Transcription
You know, biologically, I said. How much LSD should you give an elephant? (mellow music) Well, to a reasonable person, the correct answer is probably none. But what if you needed to do it for a scientific experiment?
In the 1960s, the CIA was working on a top-secret project known as MKUltra, and one of their main goals was to find out how drugs like LSD could be used to change human behavior. And this is where elephants come in because elephants are normally quite docile, but sometimes they just snap. And the hypothesis was that this change in behavior might be triggered by the release of an LSD-like substance that naturally occurs in their brains. So if that's true, then administering LSD to a docile elephant might reproduce that behavior. So the real question was, how much LSD should you give an elephant so that the dose is large enough to cause a psychological reaction, but not so large that it causes harm?
(mellow music) Well, the researchers didn't know. LSD had never been given to an animal that big, but they did know that the safe dose in cats was around 0.3 milligrams. Now, since an elephant has around a thousand times the mass of a cat, they figured, we'll give it a thousand times the dose. They received approval to perform the experiment on Tusko, an Indian elephant at the Lincoln Park Zoo in Oklahoma. And there, they injected him with nearly 300 milligrams of LSD. But within five minutes, Tusko trumpeted, collapsed, fell heavily onto his right side, defecated, and went into status epilepticus. They administered a few other drugs in an attempt to revive him, but Tusko died shortly thereafter. The mistake they made was to assume that safe drug dosage scales linearly with mass. It does not. And it turns out there are a lot of things like this that don't scale in the way you'd expect.
(mellow music) For example, take the smallest mammal by mass, the Etruscan shrew, and the largest land mammal, the African bush elephant. Which do you think has more heartbeats over the course of its entire life? Well, an African elephant has around a billion heartbeats in its lifetime, and an Etruscan shrew also has a billion heartbeats in its lifetime. What about a wallaby? Also a billion. A two-toed sloth? A billion again. Just about every mammal, no matter what environment they live in, how large they are, or even whether they live for one year or 100 years, they all get around a billion heartbeats between the day they're born and the day they die.
"Why a billion? Why not a million? Why not a thousand? It's not these other numbers."
This video is in large part based on the book *Scale* by Geoffrey West. If you want to learn more, I'll put a link to it down in the description. What's even more curious is that just by knowing a mammal's mass, you can predict a staggering number of biological traits, from its pulse rate and reproductive output to its total lifespan. The same pattern holds for cities. If you know the population and location, that allows you to forecast everything from average wages and patent filings to crime rates, disease prevalence, and even the literal speed at which pedestrians walk. So how is that possible? Well, for that, we have to go back to the case of Tusko the Elephant.
Researchers assumed that safe drug dosage is proportional to mass. Double the mass, double the dose. But it turns out that the speed at which an animal can process chemical compounds doesn't depend directly on its mass. It depends more on its metabolic rate. That is, the number of calories it uses in a given amount of time.
"There's a lot going on in your body. Your heart is pumping, that takes energy. You're moving food around. Digestion takes energy. Keeping your brain going takes energy, breathing. Everything that you do requires energy."
(mellow music) For a cat, for a single day, they require roughly 250 kilocalories of energy, used by all of their trillions of cells. But there is nothing special about a cat's cells. If you take a cat's cells and an elephant's cells and put them under a microscope, you'll find they're a similar size, similar makeup, and they perform the same sorts of functions. The same is true for other animals. In other words, the building blocks of animals are always roughly the same. So an elephant that has a thousand times the mass has about a thousand times as many cells. So you'd expect it would need a thousand times as much energy. So 250,000 kilocalories per day. But that is where you run into problems.
"So what's the issue here? If I'm an organism and I'm burning up energy all day long, that energy is radiated out in the form of body heat. So we're constantly losing heat through our surface, through our skin, to the environment."
To see why this matters, let's simplify the problem and do the standard physicist thing. Let's assume our animals are perfect spheres. To be clear, this isn't necessary for the argument to work, but it does make everything a lot easier to follow. Since the volume of an animal is proportional to its mass, our 3000-kilogram elephant has a volume a thousand times greater than our three-kilogram cat. So its radius must be 10 times larger. That's because volume is proportional to radius cubed. But surface area only grows as radius squared. So the elephant's surface area only increases by a factor of 100. Generating a thousand times as much heat while only having a hundred times the surface area to radiate it away would end very poorly for the elephant. If this were the case, it would boil alive.
