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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Symmetry breaking and non-cell-autonomous growth rates in cancer

“You can’t step in the same river twice” might seem like an old aphorism of little value, but I think it is central to making sense of the sciences. This is especially clear if we rephrase it as: “you can’t do the same experiment twice”. After all, a replication experiment takes place at a different time, sometimes a different place, maybe done by a different experimenter. Why should any of the countless rules that governed the initial experiment still hold for the replicate? But our methodology demands that we must be able to repeat experiments. We achieve by making a series of symmetry assumptions. For example: the universality or homogeneity of physical laws. We can see this with early variants of the principle of sufficient reason in Anaximander and Aristotle. It developed closer to the modern statements with Galileo, Copernicus and Newton by pushing the laws of physics outside the sublunary sphere and suggesting that the planets follows the same laws as the apple. In fact, Alfred North Whitehead considered a belief in trustworthy uniformity of physical laws to be the defining feature of western philosophy (and science) since Thales.

In this post, I want to go through some of the symmetries we assume and how to break them. And I want to discuss this at levels from grand cosmology to the petri dish. In the process, I’ll touch on the fundamental constants of physics, how men stress out mice, and how standard experimental practices in cancer biology assume a cell-autonomous process symmetry.

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Deadlock & Leader as deformations of Prisoner’s dilemma & Hawk-Dove games

Recently, I’ve been working on revisions for our paper on measuring the games that cancer plays. One of the concerns raised by the editor is that we don’t spend enough time introducing game theory and in particular the Deadlock and Leader games that we observed. This is in large part due to the fact that these are not the most exciting games and not much theoretic efforts have been spent on them in the past. In fact, none that I know of in mathematical oncology.

With that said, I think it is possible to relate the Deadlock and Leader games to more famous games like Prisoner’s dilemma and the Hawk-Dove games; both that I’ve discussed at length on TheEGG. Given that I am currently at the Lorentz Center in Leiden for a workshop on Understanding Cancer Through Evolutionary Game Theory (follow along on twitter via #cancerEGT), I thought it’d be a good time to give this description here. Maybe it’ll inspire some mathematical oncologists to play with these games.

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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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Dark selection from spatial cytokine signaling networks

Greetings, Theory, Evolution, and Games Group! It’s a pleasure to be on the other side of the keyboard today. Many thanks to Artem for the invite to write about some of our recent work and the opportunity to introduce myself via this post. I do a bit of blogging of my own over at — mostly about neat science I stumble over while figuring out my way.

I’m a biologist. I study the evolutionary dynamics within somatic tissue, or, how mutations occur, compete, accumulate, and persist in our tissues, and how these dynamics manifest as aging and cancer (Cannataro et al., 2017a). I also study the evolutionary dynamics within tumors, and the evolution of resistance to targeted therapy (Cannataro et al., 2017b).

In November 2016 I attended the Integrated Mathematical Oncology Workshop on resistance, a workweek-long intensive competitive workshop where winners receive hard-earned $$ for research, and found myself placed in #teamOrange along with Artem. In my experience at said workshop (attended 2015 and 2016), things usually pan out like this: teams of a dozen or so members are assembled by the workshop organizers, insuring a healthy mix of background-education heterogeneity among groups, and then after the groups decide on a project they devise distinct but intersecting approaches to tackle the problem at hand. I bounced around a bit early on within #teamOrange contributing to our project where I could, and when the need for a spatially explicit model of cytokine diffusion and cell response came up I jumped at the opportunity to lead that endeavor. I had created spatially explicit cellular models before — such as a model of cell replacement in the intestinal crypt (Cannataro et al., 2016) — but never one that incorporated the diffusion or spread of some agent through the space. That seemed like a pretty nifty tool to add to my research kit. Fortunately, computational modeler extraordinaire David Basanta was on our team to teach me about modeling diffusion (thanks David!).

Below is a short overview of the model we devised.

