30cm, whereas the aorta of a shrew has a radius of 0.01cm, yet both sustain the same blood pressure and velocity! 2. Let lk be the length of the network between branchings k and k − 1, and let γk = lk+1 lk . Then γ is independent of k. 3. Similarly, let nk be the number of branchings at a node. Then n is independent of k. 4. If we are filling a d−dimensional volume, γ = n1/d d and metabolism will scale like d+1 . In the three-dimensional world, this 3 gives 4 . If we were constrained to two dimensions, it would be 23 - this could be checked against organisms that are almost two dimensional, like flatworms. A final word: This model is still very new and needs to be verified against observed data. However, it seems that the model does explain existing data, and it makes very specific predictions about several other measurements. 3 References • “Scaling in Biology”, Edited by James H. Brown and Geoffrey B. West, University Press, 2000. • “Effects of Size and Temperature on Metabolic Rates”, Gillooly, J., Brown, J., West, G., Savage, V., and Charnov, E. Science, vol. 293, 2001, pp 22482251 (with a graphical erratum posted Nov 16, 2001). 4