(mellow music) So in 1838, French scientists proposed a different scaling law. Since metabolism generates heat and that heat is radiated through the surface, metabolic rate, B, should scale in proportion to the surface area, A, instead. This became known as the surface law. Now, we can rewrite this to see how metabolic rate scales as a function of mass. Surface area is proportional to radius squared, so we can swap that in, and if mass is proportional to radius cubed, then radius must be proportional to mass to the one-third. Plugging that in for R, we find that metabolic rate should scale with mass to the two-thirds.
"Really familiar example would come from thinking about cooking."
(mellow music) Maybe you want to make a big turkey for Thanksgiving. People commonly phrase it as, "How long do I need per pound?" But that, you realize, is linear thinking. The important thing is really the thickness of the bird or the roast, because the heat is coming in by conduction from the oven and it's got to do thermal diffusion into the meat. The time for the diffusion will scale like the characteristic length, which in this case would be the thickness of the meat. It'll go like the dimension squared, whereas volume of the meat is going to be like length cubed, and that's proportional to the weight of the bird or the roast. So when you put those things together, you'll get that the characteristic amount of time needed to cook the roast will go like its mass to the two-thirds power.
"So if I'm thinking about cooking a roast that weighs twice as much as another roast, how much longer? Is that two to the two-thirds the time?"
"Yeah, if you wanted to double the weight, then you only have to cook it about 60% longer, not 100% longer."
(Host) Similarly, according to this scaling, an elephant that's a thousand times as heavy as a cat should only burn a hundred times as many calories. So 25,000 instead of 250,000. And the appropriate dose of LSD for Tusko would've been just 30 milligrams. These two wildly different predictions come from different assumptions about how metabolic rate scales as a function of mass. But notice, in both cases, it's just proportional to mass raised to some power. These kinds of relationships are called power laws, and all power laws have a special property. If you take the logarithm of the X and Y values, you get a straight line. And the slope of the line is equal to the exponent of the power law. This makes it easy to identify different types of power laws. If the slope is one, that's just everyday linear scaling. If it's less than one, like the surface law, that is called sublinear scaling. And if it's larger than one, that is known as superlinear scaling.
For nearly a hundred years, biologists generally agreed that the two-thirds exponent of the surface law was the correct one for metabolic rate. But then in 1932, Swiss biologist Max Kleiber decided to put it to the test. He took the metabolic rates of different animals, from a small dove at 150 grams to a large steer at 680,000 grams, and then he plotted them against their mass on a log-log plot. And as expected, all the data did fall on a straight line. But the slope wasn't two-thirds. Instead, it was about three-quarters. This became known as Kleiber's Law. It implies that if you double an animal's mass, metabolic rate goes up by about 1.68, an increase of 68% instead of the 59% which would be predicted by the two-thirds scaling law. So according to Kleiber's Law, an elephant burns roughly 178 times as many calories as a cat, or about 45,000 kilocalories. And the actual LSD dose Tusko should have received was 53 milligrams, which is around a sixth of the dose the researchers gave him. So that explains the dosing catastrophe with Tusko the Elephant.
(mellow music) Kleiber's original work was based on a small data set, primarily made up of mammals. But if you plot the data for a larger range of mammals, as well as for other animals like birds, reptiles, and fish, you find that they all follow roughly the same scaling relationship. Now, they don't all fit perfectly onto the same line because the base metabolic rate for warm-blooded animals is higher than for cold-blooded ones, but they all scale according to the same power law: mass raised to the three-quarters. Some argue that this relationship expands all the way down to single cells and the molecular machines inside of them. If that's true, Kleiber's Law governs life spanning more than 25 orders of magnitude.
But then the question is, if an elephant has so many more cells, and each cell is a similar size to a cell in a smaller creature, and yet it's using proportionately less energy, like it's got a lower metabolic rate per pound or per kilo or per cell, that's really saying that those cells are functioning with much less energy.
"It's like there's some efficiency to being big, and it's hard to understand why. What is it that the cells are providing to each other, somehow cooperating in some way?"
