Constant-sum games as a way from non-cell autonomous processes to constant tumour growth rate

A lot of thinking in cancer biology seems to be focused on cell-autonomous processes. This is the (overly) reductive view that key properties of cells, such as fitness, are intrinsic to the cells themselves and not a function of their interaction with other cells in the tumour. As far as starting points go, this is reasonable. But in many cases, we can start to go beyond this cell-autonomous starting point and consider non-cell-autonomous processes. This is when the key properties of a cell are not a function of just that cell but also its interaction partners. As an evolutionary game theorist, I am clearly partial to this view.

Recently, I was reading yet another preprint that has observed non-cell autonomous fitness in tumours. In this case, Johnson et al. (2019) spotted the Allee effect in the growth kinetics of cancer cells even at extremely low densities (seeding in vitro at <200 cells in a 1 mm^3 well). This is an interesting paper, and although not explicitly game-theoretic in its approach, I think it is worth reading for evolutionary game theorists.

Johnson et al.'s (2019) approach is not explicitly game-theoretic because they consider their in vitro populations as a monomorphic clonal line, and thus don't model interactions between types. Instead, they attribute non-cell autonomous processes to density dependence of the single type on itself. In this setting, they reasonably define the cell-autonomous null-model as constant exponential growth, i.e. \dot{N}_T = w_TN_T for some constant fitness w_T and total tumour size N_T.

It might also be tempting to use the same model to capture cell-autonomous growth in game-theoretic models. But this would be mistaken. For this is only effectively cell-autonomous at the level of the whole tumour, but could hide non-cell-autonomous fitness at the level of the different types that make up the tumour. This apparent cell-autonomous total growth will happen whenever the type interactions are described by constant-sum games.

Given the importance of constant-sum games (more famously known as zero-sum games) to the classical game theory literature, I thought that I would write a quick introductory post about this correspondence between non-cell autonomous constant-sum games and effectively cell-autonomous growth at the level of the whole tumour.

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Effective games from spatial structure

For the last week, I’ve been at the Institute Mittag-Leffler of the Royal Swedish Academy of Sciences for their program on mathematical biology. The institute is a series of apartments and a grand mathematical library located in the suburbs of Stockholm. And the program is a mostly unstructured atmosphere — with only about 4 hours of seminars over the whole week — aimed to bring like-minded researchers together. It has been a great opportunity to reconnect with old colleagues and meet some new ones.

During my time here, I’ve been thinking a lot about effective games and the effects of spatial structure. Discussions with Philip Gerlee were particularly helpful to reinvigorate my interest in this. As part of my reflection, I revisited the Ohtsuki-Nowak (2006) transform and wanted to use this post to share a cute observation about how space can create an effective game where there is no reductive game.

Suppose you were using our recent game assay to measure an effective game, and you got the above left graph for the fitness functions of your two types. On the x-axis, you have seeding proportion of type C and on the y-axis you have fitness. In cyan you have the measured fitness function for type C and in magenta, you have the fitness function for type D. The particular fitnesses scale of the y-axis is not super important, not even the x-intercept — I’ve chosen them purely for convenience. The only important aspect is that the cyan and magenta lines are parallel, with a positive slope, and the magenta above the cyan.

This is not a crazy result to get, compare it to the fitness functions for the Alectinib + CAF condition measured in Kaznatcheev et al. (2018) which is shown at right. There, cyan is parental and magenta is resistant. The two lines of best fit aren’t parallel, but they aren’t that far off.

How would you interpret this sort of graph? Is there a game-like interaction happening there?

Of course, this is a trick question that I give away by the title and set-up. The answer will depend on if you’re asking about effective or reductive games, and what you know about the population structure. And this is the cute observation that I want to highlight.

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Abstract is not the opposite of empirical: case of the game assay

Last week, Jacob Scott was at a meeting to celebrate the establishment of the Center for Evolutionary Therapy at Moffitt, and he presented our work on measuring the effective games that non-small cell lung cancer plays (see this preprint for the latest draft). From the audience, David Basanta summarized it in a tweet as “trying to make our game theory models less abstract”. But I actually saw our work as doing the opposite (and so quickly disagreed).

However, I could understand the way David was using ‘abstract’. I think I’ve often used it in this colloquial sense as well. And in that sense it is often the opposite of empirical, which is seen as colloquially ‘concrete’. Given my arrogance, I — of course — assume that my current conception of ‘abstract’ is the correct one, and the colloquial sense is wrong. To test myself: in this post, I will attempt to define both what ‘abstract’ means and how it is used colloquially. As a case study, I will use the game assay that David and I disagreed about.

This is a particularly useful exercise for me because it lets me make better sense of how two very different-seeming aspects of my work — the theoretical versus the empirical — are both abstractions. It also lets me think about when simple models are abstract and when they’re ‘just’ toys.

