My most frequently asked question has got to be: what organism do you work with? This is a funny question because I don’t work with any particular organism. The funny thing that happens is that after I tell people I don’t work with a particular organism and describe my work, they say “Oh, so in what kinds of animals?” This could be sending me several messages: one is that people have no idea that research into theory exists at all. I find this explanation unlikely since everybody knows about theoretical physics (which actually exists, it’s not just theoretical). There’s even a TV show in which everybody’s favorite character is a theoretical physicist. The message I get, instead, is that it’s quite strange to think about a “biologist” who doesn’t work in a lab, in a field, or a greenhouse. What would such a person do? How can you be a biologist and do all your work without touching real animals or plants? This is the point in the conversation when I tell people that I’m a mathematician getting a degree in biology. Basically what I tell people is “I do math” or “I solve math problems.”

The cool part is I get to decide what those problems are. I’d like to give everyone a better idea of what I actually do in my research. It’s one reason I started this blog and so I plan to explain things about my general way of working over a few posts.

## Theoretical questions, mathematical answers

What scientists basically do is solve puzzles. These puzzles are usually posed as questions like “how does this plant grow?” or “When (in history) did this gene originate?” or something else highly empirical. When you want simply an accurate description of the world, when you want “How does this work?” you can go and get an organism and describe it. You can collect data, and you can formulate various hypotheses about the patterns in the data, then test those hypotheses by looking at other patterns in the data.

I do basically the same thing, except the questions that I handle are of a theoretical nature. That means that basically what I’m asking questions about are those hypotheses themselves, not the patterns in the data. I have a basic question about how sexual signals evolve. “How do they evolve?” means I want to know how a population of organisms, with some individuals carrying expensive organs or behaviors, would change over time. If a gene for such a signal arose, would most members of the population have it at some later time? Or would it go extinct? What does our existing theory tell us? How can I extend the theory we have so that we better understand this?

Sexual signals and behaviors appear to be costly, and so Darwin reasoned that they couldn’t come about by natural selection. He solved that puzzle by supposing a different form of selection that worked by mate competition, instead of simple competition to stay alive. Darwin was the first evolutionary theorist!

What Darwin didn’t have was a mathematical theory of how populations evolve. In evolutionary biology, we’re lucky to have such a theory, developed by the “Big Three” of Sewall Wright, Ronald Fisher and J.B.S. Haldane. Wright, Fisher and Haldane formulated the theory of population genetics as a mathematical description of how the frequencies of alleles change in populations over time. Just like Darwin, we don’t have to use math to answer these questions: we can take a guess or hypothesize, and go straight to testing that idea on real organisms. However, we can use mathematical tools to answer these questions, and mathematical answers are the most effective. There are several different ways to do this.

## Some nuts and bolts

Everything I do can be called “mathematical modeling.” However, some of the time it is less mathematical, or more computational. If I am asking a question of how a gene frequency changes over time, then one way is the analytical approach: I write down a set of equations describing how the gene changes its frequency over time, and then I describe properties of those equations. The particular biological model I’m interested in will determine how the equations come out: all the information I need to say how the population will change should be in these dynamical equations.

The most important properties of dynamical equations are its equilibrium and the stability of that equilibrium. An equilibrium is a state (e.g. a gene frequency) that doesn’t change over time. So for example, in standard population genetics, if an allele’s frequency is zero, then it won’t change, unless there’s a mutation. Then zero is an equilibrium as long as I don’t allow mutation in the model. Stability means that when a population is close to equilibrium, it moves toward equilibrium. This is just like rolling a ball down a hill: when it gets to the bottom, it stops. That’s a stable equilibrium: if you throw the ball back up the hill, it rolls back down. This is important because we expect to find these equilibria in nature: we expect populations to get to a certain point and stay relatively close, and that’s what we should be able to see much of the time in nature. This is incredibly simplistic, but we have to start somewhere.

Another kind of dynamical model is a *computational* or simulation model. In this sort of modeling we input certain calculations as a computer program. Instead of writing out equations for how the variables change over time, we let the computer iterate a set of calculations to simulate how they would change. I could have a very good idea of the life cycle of a theoretical organism, and I program each piece (birth, growing up, mating and death) into a program, and then calculate how the gene or phenotype frequencies in the population(s) change over time. There’s two reasons to do things this way: the model of the life cycle might be extremely complicated, so that each piece might be a paper in itself before I solve my big question; or, we might explicitly mathematically know that there is no simple solution to the problem. In that case using a computer helps us come up with an approximation to the solution.

Both the analytical model I described and the computer model describe change over time: they are *dynamical* models. I like doing things this way. However, there’s another way, which is simply to mathematically describe how things are at a particular point in a population, or to describe the way an organism works. The solution to such a model is usually to say what the optimal phenotype of an organism is, given certain conditions. Evolutionary game theory and foraging theory are examples of this “phenotypic” approach. These approaches can be used to answer similar questions, but they do tell you fundamentally different things in most cases. A phenotypic model cannot tell you how something evolved, only the circumstances under which you could expect it to evolve. Phenotypic models are very popularly interpreted in behavioral ecology, where the researcher’s task is finding out how a particular strategy affects reproductive success of individuals. It’s not necessary to care, in that case, how something evolved: it’s just there and bears questioning.

The biological interpretation comes in when we consider the *parameters* of a particular model. Usually these special quantities determine if an equilibrium exists, and if it is stable or not, in a particular model. For example a particular model might have a mutation rate parameter, or a parameter describing the rates at which males and females meet, or a parameter describing the intensity of selection against a trait. Models of disease transmission have parameters describing how often a virus has a chance to move from one person to another. Parameters usually correspond to real biological quantities that could be measured, and that’s how hypotheses get formed and tested.

## What is this all for?

The real point of doing all this is for somebody, a field or lab researcher, to be able to apply these ideas. If I can come up with a satisfactory solution to a theoretical puzzle, another researcher can take that theory to form a hypothesis and then test that hypothesis using fruit flies, or peacocks or some other real organism. My favorite kinds of applications look for patterns across large groups of organisms, many different species, and try to apply these ideas very broadly. Let’s say this species supports the conclusions of a model, and so does its closely-related species, since they correspond to different parameter values. Most of the time, however, theories get tested one organism at a time, and the research either supports the theory, or fails to support it in some way. This is the process of refinement, which is a lot of fun. For example, a theory could be supported in fruit flies, but not in cockroaches, but we find out that cockroaches are not a good species to use for testing the idea. This reveals a deficiency in the idea, since it doesn’t include cockroaches. Then I get to do some more work, and write another paper.

###### Related articles

- Research finds that maths-heavy evolutionary biology research papers are less often cited (aperiodical.com)
- The Good Fight (blogs.scientificamerican.com)
- BBC documentary: “What Darwin Didn’t Know” (whyevolutionistrue.wordpress.com)

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