"I just put in my calculator 100 to the three-quarters, so if you got an organism that's 100 times bigger, then according to this, only 31.6 times the metabolic rate. So it seems like a big savings."
"It's a massive improvement, right? It's a big savings. It's a biological fact, but people have been arguing now for a century what accounts for this three-quarters."
(mellow music) In the decades following Kleiber's observation, the mystery only deepened. Researchers discovered that brain size also roughly scales as mass to the three-quarters, and so does an animal's growth rate and the amount of blood pumped per minute. But not every property scales as mass to the three-quarters.
"If you ask how long a creature will live, a mammal, that tends to be proportional to its mass to the one-quarter power. One over four, not three over four."
(Host) So that means if you double the mass of a mammal, then on average, its lifespan is around 19% longer. Similarly, blood circulation time also scales as roughly mass to the one-quarter, while breathing rate and heart rate both scaled to the negative one-quarter. They're not all three-quarters power laws, but they are all multiples of a quarter. So the question on everyone's mind was, where are these quarter-power scaling laws coming from?
One popular theory emerged in the 1990s.
(mellow music) Brian Enquist was studying for an undergraduate degree in biology.
"Most of the biology classes then that you take, you learn about the Krebs Cycle and then you have to memorize all the different parts of a flower and then all... They have a different terminology for everything."
(Henry) But then one day he took a zoology class where they showed him those quarter-power scaling plots.
(Brian) "I just couldn't believe it. I was like, 'You got to be kidding me.' You know, this is kind of like something fundamental that's kind of like underlying biological diversity. And I knew immediately that I wanted to kind of quit my specialized kind of plant physiology research and do something associated with scaling."
(mellow music) (Henry) So Enquist started studying for his PhD, under Professor James Brown, who had been thinking about scaling laws for years. So they knew that often when power laws appear, there's some form of self-similarity in the underlying system. So they wondered, what could that self-similarity be? And they suspected that it might have something to do with the way resources are transported through the body, specifically the networks that do this. They had the biological intuition, but to get a complete theory, they needed a formal mathematical framework. In other words, they needed a mathematician or a theoretical physicist.
"And at the time, Jim was associated with the Santa Fe Institute, and he started asking around with, is there anyone up here that's interested in these biological scaling relationships? And then the president at the time said, 'You know what? I know of this physicist who's up at Los Alamos who was talking about biological scaling relationships.' And so we met Geoffrey and it was like immediately, it's like we'd been talking about the same things for like years. It's like, you know, finding someone who's been like speaking your language, but no one could understand you."
(Henry) So West, Brown, and Enquist teamed up to try and find a compelling explanation for Kleiber's Law.
They started by assuming three simple premises. The first premise is that the networks that distribute resources are space-filling, since they need to reach every cell in the body. The second premise is that the terminal units of those networks, the thinnest segments on the outer periphery of the delivery system, have the same width regardless of the size of the organism. That is, the outermost blood vessels that carry nutrients to an elephant's skin cells are about as thick as the ones in a mouse. The elephant just has many more of them. And the third premise is that over time, evolution has driven these transport networks toward an efficient design. So what should such a network look like?
(mellow music) Well, intuitively, to get fuel from one place to another as efficiently as possible, you want the path to be as short as possible. So basically, a straight line. But because fuel needs to reach every part of the body, you would also need many different paths. And as you go to larger and larger organisms, the networks inside need to reach a larger and larger volume. One way to do this is to stretch all the paths and match the growth of the animal. In this case, the volume of the animal should scale as these internal path lengths cubed. Or if we rearrange that, internal path length should scale as volume to the one-third, just like the overall length of the animal does. But this design is incredibly wasteful. Take these two regions. The two vessels that are bringing blood here go through almost the same path in the body, and they only split up just before they reach their destinations. If instead we had just one vessel up to this point and split it only when the paths needed to diverge, we could serve the two regions using a lot less vessel material and a lot less blood to fill the vessels. Of course, you can extend this logic for all the blood vessels in the body, and you end up with a much more efficient design of branching blood vessels. But now we run into another issue because each branching point provides an opportunity for some of the blood to bounce back, that is, reflect. If you have many reflections, that would mean it costs significantly more energy to pump blood around. So next, they argue that nature should favor structures that minimize reflections. And as it turns out, this happens if the cross-sectional area of the vessels stays the same before and after the branching. So if you've got a cross-sectional area of two centimeters squared for the main vessel, then each of the two branches need to have an area of one centimeter squared each, for large vessels at least. For smaller ones, daughter branches can be a bit thicker to allow blood to slow down and exchange resources with the tissue it's reaching. If you keep repeating this pattern across the network, you end up with this: a branching, self-similar fractal. And if you look at the actual shape of the circulatory system, it has this geometry. So it seemed like they were onto something.
