## Quick introduction: Generalizing the NK-model of fitness landscapes

As regular readers of TheEGG know, I’ve been interested in fitness landscapes for many years. At their most basic, a fitness landscape is an almost unworkably vague idea: it is just a mapping from some description of organisms (usually a string corresponding to a genotype or phenotype) to fitness, alongside some notion of locality — i.e. some descriptions being closer to each other than to some other descriptions. Usually, fitness landscapes are studied over combinatorially large genotypic spaces on many loci, with locality coming form something like point mutations at each locus. These spaces are exponentially large in the number of loci. As such, no matter how rapidly next-generation sequencing and fitness assays expand, we will not be able to treat a fitness landscape as simply an array of numbers and measure each fitness. At least for any moderate or larger number of genes.

The space is just too big.

As such, we can’t consider an arbitrary mapping from genotypes to fitness. Instead, we need to consider compact representations.

Ever since Julian Z. Xue first introduced me to it, my favorite compact representation has probably been the NK-model of fitness landscapes. In this post, I will rehearse the definition of what I’d call the classic NK-model. But I’ll then consider how the model would have been defined if it was originally proposed by a mathematician or computer scientists. I’ll call this the generalized NK-model and argue that it isn’t only mathematically more natural but also biologically more sensible.

## Quick introduction: Evolutionary game assay in Python

It’s been a while since I’ve shared or discussed code on TheEGG. So to avoid always being too vague and theoretical, I want to use this post to explain how one would write some Python code to measure evolutionary games. This will be an annotated sketch of the game assay from our recent work on measuring evolutionary games in non-small cell lung cancer (Kaznatcheev et al., 2019).

The motivation for this post came about a month ago when Nathan Farrokhian was asking for some advice on how to repeat our game assay with a new experimental system. He has since done so (I think) by measuring the game between Gefitinib-sensitive and Gefitinib-resistant cell types. And I thought it would make a nice post in the quick introductions series.

Of course, the details of the system don’t matter. As long as you have an array of growth rates (call them yR and yG with corresponding errors yR_e and yG_e) and initial proportions of cell types (call them xR and xG) then you could repeat the assay. To see how to get to this array from more primitive measurements, see my old post on population dynamics from time-lapse microscopy. It also has Python code for your enjoyment.

In this post, I’ll go through the two final steps of the game assay. First, I’ll show how to fit and visualize fitness functions (Figure 3 in Kaznatcheev et al., 2019). Second, I’ll transform those fitness functions into game points and plot (Figure 4b in Kaznatcheev et al., 2019). I’ll save discussions of the non-linear game assay (see Appendix F in Kaznatcheev et al., 2019) for a future post.

## Abstracting evolutionary games in cancer

As you can tell from browsing the mathematical oncology posts on TheEGG, somatic evolution is now recognized as a central force in the initiation, progression, treatment, and management of cancer. This has opened a new front in the proverbial war on cancer: focusing on the ecology and evolutionary biology of cancer. On this new front, we are starting to deploy new kinds of mathematical machinery like fitness landscapes and evolutionary games.

Recently, together with Peter Jeavons, I wrote a couple of thousand words on this new machinery for Russell Rockne’s upcoming mathematical oncology roadmap. Our central argument being — to continue the war metaphor — that with new machinery, we need new tactics.

Biologist often aim for reductive explanations, and mathematical modelers have tended to encourage this tactic by searching for mechanistic models. This is important work. But we also need to consider other tactics. Most notable, we need to look at the role that abstraction — both theoretical and empirical abstraction — can play in modeling and thinking about cancer.

The easiest way to share my vision for how we should approach this new tactic would be to throw a preprint up on BioRxiv or to wait for Rockne’s road map to eventually see print. Unfortunately, BioRxiv has a policy against views-like articles — as I was surprised to discover. And I am too impatient to wait for the eventual giant roadmap article.

Hence, I want to share some central parts in this blog post. This is basically an edited and slightly more focused version of our roadmap. Since, so far, game theory models have had more direct impact in oncology than fitness landscapes, I’ve focused this post exclusively on games.