How Strong is Natural Selection? Stitching Together Linear and Nonlinear Selection on a Single Scale

Post provided by Robert May Prize Winner Jonathan Henshaw

Some individuals survive and reproduce better than others. Traits that help them do so may be passed on to the next generation, leading to evolutionary change. Because of this, evolutionary biologists are interested in what differentiates the winners from the losers – how do their traits differ, and by how much? These differences are known as natural selection.

Linear and Nonlinear Selection

Traditionally, natural selection is separated into linear selection (differences in average trait values) and nonlinear selection (any other differences in trait distributions between winners and the rest). For example, successful individuals might be unusually close to average: this is known as stabilizing selection. Alternatively, winners might split into two camps, some with unusually high trait values, and others with unusually low trait values. This is disruptive selection (famously thought to explain the ur-origin of sperm and eggs). Stabilizing and disruptive selection are important types of nonlinear selection. In general, though, the trait distribution of successful individuals can differ from the general population in arbitrarily complicated ways.

When individuals with larger trait values have higher fitness on average (left panel), the trait distribution of successful individuals is shifted towards the right (right panel, orange curve). The difference in mean trait values between the winners and the general population is called linear selection.

When individuals with larger trait values have higher fitness on average (left panel), the trait distribution of successful individuals is shifted towards the right (right panel, orange curve). The difference in mean trait values between the winners and the general population is called linear selection.

When individuals with average trait values have higher fitness than those with extreme trait values (left panel), the trait distribution of successful individuals is shifted towards the middle (right panel, orange curve), resulting inknown as stabilizing selection.

When individuals with average trait values have higher fitness than those with extreme trait values (left panel), the trait distribution of successful individuals is shifted towards the middle (right panel, orange curve), known as stabilizing selection.

The standard approach, developed by Russell Lande and Stevan J. Arnold, quantifies linear and nonlinear selection in two separate statistical frameworks. We don’t seem to appreciate the weirdness of this. Imagine you want to understand selection on wing length in a species of bird. If longer-winged individuals tend to do better than shorter-winged individuals, you can quantify that using the framework for linear selection. But if medium-winged individuals tend to do better, you need the nonlinear framework. And there is no easy way to compare the numbers that these two approaches spit out!

This problem first hit me during peer review of a previous paper, when a reviewer criticized our method for only capturing linear selection. Unfortunately, there were no methods that captured all types of selection, regardless of its shape.

Thinking about this problem led me to optimal transport theory – the mathematical study of how to move mass around efficiently. It also led me to Yoav Zemel, a statistician at EPFL in Lausanne (now at the University of Göttingen), who became a co-author on ‘A unified measure of linear and nonlinear selection on quantitative traits’.

The Distributional Selection Differential (DSD): A New Tool for an Old Problem

Our paper develops the first unified measure of the total strength of natural selection, which includes both linear and nonlinear selection. The basic idea is simple: quantify the difference in trait distributions before selection (everybody) and after selection (just the winners). Imagine both trait distributions as piles of sand – how much work would it take to transform one distribution into the other by shifting sand? We formalised this idea into a mathematical definition of the strength of natural selection, which works no matter what shapes the sand piles take. We call this the distributional selection differential or DSD. You can find R code to calculate the DSD on my GitHub page.

How much work would it take to transform the trait distribution before selection (left panel) to the distribution after selection (right panel)?

How much work would it take to transform the trait distribution before selection (left panel) to the distribution after selection (right panel)?

Surprisingly, this simple definition is mathematically equivalent to two other formulations that at first glance seem totally unrelated. First, you might be familiar with the idea that linear selection can be expressed as a covariance between trait values and fitness. The DSD can also be expressed as a covariance, in this case between fitness and a special function of trait values that we call a maximizer. Second, the cumulative distribution function for a trait distribution is just a curve showing the proportion of trait values that fall below any given value. The DSD is equal to the area between the two cumulative distribution functions for the trait distributions before and after selection.

What Can You Do with the DSD?

In the medium ground finch, beak size shows two distinct fitness peaks, possibly best suited for feeding on two different sizes of seeds. ©putneymark

In the medium ground finch, beak size shows two distinct fitness peaks, possibly best suited for feeding on two different sizes of seeds. ©putneymark

You can do three types of analyses with the DSD that were impossible in the traditional Lande-Arnold framework. The first is to quantify the strength of selection on a single scale, regardless of the type of selection. In other words, no matter whether selection is linear, stabilizing, disruptive, or whatever, you can quantify its strength with a standardized number. This makes the method ideal for comparative and meta-analyses. Our approach also lets you compare the strength of linear and nonlinear selection on a single trait.

The third new use of the DSD is to quantify the strength of selection when the relationship between trait values and fitness is complicated. For example, in Darwin’s medium ground finch, beak size shows two distinct fitness peaks. The Lande-Arnold approach doesn’t tell you how strongly selection is acting on beak size in this species, but our approach does.

We believe the DSD nicely complements existing approaches to analyzing selection, providing some new angles on an old question. We hope that it will be widely used in evolutionary biology.

To find out more about the distributional selection differential, read our Robert May Prize winning article ‘A unified measure of linear and nonlinear selection on quantitative traits’ (No Subscription Required).

Click here to find out about the articles that were shortlisted for the 2017 Robert May Early Career Researcher Prize and the ones that were highly commended.

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