Skip to 0 minutes and 14 secondsSo now that we have a fair amount of theory behind us, let's have a look at some interesting applications related to the connection between biology and mathematics. So in recent decades, biologists have become much more closely connected with mathematics. Mathematics is occurring in biology in terms of geometry of proteins. The way proteins are shaped is now understood to be intimately connected with their functionality. The combinatorics of DNA-- a lively subject these days-- is highly mathematical. Statistics, of course, is becoming more and more important as we observe large populations and try to gather data about what's actually happening in a whole range of different kinds of questions.

Skip to 1 minute and 1 secondAnd what we're going to be concentrating on is how scaling laws are becoming very important in biology. These are exactly the kinds of laws that we've been talking about, typically power laws that are being discovered to explain interesting relationships between biological objects. So, for example, a very interesting and topical issue is how lifespan of an animal depends on the size of that animal. It's been observed a long time ago that bigger animals tend to live longer. But in recent decades, we've been able to quantify that and get rough ideas of the relationship. And it's now understood that, very roughly, lifespan is proportional to mass to the 1/4.

Skip to 1 minute and 53 secondsSo this is an example of a power law involving a fractional power up there. So the 1/4 function is like that 1/2 function, like 1/2 of 1/2, the square root of the square root. So it grows less quickly than the square root does, but it's still grows.

Skip to 2 minutes and 15 secondsAnother interesting relation is that between size and heart rate. So it turns out that as animals get bigger, their heart rates generally slow down. Small animals have quick heart rates. And we've been able to quantify that in terms of relationship, of power law, which surprisingly has a very similar kind of form as the previous one, except there's a mass to the minus 1/4 now, reflecting the fact that this is going down. As the mass increases, the heart rate goes down.

Skip to 2 minutes and 52 secondsAn interesting consequence of these two put together is that if we multiply lifespan times heart rate, the mass to the 1/4 and the mass to the minus 1/4 will cancel out. And we'll just get that this is more or less constant, independent of the mass. There's a remarkable observation that you take very big animals or you take very small animals, it doesn't really matter. If you multiply their heart rate by their lifespan, you get the same number, which is to say that roughly we have 1 billion ticks of our heart. One billion heartbeats is roughly what most animals are allocated to, very roughly. Whether it's a very small animal or a very big animal, you get one billion heart beats.

Skip to 3 minutes and 44 secondsAn elephant, heart beats slowly, and it lives a reasonably long time. A mouse, heart beats very quickly, and it doesn't live very long. So it's absolutely remarkable aspect of-- the mathematical aspect of biology. So this subject in biology of studying relationships, especially power laws, is called allometry. And one of the important tools to try to discover what these exponents are likely to be in any given situation is to plot things in terms of a log-log plot. So we'll have to have a little bit of a discussion about log-log plots. We already talked a little bit about logs, so you know a little bit about that. We'll see how to use those and how that connects with allometry.

Skip to 4 minutes and 30 secondsAnd we'll also have a look at Kleiber's law, which is in the same spirit as this kind of discussion, but it connects metabolic rate with the size of animal. So I believe this more quantitative introduction of power laws into biology is a very interesting direction, and probably, we're going to see more precise formulations of these kinds of relations in the future.

# Lifespans and heartbeats

In this video we explain the mathematical power laws that relate animal mass to heart rate and lifespan.

## Allometry

In biology, the application of power laws is part of *allometry*, the study of how body size affects biological parameters, and how various physical and biological attributes of animals scale with size. Since there are many kinds of animals, we are looking for general laws that have wide applicability.

Good examples include the relations between mass of an animal and lifespan, between mass and heartrate, and between mass and total metabolic rate. These are generally speaking governed by power laws, but there are exceptions – often involving birds, fish, or man himself.

This kind of discussion brings a mathematical aspect to issues of considerable interest to humans: after all, living a long and fulfilling life is something we all hope for.

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