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Token vs type fitness and abstraction in evolutionary biology

There are only twenty-six letters in the English alphabet, and yet there are more than twenty-six letters in this sentence. How do we make sense of this?

Ever since I first started collaborating with David Basanta and Jacob Scott back in 2012/13, a certain tension about evolutionary games has been gnawing at me. A feeling that a couple of different concepts are being swept up under the rug of a single name.[1] This feeling became stronger during my time at Moffitt, especially as I pushed for operationalizing evolutionary games. The measured games that I was imagining were simply not the same sort of thing as the games implemented in agent-based models. Finally this past November, as we were actually measuring the games that cancer plays, a way to make the tension clear finally crystallized for me: the difference between reductive and effective games could be linked to two different conceptions of fitness.

This showed a new door for me: philosophers of biology have already done extensive conceptual analysis of different versions of fitness. Unfortunately, due to various time pressures, I could only peak through the keyhole before rushing out my first draft on the two conceptions of evolutionary games. In particular, I didn’t connect directly to the philosophy literature and just named the underlying views of fitness after the names I’ve been giving to the games: reductive fitness and effective fitness.

Now, after a third of a year busy teaching and revising other work, I finally had a chance to open that door and read some of the philosophy literature. This has provided me with a better vocabulary and clearer categorization of fitness concepts. Instead of defining reductive vs effective fitness, the distinction I was looking for is between token fitness and type fitness. And in this post, I want to discuss that distinction. I will synthesize some of the existing work in a way that is relevant to separating reductive vs. effective games. In the process, I will highlight some missing points in the current debates. I suspect this points have been overlooked because most of the philosophers of biology are focused more on macroscopic organisms instead of the microscopic systems that motivated me.[2]

Say what you will of birds and ornithology, but I am finding reading philosophy of biology to be extremely useful for doing ‘actual’ biology. I hope that you will, too.

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Replicator dynamics and the simplex as a vector space

Over the years of TheEGG, I’ve chronicled a number of nice properties of the replicator equation and its wide range of applications. From a theoretical perspective, I showed how the differential version can serve as the generator for the action that is the finite difference version of replicator dynamics. And how measurements of replicator dynamics can correspond to log-odds. From an application perspective, I talked about how replicator dynamics can be realized in many different ways. This includes a correspondance to idealized replating experiments and a representation of populations growing toward carrying capacity via fictitious free-space strategies. These fictitious strategies are made apparent by using a trick to factor and nest the replicator dynamics. The same trick can also help us to use the symmetries of the fitness functions for dimensionality reduction and to prove closed orbits in the dynamics. And, of course, I discussed countless heuristic models and some abductions that use replicator dynamics.

But whenever some object becomes so familiar and easy to handle, I get worried that I am missing out on some more foundational and simple structure underlying it. In the case of replicator dynamics, Tom Leinster’s post last year on the n-Category Cafe pointed me to the simple structure that I was missing: the vector space structure of the simplex. This allows us to use linear algebra — the friendliest tool in the mathematician’s toolbox — in a new way to better understand evolutionary dynamics.

A 2-simplex with some of its 1-dimensional linear subspaces drawn by Greg Egan.

Given my interest in operationalization of replicator dynamics, I will use some of the terminology and order of presentation from Aitchison’s (1986) statistical analysis of compositional data. We will see that a number of operations that we define will have clear experimental and evolutionary interpretations.

I can’t draw any real conclusions from this, but I found it worth jotting down for later reference. If you can think of a way to make these observations useful then please let me know.

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Ontology of player & evolutionary game in reductive vs effective theory

In my views of game theory, I largely follow Ariel Rubinstein: game theory is a set of fables. A collection of heuristic models that helps us structure how we make sense of and communicate about the world. Evolutionary game theory was born of classic game theory theory through a series of analogies. These analogies are either generalizations or restrictions of the theory depending on if you’re thinking about the stories or the mathematics. Given this heuristic genealogy of the field — and my enjoyment of heuristic models — I usually do not worry too much about what exactly certain ontic terms like strategy, player, or game really mean or refer to. I am usually happy to leave these terms ambiguous so that they can motivate different readers to have different interpretations and subsequently push for different models of different experiments. I think it is essential for heuristic theories to foster this diverse creativity. Anything goes.

However, not everyone agrees with Ariel Rubinstein and me; some people think that EGT isn’t “just” heuristics. In fact, more recently, I have also shifted some of my uses of EGT from heuristics to abductions. When this happens, it is no longer acceptable for researchers to be willy-nilly with fundamental objects of the theory: strategies, players, and games.

The biggest culprit is the player. In particular, a lot of confusion stems from saying that “cells are players”. In this post, I’d like to explore two of the possible positions on what constitutes players and evolutionary games.