But how do you get from this to quarter-power scaling laws? Well, mathematician Felix Hausdorff discovered that self-similar fractals have an interesting property.
(mellow music) Take a straight line segment. It's completely one-dimensional and not a fractal at all. Hausdorff assigned this line segment a value of 1.0. But now imagine adding a few bends and more bends to those bends. If you keep doing this, the line becomes more and more fractal-like. Eventually, if you keep applying the right kind of contortions, that one-dimensional line segment fills up an entire region of the 2D plane. Hausdorff assigned these space-filling fractal curves a value of 2.0, corresponding to their dimensionality. The same ideas apply to a 2D surface.
"So think about a piece of paper, right? Two-dimensional. All then the fractal network is doing is crumpling up that sheet of paper, and it effectively fills a ball of volume. So you can now describe that sheet of paper as a sphere instead of a two-dimensional sheet of paper."
(Henry) Repeat the right pattern of folds at smaller and smaller scales, and a 2D surface fills more and more of a 3D volume. In the mathematical limit, it becomes space-filling with Hausdorff dimension of 3.0.
(Brian) "You know, biologically, I said, 'Well, what does that mean?' That enables an organism for a given size to pack in more of these metabolic surface areas than would be expected. And it's because of this fractal-like structure that enables you then to pack in and have all of these folds and convolutions and on top of each other to pack in an enormous amount of membrane surfaces."
As a result, the Hausdorff dimension of the surface of the circulatory system is roughly three, meaning its surface area doesn't scale as its length squared, but it's length cubed. And since the metabolic rate hinges on how fast resources can be exchanged across the surface area, well, it must also scale like length cubed. But remember, West, Brown, and Enquist wanted to explain Kleiber's Law. So they needed to know how metabolic rate scales with mass.
(mellow music) Since every cell needs to be served by the network, this means that the volume around the network should be proportional to the animal's mass. And since volume is just surface area times length and surface area is proportional to length cubed, that means both volume and mass must be proportional to length to the fourth, which can be rewritten to show that length is proportional to mass raised to the one-quarter. And if you plug that into the equation for the metabolic rate, you find that the metabolic rate must be proportional to mass to the three-quarters, exactly as Max Kleiber had found.
West, Brown, and Enquist published their work in 1997, and it soon came to be known as WBE Theory, after their initials.
"And the theory is quite rigid. It makes very specific predictions. This is good science. Okay? This is sticking your neck out, and it's an incredibly beautiful theory."
(mellow music) (Derek) Take a look at this table from Geoffrey West's book. These are the scaling exponents WBE theory predicts, including many that are not multiples of a quarter, but all follow from the same theory. For example, the radius of an animal's aorta should scale with its mass to the three-eighths, or 0.375. And the area of its lungs should scale with mass to the 11-12ths, or about 0.92. In total, this chart makes 26 different predictions. Now, these are the observed data. Radius of the aorta, 0.36. Lung area, 0.95.
(Steven) "That's the really shocking thing about what they did. They had a table with something like, I don't know, 20 or 30 predictions of exotic exponents, and that's what you really see in the data. So this one theory accounts not only for the three-quarters power of metabolism, but for literally dozens of other things that biologists have measured."
Some of the scaling laws are easy to explain once you've got the three-quarters law for metabolism. Take a mammal's heart rate, for instance. Heart rate is equal to the blood flow rate over the amount or volume of blood in every beat.
