It had to happen, I suppose, and I get the feeling you’ve been waiting for it. There is now a mathematical model of defecation.

But hold it in for a moment. The research that produced that model follows in the footsteps of a Law of Urination, which tells us how long mammals take to urinate. Now my wife once watched a giraffe spend over a minute peeing, but if this Law is to be believed, that was an outlier. Researchers David Hu, Patricia Yang, Jerome Choo and Jonathan Pham of the Georgia Institute of Technology went to the Atlanta Zoo with sophisticated cameras and other equipment and filmed animals, small to large, as they urinated. They even dug out several YouTube videos of animals, jaguars to tapirs to rhinos, as they urinated.

Their conclusion in a 2014 paper (Law of Urination: all mammals empty their bladders over the same duration, published on the online repository arXiv, 26 March 2014, bit.ly/2PniwQy): the animals emptied their bladders in a “nearly constant duration" between 8 and 34 seconds. (If that doesn’t seem particularly “constant", what they mean is that the average time to empty was 21 seconds, and the great majority of the measurements were between 8 and 34 seconds.)

How do we explain this consistency across all manner of animals? Hu and colleagues argue that “larger animals have longer urethras, thus higher gravitational force and flow speed." Smaller ones “are challenged during urination due to high viscous and surface tension forces that limit their urine to single drops." But “the urethra constitutes as a flow-enhancing device enabling the urinary system to be scaled up without compromising its function." All of which are factors in producing a mathematical model for urination.

Why would anyone want such a model, though? Besides helping in diagnosing urinary problems, Hu and colleagues also think it might “inspire the design of scalable hydrodynamic systems based on those in nature." Briefly, a hydrodynamic system is one in which fluids are in motion and that motion itself transfers energy; an automatic transmission in a car is a good example. Who’d have thought that the way animals urinate might just have lessons for how we can shift gears?

But urination done with, Hu et al moved on to defecation. Again they visited the Atlanta Zoo. Again they filmed animals—cats to elephants—in the act. Again they dug through YouTube for 19 videos of animals defecating. And among other things in their 2017 paper (Hydrodynamics of Defecation", Soft Matter, 2017, b.gatech.edu/2Suwty1), again they found that animals “defecate within a nearly constant duration"—this time, between 5 and 19 seconds.

Is it surprising that defecation is more swiftly accomplished than urination? I’ll leave that for you to speculate about.

Of course faeces, being solid, doesn’t move through the intestine in the same way as urine moves through the urethra. The mathematical model for defecation suggests that faeces moves along “similar to a sled sliding down a chute". Hu et al explain that the model also “accounts for the shorter and longer defecation times associated with diarrhoea and constipation, respectively", thank you very much.

En route in the intestine, the motion of faeces is aided by a layer of mucus, which lubricates the movement and helps when it finally has to be ejected from the body. As you might expect, the larger the animal, the greater the amount of faeces and the thicker the mucus layer it needs. But what’s surprising is that larger animals don’t take a longer time to defecate than smaller ones—in fact, the larger the animal, the very slightly faster it defecates. How can we explain this? After all, the greater quantity of faeces must also make its way out through longer rectums—compare the elephant’s 40 cm-long rectum to a cat’s mere 4 cm.

But there’s more to the picture. On average, an animal ejects two pieces of faeces. The length of each is about as long as the rectum, and about five times the diameter of the rectum, but the “faecal and rectal diameters match". What this suggests is that the faeces is shaped in the intestine and then pushed through the rectum by muscular contractions, or rectal pressure. This is a different process from, for example, squeezing out toothpaste, which is only shaped once in the nozzle. In contrast, if faeces is to be shaped in the rectum, it would need much more force than animals exert during defecation.

With all this considered, Hu and his colleagues arrived at their defecation model. It expresses defecation time in terms of the dimensions of the rectum, the rectal pressure, and the viscosity of the mucus the faeces picks up from the intestine. I’ll spare you the equation itself.