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Spatializing the Go-vs-Grow game with the Ohtsuki-Nowak transform

Recently, I’ve been thinking a lot about small projects to get students started with evolutionary game theory. One idea that came to mind is to look at games that have been analyzed in the inviscid regime then ‘spatialize’ them and reanalyze them. This is usually not difficult to do and provides some motivation to solving for and making sense of the dynamic regimes of a game. And it is not always pointless, for example, our edge effects paper (Kaznatcheev et al, 2015) is mostly just a spatialization of Basanta et al.’s (2008a) Go-vs-Grow game together with some discussion.

Technically, TheEGG together with that paper have everything that one would need to learn this spatializing technique. However, I realized that my earlier posts on spatializing with the Ohtsuki-Nowak transform might a bit too abstract and the paper a bit too terse for a student who just started with EGT. As such, in this post, I want to go more slowly through a concrete example of spatializing an evolutionary game. Hopefully, it will be useful to students. If you are a beginner to EGT that is reading this post, and something doesn’t make sense then please ask for clarification in the comments.

I’ll use the Go-vs-Grow game as the example. I will focus on the mathematics, and if you want to read about the biological or oncological significance then I encourage you to read Kaznatcheev et al. (2015) in full.
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Cataloging a year of cancer blogging: double goods, measuring games & resistance

Happy year of the Rooster and 2017,

This month marks the start of the 7th calendar year of updates on TheEGG. Time to celebrate and summarize the posts of the year past. In 2016 there was the same number of posts as 2015, but instead of being clustered in a period of <7 months, they were more uniformly distributed across the calendar. Every month had at least one new post, although not necessarily written by me (in the case of the single post by Abel Molina in October). There were 29 entries, one linkdex cataloging 2015, and two updates on EGT reading group 51 – 55 & 56 – 60.

In September, as part of my relocation from Tampa to Oxford, I attended the 4th Heidelberg Laureate Forum. I wrote two pieces for their blog: Alan Turing and science through the algorithmic lens and a spotlight on Jan Poleszczuk: from HLF2013 to mathematical oncology. You can read those (and more posts coming this year) on their blog. I won’t go into more detail here.

As before, this post is meant to serve as an organizing reference and a way to uncover common themes on TheEGG. A list of TL;DRs from 2016. The year was split up into four major categories: cancer, complexity & evolution, other models, and philosophy. The cancer posts make up almost half the articles from last year, and are further subdivided into three subsections: double goods game, experimental game theory, and therapy resistance. I want to focus on these cancer posts for this linkdex, and the other three categories in the next installment.

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Chemical games and the origin of life from prebiotic RNA

From bacteria to vertebrates, life — as we know it today — relies on complex molecular interactions, the intricacies of which science has not fully untangled. But for all its complexity, life always requires two essential abilities. Organisms need to preserve their genetic information and reproduce.

In our own cells, these tasks are assigned to specialized molecules. DNA, of course, is the memory store. The information it encodes is expressed into proteins via messenger RNAs.Transcription (the synthesis of mRNAs from DNA) and translation (the synthesis of proteins from mRNAs) are catalyzed by polymerases necessary to speed up the chemical reactions.

It is unlikely that life started that way, with such a refined division of labor. A popular theory for the origin of life, known as the RNA world, posits that life emerged from just one type of molecule: RNAs. Because RNA is made up of base-complementary nucleotides, it can be used as a template for its own reproduction, just like DNA. Since the 1980s, we also know that RNA can act as a self-catalyst. These two superpowers – information storage and self-catalysis – make it a good candidate for the title of the first spark of life on earth.

The RNA-world theory has yet to meet with empirical evidence, but laboratory experiments have shown that self-preserving and self-reproducing RNA systems can be created in vitro. Little is known, however, about the dynamics that governed pre- and early life. In a recent paper, Yeates et al. (2016) attempt to shed light on this problem by (1) examining how small sets of different RNA sequences can compete for survival and reproduction in the lab and (2) offering a game-theoretical interpretation of the results.

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Drug holidays and losing resistance with replicator dynamics

A couple of weeks ago, before we all left Tampa, Pranav Warman, David Basanta and I frantically worked on refinements of our model of prostate cancer in the bone. One of the things that David and Pranav hoped to see from the model was conditions under which adaptive therapy (or just treatment interrupted with non-treatment holidays) performs better than solid blocks of treatment. As we struggled to find parameters that might achieve this result, my frustration drove me to embrace the advice of George Pólya: “If you can’t solve a problem, then there is an easier problem you can solve: find it.”

IMO6 LogoIn this case, I opted to remove all mentions of the bone and cancer. Instead, I asked a simpler but more abstract question: what qualitative features must a minimal model of the evolution of resistance have in order for drug holidays to be superior to a single treatment block? In this post, I want to set up this question precisely, show why drug holidays are difficult in evolutionary models, and propose a feature that makes drug holidays viable. If you find this topic exciting then you should consider registering for the 6th annual Integrated Mathematical Oncology workshop at the Moffitt Cancer Center.[1] This year’s theme is drug resistance.
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