(mellow music) The volume of blood per beat has been found to scale in direct proportion to an animal's mass. So that's just M. And the blood flow rate? Well, remember that metabolism is all about how nutrients get distributed around the body. So most biologists agree metabolic rate and blood flow rate are directly proportional to one another. So heart rate should scale as metabolic rate over mass. Swapping in the scaling law Kleiber had found, that gives us M to the minus one quarter, meaning bigger animals should have slower heartbeats than smaller ones. And this is exactly what we observe in nature. The world's smallest mammal, the Etruscan Shrew, has an extraordinary heart rate of 1200 beats per minute. That's 20 beats per second. Whereas the biggest land mammal, the African bush elephant, has a typical heart rate of only 30 beats per minute.
And we can do something similar for lifespan. One of the leading theories is that an animal's lifespan is based on the accumulation of metabolic damage. That is, as each chunk of tissue in an organism processes nutrients over time, this causes damage to accumulate, and that over time causes the animal to die. So the rate at which an animal accumulates damage is its metabolic rate per unit of mass, and its lifespan should be the inverse of that rate. If damage accumulates faster, it dies sooner. If it accumulates slower, it lives longer. So lifespan is proportional to M over B, or substituting in Kleiber's Law, M to the one quarter. So lifespan should increase with mass. And you do see this in nature.
(mellow music) A shrew only lives for one to two years in the wild, while a mighty African elephant can live up to 70 years. So if you're a small mammal, you have many heartbeats per minute, but you live a relatively short life. Conversely, if you're a larger mammal, your heart beats much slower and you live a lot longer.
"So it's the, you know, 'live fast and burn out and die young,' right? Or 'spend it frugally and live a really long life.'"
But you might have also noticed something else. Heart rate scales as B over M. So it equals B over M times some constant. And lifespan scales as M over B. They're inverses of each other. They scale in equal and opposite directions. Now, the total number of heartbeats in an animal's life is just the heart rate multiplied by the lifespan. So when you multiply these two terms, they cancel out, leaving you with just a constant. So that suggests that no matter what mammal you're talking about, it should have roughly the same number of heartbeats. You can find that number by just plugging in some examples. Let's start with the Etruscan Shrew. The shrew's 1200 beats per minute multiplied by a lifespan of around 1.5 years gives you around 950 million heartbeats in the course of the shrew's life. Meanwhile, an African elephant's 30 beats per minute multiplied by a lifespan of around 65 years gives you a little over a billion heartbeats. And we could keep going.
(upbeat music) But for nearly every mammal you look at, you keep landing at the same figure of around a billion heartbeats. This is why nearly every mammal from a tiny field mouse to a gazelle, from a cheetah to a hippopotamus, they all get around a billion heartbeats between the day they're born and the day they die. But there is one major outlier, one mammal that gets significantly more than a billion heartbeats. And that is us, humans.
(mellow music) We are the lucky ones. Three centuries ago, humans were much closer to the standard value of one billion heartbeats. But around the mid-1800s, germ theory and better sanitation methods became widespread, causing a stark decrease in the number of child mortalities and deaths from disease. So life expectancy began to climb up. And with it, the average number of heartbeats in a lifetime. If you look closely, you can also see some significant drops, like this 1918 dip from the Spanish flu pandemic or over here, what appears to be the impact of the Second World War. But overall, the trend is clear. We have systematically been increasing the number of heartbeats we get in our lifetime to the point where now the average human gets nearly three billion heartbeats before they die. I don't know if there's a better argument for science and technology than this. It has literally given the average human more than a full extra life. And it's not just humans. Other mammals have been observed to have much longer lives in captivity when they're away from the hazards they would naturally encounter in the wild. Or if you look at it purely from the number of years we get, we now have the lifespan of a much larger mammal, somewhere between an elephant and a whale.
But there is one curious thing about this trend.
(mellow music) Take a look at this graph. It looks surprisingly similar to the graph from before. In fact, if you overlay them, they look remarkably similar. Now I want you to take a guess at what this graph is. Have you got your answer? It is the number of people living in cities. For these two charts, we're using data from parts of the United Kingdom where the record-keeping goes back several centuries, but the rest of the world has followed similar trends. Of course, that doesn't mean cities cause people to live longer, but it goes against the perception of cities as being full of pollution and breeding grounds for disease. Already in 1889, a medical doctor wrote, "The poisonous germs and pollutions of the city, its impure air and water, bad sewage, and endless nuisances." So how do you reconcile these two views? Or more specifically, is there any quantitative data on how smaller cities compare to larger ones?
