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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Danger of motivatiogenesis in interdisciplinary work

Randall Munroe has a nice old xkcd on citogenesis: the way factoids get created from bad checking of sources. You can see the comic at right. But let me summarize the process without direct reference to Wikipedia:

1. Somebody makes up a factoid and writes it somewhere without citation.
2. Another person then uses the factoid in passing in a more authoritative work, maybe sighting the point in 1 or not.
3. Further work inherits the citation from 2, without verifying its source, further enhancing the legitimacy of the factoid.
4. The cycle repeats.

Soon, everybody knows this factoid and yet there is no ground truth to back it up. I’m sure we can all think of some popular examples. Social media certainly seems to make this sort of loop easier.

We see this occasionally in science, too. Back in 2012, Daniel Lemire provided a nice example of this with algorithms research. But usually with science factoids, it eventually gets debuked with new experiments or proofs. Mostly because it can be professionally rewarding to show that a commonly assumed factoid is actually false.

But there is a similar effect in science that seems to me even more common, and much harder to correct: motivatiogenesis.

Motivatiogenesis can be especially easy to fall into with interdisiplinary work. Especially if we don’t challenge ourselves to produce work that is an advance in both (and not just one) of the fields we’re bridging.

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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.
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Causes and costs in biological vs clinical resistance

This Wednesday, on These few lines, Rob Noble warned of the two different ways in which the term de novo resistance is used by biologists and clinicians. The biologist sees de novo resistance as new genetic resistance arising after treatment has started. The clinician sees de novo resistance as a tumour that is not responsive to treatment from the start. To make matters even more confusing, Hitesh Mistry points to a further interpretation among pharmocologists: they refer to the tumour remaining after a partial but incomplete response to treatment as de novo resistant. Clearly this is a mess!

But I think this is an informative mess. I don’t think it is a matter of people accidentally overloading the same word. Instead, I think it reflects a conceptual difference in how biologists and clinicians think about resistance. A difference that is a bit akin to the difference between reductive and effective theories. It is also a difference that I had to deal with during the revisions of our recent work on measuring the games played by treatment sensitive and treatment resistance non-small cell lung cancer (Kaznatcheev et al., 2018).

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Methods and morals for mathematical modeling

About a year ago, Vincent Cannataro emailed me asking about any resources that I might have on the philosophy and etiquette of mathematical modeling and inference. As regular readers of TheEGG know, this topic fascinates me. But as I was writing a reply to Vincent, I realized that I don’t have a single post that could serve as an entry point to my musings on the topic. Instead, I ended up sending him an annotated list of eleven links and a couple of book recommendations. As I scrambled for a post for this week, I realized that such an analytic linkdex should exist on TheEGG. So, in case others have interests similar to Vincent and me, I thought that it might be good to put together in one place some of the resources about metamodeling and related philosophy available on this blog.

This is not an exhaustive list, but it might still be relatively exhausting to read.

I’ve expanded slightly past the original 11 links (to 14) to highlight some more recent posts. The free association of the posts is structured slightly, with three sections: (1) classifying mathematical models, (2) pros and cons of computational models, and (3) ethics of models.

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Overcoming folk-physics: the case of projectile motion for Aristotle, John Philoponus, Ibn-Sina & Galileo

A few years ago, I wrote about the importance of pairing tools and problems in science. Not selecting the best tool for the job, but adjusting both your problem and your method to form the best pair. There, I made the distinction between endogenous and exogenous questions. A question is endogenous to a field if it is motivated by the existing tools developed for the field or slight extensions of them. A question is exogenous if motivated by frameworks or concerns external to the field. Usually, such an external motivating framework is accepted uncritically with the most common culprits being the unarticulated ‘intuitive’ and ‘natural’ folk theories forced on us by our everyday experiences.

Sometimes a great amount of scientific or technological progress can be had from overcoming our reliance on a folk-theory. A classic examples of this would be the development of inertia and momentum in physics. In this post, I want to sketch a geneology of this transition to make the notion of endogenous vs exogenous questions a bit more precise.

