# Idealization vs abstraction for mathematical models of evolution

This week I was in Turku, Finland for the annual congress of the European Society for Evolutionary Biology. I presented in the symposium on mathematical models in evolutionary biology organized by Guy Cooper, Matishalin Patel, Tom Scott, and Asher Leeks. It was a fun. It was also a big challenge given the short ten minute format. I decided to use my ten minutes to try to convince the audience that we should consider not just idealized models but also abstractions. So after my typical introduction of computational vs algorithmic biology, I switched to talking about triangles. If you would like, dear reader, then you can watch the whole session online (or grab my slides as pdf). In this post, I just want to focus on the distinction between idealized vs. abstract models.

Just as in my ESEB talk, I’ll use triangles to explain the distinction between idealized vs. abstract models.

So let us imagine evolutionary biology as a bunch of triangles. We can think of each of these triangles as representing a different biological process that implements evolution. In particular, let us think of each triangle as a different population with its own structure, demography, standing genetic variation, etc. and thus its own corresponding evolutionary dynamic. In the top right corner, the green triangle might correspond to a bee colony with a very specific and strange sex ratio. Oner on the bottom right, the orange triangle might be a biofilm of slime mold with their complicated spatial structure. Maybe the blue triangle is a population of antelope undergoing range expansion. We could go on and on. For every empirical population studied at ESEB, we might imagine a corresponding triangle.

The point is that they each have some very particular details. These can be very different for each evolutionary dynamic. So how do we usually deal with this complexity?

We deal with this complexity by idealizing.

We pick a particularly simple or convenient evolutionary dynamic. One that we think is general. Something like using an equilateral triangle as a stand in for the mess of real triangles. We will often argue that this particularly simple model gets at the ‘essence’ of all evolutionary dynamics. But in reality, our choice is often guided by our methods. We pick the equilateral triangle — or the strong-selection weak-mutation dynamics — because we have the mathematical skills necessary to analyze it. And from then on we suppose that evolutionary dynamics is an equilateral triangle and analyze it as such.

If we end up talking with more experimentally oriented colleagues, we might say: “oh yeah, this is kind of like the green bee colony”. But our colleague might study slime molds and we would have to admit that it is not so much like the orange slime mold triangle. At that point, the resourceful modeler might offer to deform their idealized triangle to get one that looks more like the slime molds. So we end up endlessly modifying our idealized models with various features that we want to add or take consideration of. Of course, in practice this is made extra difficult but our lack of knowledge about what kinds of triangles occur in nature.

I think that this idealization approach is the standard approach in theoretical and mathematical biology.

But it isn’t the only approach that we can take.

Instead of making an idealization, we can follow the route of abstraction. We can just note that all the shapes we drew are triangles. And so let us see what we can conclude from properties that all triangles have in common.

Unfortunately, abstraction comes with some downsides. First, it means that we cannot get certain specific results. We can say much more about a specific equilateral triangle than we can about an arbitrary triangle. Second, we lose some things. An equilateral triangle is a concrete triangle, it ‘looks’ like a triangle. An equilateral triangle ‘resembles’ the triangles it is modeling. The concept of triangle, however, is not a concrete triangle. It doesn’t ‘look’ like anything. It doesn’t ‘resemble’ the system it models. Rather, it specifies a language in which that system can be expressed. Thus, the abstraction can be of a different type than the things it abstracts over. And we need different tools for dealing with this.

This is where the tools of theoretical computer science come in.

How do we reason about arbitrary triangles? Or in our case: arbitrary evolutionary dynamics with arbitrary population structures, etc. For this, I use theoretical abstraction.

As I’ve discussed several times before on TheEGG, all these various evolutionary dynamics are still algorithms. Thus, they are subjects to the laws of computational complexity. We can use these laws to establish general results like the difficulty of reaching local fitness peaks. See Kaznatcheev (2019) for more.

But abstraction can also help with experiment, not just theory. Or as I’ve written before: abstract is not the opposite of empirical.

In the language of triangles, we might care about some specific property of triangles like their area. Normally, we would find this area by measuring all three sides or measuring two sides and the angle between them. Then from these reductive measurements we would compute the effective area. In the context of evolutionary dynamics, especially in the context of evolutionary game theory — this might correspond to knowing the pairwise interaction between strategies and then running that interaction over some spatial structure to get some surprising prediction about which strategy comes to dominate the population following this particular spatially structured evolutionary dynamic. As I discuss in Kaznatcheev (2018), this direction from reductive to effective has been the standard approach in much of evolutionary game theory.

But do we need to always measure these reductive details that identify a particular triangle? After all, many triangles have the same area. And if we only care about the area then we don’t need to know which particular combination of side-lengths resulted in our area. Especially if we can come up with a clever way to measure area directly without measuring side lengths.

I don’t know how to do this for the area of triangles, but I do know how to do this for effective games (Kaznatcheev 2017; Kaznatcheev et al., 2019). In the context of evolutionary games, the clearest example of multiple-realizability is due to spatial structure. I highlight this in the figure below.

Consider some effective, population level game. Say the Leader game we measured in non-small cell lung cancer: $G_\text{eff} = \begin{pmatrix} 2.6 & 3.5 \\ 3.1 & 3.0 \end{pmatrix}$.

$latex G_\text{eff}$ is like the area of a triangle. It can be implemented by many different side measurements, which in this case correspond to different reductive games.

For example, with an idealized inviscid population structure, $G_\text{eff}$ is implemented by a reductive game that has the same numeric values; i.e. $G_\text{red} = G_\text{eff}$. But in an idealized spatialized population like a death-birth 3-regular random graph, it is instead implemented by a qualtitatively different Hawk-Dove game; i.e. $G_\text{red} = \begin{pmatrix} 2.6 & 3.7 \\ 2.9 & 3.0 \end{pmatrix}$.

But most important is the case where we have nature implementing our game. It is some experimental triangle that has some specific side-lengths that give us our area. The experimental process of the game assay calculates this ‘area’ — i.e. effective game. But we don’t need to know the details of this reductive game and the transformation if all we care about is the effective game. In other words, we can be happy with $G_\text{red} = ?$.

This can be useful if we care about something like the outcome for a patient — a global effective property — but don’t know the details of the interactions going on within the tumour — the local reductive game. In this case, we want a process like the game assay to measure the global effective game without first having to learn the reductive game and the details of the population structure that transforms it.

Hopefully thinking about abstraction and multiple realizability can help us go after these kind of goals more clearly.

### References

Kaznatcheev, A. (2017). Two conceptions of evolutionary games: reductive vs effective. bioRxiv, 231993.

Kaznatcheev, A. (2018). Effective games and the confusion over spatial structure. Proceedings of the National Academy of Sciences, 115(8): E1709-E1709.

Kaznatcheev, A. (2019). Computational complexity as an ultimate constraint on evolution. Genetics, 212(1): 245-265

Kaznatcheev, A., Peacock, J., Basanta, D., Marusyk, A., & Scott, J. G. (2019). Fibroblasts and alectinib switch the evolutionary games played by non-small cell lung cancer. Nature Ecology & Evolution, 3(3): 450.