It turns out this is something that Geoffrey West pursued after his work with Brown and Enquist. He and collaborators like Luis Bettencourt and others have looked at scaling laws in cities.
(mellow music) (Host) They looked at how different properties like the amount of crime scale as the population of the city increases, and what they found is that if you plot serious crimes on a log-log plot, the data clusters around a straight line with a slope of 1.15, meaning crime grows faster than linear or superlinear. So for every doubling of a city's population, you get around 2.2 times as many criminal cases, or around 120% more crime, as opposed to the 100% you might naively expect. To make matters worse, researchers found that the same general pattern holds for the amount of wastewater and even the number of AIDS cases. The exact exponents vary a little, but overall, as cities grow larger, you systematically get more of each. So it seems like that medical doctor was onto something. And you might think life on Earth would be better off if we all lived in small towns instead, but that might not be the case.
(mellow music) In 2006, Dirk Helbing, Christian Kuhnert, and Geoffrey West looked at how the number of gas stations scales as a function of population.
"If a city is twice as big, does it need twice as many gas stations? Because, you know, we have to supply not exactly nutrients, but energy, gas, for all those cars."
(Host) The naive expectation is that if you double the number of cars, you're going to need to double the amount of fuel, so double the gas stations. To find out whether this was true, they plotted the data on a log-log plot and found a straight line. But the exponent wasn't one, it was about 0.8. This means that for every doubling, you only need around 74% more gas stations, which is a decent savings.
(mellow music) The amount of roads and electrical cables also scale in roughly the same way. The rough figure that West gives in his book is that they all have scaling exponents of around 0.85.
"It is interesting that some of the things that we can share, like you can drive on the road, but so can I. When it's shared resources, yes, cities can be surprisingly green. The argument is that cities can be even greener than you might think than living out in the middle of nowhere."
But cities have even bigger benefits. Things like total wages, GDP, and the number of patents all scale superlinearly, with exponents that cluster somewhere around 1.15, meaning that for every doubling in size, you get around 120% more of each. All of this becomes especially significant when you compare a small town of say 50,000 people to a city 100 times its size, because infrastructure needs only need to go up by a factor of about 50, while total wages, GDP, patents, and inventions, they all go up by a factor of 200. Unfortunately, disease and crime also go up by the same factor. Or look at it this way, on a per-person basis, you'd only need about half the infrastructure, while you get double all the socioeconomic factors. So cities, far from being detrimental to the world, they might actually be one of our best inventions and an indirect driver of a lot of scientific and technological progress. Perhaps this is also why people often say that life in the city feels faster. A feeling that seems justified because researchers looked at how fast people walk in cities of different sizes. And they found that people literally do walk faster in larger cities.
"That turns out to depend on city size. It's not just that the sidewalks are congested or not. It's just like the vibe gets people amped up. They move faster in cities."
So the pace of life seems to be increasing, but that may come at a cost because as one person put it, "Everything nowadays is ultra. Everything is being transcended continually in thought as well as in action. No one knows himself any longer. Young people are stirred up much too early in life and then carried away in the world of the times. Wealth and rapidity are what the world admires." Could it be that life is accelerating so fast that humans won't be able to keep up? Well, probably not, because this quote was written in 1825 by Wolfgang von Goethe. For the past 200 years, and probably longer, almost every generation has felt like life was accelerating. And yet, every generation has managed. So I think it's likely we can continue to adapt indefinitely. And as cities continue to grow in size, all of us will continue to reap the benefit of economies of scale in much the same way that mammals benefit from being larger. But while WBE Theory seems to predict where the exponents in biological scaling laws come from, for cities, there is no widely accepted explanatory theory yet. Trying to explain where those exponents of 0.85 and 1.15 come from is one of the big goals for theorists. Although even WBE Theory is not universally accepted. For one, the fact it predicts the right exponents doesn't necessarily mean the theory itself is correct. There are a few other theories that predict some of the same scaling exponents, and there are also some other critiques.