How was the folk-physics of projectile motion abandoned?

In the process, I’ll get to touch briefly on two more recent threads on TheEGG: The elimination of the ontological division between artificial and natural motion (that was essential groundwork for Darwin’s later elimination of the division between artificial and natural processes) and the extraction and formalization of the tacit knowledge underlying a craft.
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Looking for species in cancer but finding strategies and players

Sometime before 6 August 2014, David Basanta and Tamir Epstein were discussing the increasing focus of mathematical oncology on tumour heterogeneity. An obstacle for this focus is a good definitions of heterogeneity. One path around this obstacle is to take definitions from other fields like ecology — maybe species diversity. But this path is not straightforward: we usually — with some notable and interesting examples — view cancer cells as primarily asexual and the species concept is for sexual organisms. Hence, the specific question that concerned David and Tamir: is there a concept of species that applies to cancer?

I want to consider a couple of candidate answers to this question. None of these answers will be a satisfactory definition for species in cancer. But I think the exercise is useful for understanding evolutionary game theory. With the first attempt to define species, we’ll end up using the game assay to operationalize strategies. With the second attempt, we’ll use the struggle for existence to define players. Both will be sketches that I will need to completely more carefully if there is interest.

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John Maynard Smith on reductive vs effective thinking about evolution

“The logic of animal conflict” — a 1973 paper by Maynard Smith and Price — is usually taken as the starting for evolutionary game theory. And as far as I am an evolutionary game theorists, it influences my thinking. Most recently, this thinking has led me to the conclusion that there are two difference conceptions of evolutionary games possible: reductive vs. effective. However, I don’t think that this would have come as much of a surprise to Maynard Smith and Price. In fact, the two men embodied the two different ways of thinking that underlay my two interpretations.

I was recently reminded of this when Aakash Pandey shared a Web of Stories interview with John Maynard Smith. This is a 4 minute snippet of a long interview with Maynard Smith. In the snippet, he starts with a discussion of the Price equation (or Price’s theorem, if you want to have that debate) but quickly digresses to a discussion of the two kinds of mathematical theories that can be made in science. He identifies himself with the reductive view and Price with the effective. I recommend watching the whole video, although I’ll quote relavent passages below.

In this post, I’ll present Maynard Smith’s distinction on the two types of thinking in evolutionary models. But I will do this in my own terminology to stress the connections to my recent work on evolutionary games. However, I don’t think this distinction is limited to evolutionary game theory. As Maynard Smith suggests in the video, it extends to all of evolutionary biology and maybe scientific modelling more generally.

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Personal case study on the usefulness of philosophy to biology

At the start of this month, one of my favourite blogs — Dynamic Ecology — pointed me to a great interview with Michela Massimi. She has recently won the Royal Society’s Wilkins-Bernal-Medawar Medal for the philosophy of science, and to celebrate Philip Ball interviewed her for Quanta. I recommend reading the whole interview, but for this post, I will focus on just one aspect.

Ball asked Massimi how she defends philosophy of science against dismissive comments by scientists like Feynman or Hawking. In response, she made the very important point that for the philosophy of science to be useful, it doesn’t need to be useful to science:

Dismissive claims by famous physicists that philosophy is either a useless intellectual exercise, or not on a par with physics because of being incapable of progress, seem to start from the false assumption that philosophy has to be of use for scientists or is of no use at all.

But all that matters is that it be of some use. We would not assess the intellectual value of Roman history in terms of how useful it might be to the Romans themselves. The same for archaeology and anthropology. Why should philosophy of science be any different?

Instead, philosophy is useful for humankind more generally. This is certainly true.

But even for a scientist who is only worrying about getting that next grant, or publishing that next flashy paper. For a scientist who is completely detached from the interests of humanity. Even for this scientist, I don’t think we have to concede the point on the usefulness of philosophy of science. Because philosophy, and philosophy of science in particular, doesn’t need to be useful to science. But it often is.

Here I want to give a personal example that I first shared in the comments on Dynamic Ecology.
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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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