Scaling Laws for Waste

It has been a long while since I last wrote a blog post.  In the meanwhile, a lot of things have happened, and apropos to the previous post, I have switched from the schedule of a research staff member to the schedule of a starting assistant professor.  Definitely some different elements to it.  More time teaching and less time working on a food truck, to say the least.

Since last I’ve posted, I’ve also gotten much more into tweeting, which perhaps does take away some of my impetus for blogging: limited attention, information overload, and all that.  As one of the great new communication media, I’m also getting interested in Twitter as an object of research study.  Indeed, some of my undergraduate researchers this summer will be looking at social media analytics.

Since the last post, I’ve also gotten further interested in resource recovery and other problems of environmental engineering, though not at all versed in the subject yet. One of the most valuable resources from which to recover energy, nutrients, water, and solids is animal waste.  Indeed, there have even been wars over the control of guano.  

I’ve had some longstanding interest in allometric scaling laws for various things, and I suppose I’ve made you at least somewhat interested.  When I was visiting Santa Fe last summer, I feel like my interest in this topic was renewed, largely due to the enthusiasm of Luis Bettencourt on scaling laws for cities.  As it turns out, there are a lot of parallels to neurobiological scaling.

With all that as preface, do you have any idea how the amount of waste produced by an animal scales with the size of the animal?  Do you think it would be allometric scaling?

In fact this question for urine has been studied in the literature more extensively than I would have expected.  In the paper, “Scaling of Renal Function of Mammals,” Edwards takes data on the mass [kg] and the urine volume [mL/24 hours] for 30 mammalian species and finds an allometric relation with power law exponent 0.75, which is the same power-law exponent as for metabolic rate as given by Kleiber’s Law. (One theoretical derivation based on elasticity is due to McMahon.)

The same urine volume exponent is presented in a paper, “Scaling of osmotic regulation in mammals and birds,” by Calder and Braun. Turning to water loss through feces, Calder and Braun say:

Fecal losses should, in absence of size-related differences in food quality, digestive efficiency, and/or reabsorption, scale in parallel to the intake that supplies metabolic requirements, but the only allometric expression we have found in the literature has M^0.63 scaling [in mL/day].

where the scaling law is quoted from a paper by Blueweiss, Fox, Kudzma, Nakashima, Peters, and Sams, “Relationships between body size and some life history parameters,” from the journal Oecologia.  The original statement in that paper regarding defecation is measured in the units g/g/day and gives a power-law exponent -0.37 based on data from mammals, but this measure of [g/g/day] already normalizes once by body weight, which is why there is no issue, 1 – 0.37 = 0.63, assuming constant liquid content.  The original data used by Blueweiss, et al. is said to be from a paper by Sacher and Staffeldt, “Relation of gestation time to brain weight for placental mammals: implications for the theory of vertebrate growth,” though I didn’t see it in there.

A contributing factor to all of this is of course food intake, via assimilation efficiency of various foods.  In a paper “Allometry of Food Intake in Free-Ranging Anthropoid Primates” in Folia Primatologica, Barton reports the power-law exponent for daily intake in grams dry weight per 24 hours as probably a little bit more than 0.75 using limited data on 9 species of primates (including humans).  For cattle, Illius in a paper, “Allometry of food intake and grazing behaviour with body size in cattle,” talks about intake with exponent a little bit less than 0.75.

Having talked about urine and feces, what about CO₂?  A paper “Direct and indirect metabolic CO₂ release by humanity” by Prairie and Duarte quotes allometric laws on respiration and defecation from the book The Ecological Implications of Body Size by Peters, which maybe I should read.

So what does this have to do with information?  I wonder if there is a notion of information metabolism with an associated scaling law like Kleiber’s.  There is a notion of a ‘garbage tape’ in the thermodynamics of computation following Landauer, and so I wonder what fraction of information is put into the garbage tape as a function of the size of the computation.  

Anyway, good to get back into blogging, hopefully without too much garbage.  After all, we don’t want too much information pollution, nor municipal solid waste in cities for that matter.