"There's a lot of discussion. It may look convincing. And personally, I tend to think it is very impressive. But I have very good colleagues like Peter Dodds at the University of Vermont, and he thinks that a lot of the data analysis is either not done exactly right or that the data are so noisy that you shouldn't really take this so seriously."
Dodds even argues that Kleiber's Law itself might not be true.
(mellow music) "In the 1960s, there's a symposium on energy metabolism, some name like this, in animals, and at the end of it, they vote 29 to zero that it's going to be three-quarters, right? Because, you know, you got to set some rules. If you go back and look at the data, which no one is really doing anymore, right, the data does not work. Like it doesn't work."
(mellow music) (Henry) This is a graph of metabolic rate as a function of mass based on a study that looked at 391 species of mammals, much larger than Kleiber's range. And it looks like the three-quarter slope fits quite well. But this is just the top part of the chart, the biggest mammals. If you zoom out, you see that the rest of the mammals appear to fall on a line that's closer to two-thirds. And in recent studies of bird metabolism, you also find a slope that appears closer to two-thirds than three-quarters. Could it be that those French scientists from centuries ago were right, that metabolic rate really does just scale with surface area? Well, not so fast.
(mellow music) For one, many recent studies of cold-blooded animals find slopes that are significantly higher than two-thirds. The bigger problem is that measuring metabolic rates is difficult. It generally involves putting animals in a container and taking very precise measurements of their heat production or oxygen consumption, all while ensuring that the animal is in an unstressed, low-activity resting state. Unsurprisingly, this is particularly difficult to pull off with large animals. And as a result, many studies have found scaling exponents where the error bars include both two-thirds and three-quarters. Today, the research community is split. Many uphold Kleiber's Law and the three-quarter scaling, while others think it's two-thirds. But a growing number suspect that there is no universal scaling exponent across all of life. In fact, it may very well be the case that the metabolic rate of larger mammals scales as their mass to the three-quarters, while for smaller ones, it's their mass to the two-thirds.
"You know, it's like everything in science that there are people arguing, and that's good."
"I guess the bigger exhortation would be someone to really measure it, measure things beautifully. You know, we're in 2026, has to be not just one elephant at one zoo, it needs to be measured again really well."
(Host) What everyone does agree on is that scaling laws are real.
Life is not linear all the time. There are deals to be had. There's often in real life departures from proportionality, and you have to be aware of it. Sometimes things punch above their weight, and as you get bigger, you get more efficient. Sometimes there are detriments to size.
So it pays to know how things scale. Clearly, from an energy efficiency perspective, animals benefit from being larger. Similarly, all of us potentially stand to gain from having more people living in larger cities. It could lead to more discoveries and inventions, and in doing so, improve the standard of living for all of us, which might even give us more heartbeats in our lifetimes. From the surface law to Kleiber's Law to WBE Theory, how metabolic rate scales with mass has been one of biology's biggest debates for centuries. By collecting better data and analyzing it carefully, there's a good chance that the next generation of researchers could be the ones to finally put this debate to rest. It could be one of you watching or a student that you know. And today's sponsor, Brilliant, is helping to train the next generation of researchers, preparing their minds to solve the huge questions they'll face. Brilliant is a cutting-edge personal tutor that helps any student excel in math, science, and coding. When it comes to learning, one-on-one tutoring really is the gold standard, but this can be expensive and logistically difficult. As a substitute, students often turn to AI for guidance. But most AI platforms hinder learning by simply just giving away the answers. So instead, Brilliant is designed to guide students to their own aha moments by asking probing questions and by breaking problems down step by step. For instance, check out what happened when I launched one of Brilliant's lessons on rotating coordinates and told its tutoring tool that I was struggling. It asked me a specific follow-up question and even labeled relevant points on the graph to give me a nudge. Brilliant is an excellent aid for the young math and science learners in your life. It's the kind of tool I wish I had when I was in school. So click the link below or scan the QR code to get started with Brilliant's tutor for free, or upgrade to premium to unlock all courses. And right now, Veritasium viewers can save 20% off an annual subscription at brilliant.org/veritasium. So I want to thank Brilliant for sponsoring this video, and I want to thank you for watching.