But in particular, there’s no place in it for the mass of the animal. This is why the research shows defecation time “within a nearly constant duration" regardless of the size of the animal. (The actual slight decrease in time, mentioned above, is explained by the greater viscosity of a larger animal’s mucus).

There’s more of interest in their paper, including at least one possibly inadvertent turn of phrase that I’ll return to. For now, I want to point out that the work Hu et al have done investigating the movement of faeces through intestines has led them to another fascinating discovery. This has to do with the wombat, which is one of those overwhelmingly cute Australian animals. Now in pursuing their wombat studies, Hu et al didn’t visit the zoo or leaf through YouTube videos. Instead, they got their results from examining “veterinary euthanized (wombat) individuals following motor vehicle collisions in Tasmania, Australia."

The scientists took the intestines from these dead wombats, emptied them and then inflated them “with a long balloon". They found that the balloon was subject to different strains at different points in the intestine. This showed that the wall of the intestine has “azimuthally varying elastic properties"—that is, the wall varies in elasticity, stretching more at some places than others. Which is all very well, but what does this mean for wombat faeces? Something amazing indeed: as the faeces progresses through this intestine of varying elasticity, it is shaped into … wait for it … cubes. (“How do wombats make cubed poo?", 71st Annual Meeting of the APS Division of Fluid Dynamics, 18 November, bit.ly/2rpNIFn).

Cube-like pieces of faeces, at any rate, with definite edges and corners. Think of that. There is no other animal in this world that produces cubic faeces, and how the wombat does it has forever been a mystery. (They certainly don’t have square anuses). Hu and his team think they have an explanation—that the less elastic nature of the final stretch of intestine gives edges to the faeces just before the wombat defecates. Presto: cubic faeces.

But why does the wombat defecate cubes? Like many other animals, it uses faeces to mark territory, and wombats like to defecate on top of logs or rocks. If they produce the more usual cylindrical faeces, those would likely roll off their rock-top perch. Produce cubes, though, and they won’t roll off that easily. Clearly this is an advantage while staking out your particular piece of real estate. And why study how wombats do this? Well, we humans only know how to produce shapes like cubes via moulding and extrusion techniques. Learning how the wombat’s intestine works might give us, as Hu’s paper says, “insight into new manufacturing techniques (for such structures) using soft tissues."

Finally, two more faeces-related scientific findings.

The first is from, yet again, Hu and his colleagues. They have estimated the amount of animal and human faeces produced annually on the planet: 3.9 trillion kg in 2014, increasing at 52 billion kg/year. Why do this calculation? Because “human and animal faeces present persistent threats to global public health and also opportunities for recovery of resources." (“Estimation of global recoverable human and animal faecal biomass", Nature, 13 November, go.nature.com/2UhIybE).

The second is from a team of English and Australian healthcare professionals who run a blog called “Don’t Forget the Bubbles" (DFTB). They know that children swallow small objects alarmingly often, causing parents much distress. So they got several researchers to swallow a Lego piece each and found that, on average, it took 1.71 days for it to appear in their faeces. Their research features scores for “Stool Hardness and Transit" as well as for “Found and Retrieved Time", and I’ll leave you to figure out those particular acronyms (“Everything is awesome: Don’t forget the Lego", Journal of Paediatrics and Child Health, 22 November 2018, bit.ly/2G1ueAR).

All in the service of science. Which is why I also want to point out that in the Hydrodynamics of Defecation paper, the authors tell us about a particular characteristic that “explains why dog faeces feel slippery when stepped on." And in their Lego paper, the DFTB folks report: “There was some evidence that females may be more accomplished at searching through their stools than males, but this could not be statistically validated."

Think of everything all this could set in motion.

Once a computer scientist, Dilip D’Souza now lives in Mumbai and writes for his dinners. His Twitter handle is @DeathEndsFun

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