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Ratcheting and the Gillespie algorithm for dark selection

In Artem’s previous post about the IMO workshop he suggests that “[s]ince we are forced to move from the genetic to the epigenetic level of description, it becomes important to suggest a plausible mechanism for heritable epigenetic effects. We need to find a stochastic ratcheted phenotypic switch among the pathways of the CMML cells.” Here I’ll go into more detail about modeling this ratcheting and how to go about identifying the mechanism. We can think of this as a potential implementation of the TYK bypass in the JAK-STAT pathway described experimentally by Koppikar et al. (2012). However, I won’t go into the specifics of exact molecules, keeping to the abstract essence.

After David Robert Grime’s post on oxygen use, this is the third entry in our series on dark selection in chronic myelomonocytic leukemia (CMML). We have posted a preprint (Kaznatcheev et al., 2017) on our project to BioRxiv and section 3.1 therein follows this post closely.

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Identifying therapy targets & evolutionary potentials in ovarian cancer

For those of us attending the 7th annual Integrated Mathematical Oncology workshop (IMO7) at the Moffitt Cancer Center in Tampa, this week was a gruelling yet exciting set of four near-all-nighters. Participants were grouped into five teams and were tasked with coming up with a new model to elucidate a facet of a particular type of cancer. With $50k on the line and enthusiasm for creating evolutionary models, Team Orange (the wonderful team I had the privilege of being a part of) set out to understand something new about ovarian cancer. In this post, I will outline my perspective on the initial model we came up with over the past week.

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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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Oxygen fueling dark selection in the bone marrow

While November 2016 might be remembered for the inauspicious political upset likely to leave future historians as confused as we are, a more positive event transpired in tandem – the 6th Integrated Mathematical Oncology (IMO) Workshop. I was honoured to take part as a member of Team Orange, where we were tasked with investigating the emergence of treatment resistance in chronic myelomonocytic leukemia (CMML).

Unlike many other cancers where the evolution of resistance to treatment is well understood, CMML is something of an enigma as the efficacy of treatment flounders even though the standard treatment doesn’t directly impinge upon tumour cells themselves.  This raises a whole host of questions, and Artem has already eloquently laid out both why this question captivated us, and the combined approach we took to probing it. In this blog post, I’ll focus on exploring one of our mechanistic hypotheses – the potential role of oxygen in treatment resistance.

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Antoni Gaudi and learning algorithms from Nature

Happy holidays.

A few days ago, I was exploring Barcelona. This means that I saw a lot of architecture by Antoni Gaudi. His works have a very distinct style; their fluid lines, bright colours, myriad materials, and interface of design and function make for very naturesque buildings. They are unique and stand in sharp contrast to the other — often Gothic revival and Catalan Modernisme — architecture around them. The contrast is conscious; when starting out, Gaudi learned the patterns of the neo-Gothic architecture then in vogue and later commented on it:

Gothic art is imperfect, only half resolved; it is a style created by the compasses, a formulaic industrial repetition. Its stability depends on constant propping up by the buttresses: it is a defective body held up on crutches. … The proof that Gothic works are of deficient plasticity is that they produce their greatest emotional effect when they are mutilated, covered in ivy and lit by the moon.

His buildings, however, do not need to be overgrown by ivy, for Gaudi already incorporates nature in their design. I felt this connection most viscerally when touring the attic of Casa Mila. The building was commissioned as an apartment for local bourgeois to live comfortably on the ground floor off the rents they collected from the upper floors. And although some of the building is still inhabited by businesses and private residence, large parts of it have been converted into a museum. The most famous part among tourists is probably the uneven organic roof with its intricate smoke stacks, ventilation shafts, and archways for framing other prominent parts of Barcelona.

This uneven roof is supported by an attic that houses an exhibit on Gaudi’s method. Here, I could see Gaudi’s inspiration. On display was a snake’s skeleton and around me were the uneven arches of the attic — the similarity was palpable (see below). The questions for me were: was Gaudi inspired by nature or did he learn from it? Is there even much of a difference between ‘inspired’ and ‘learned’? And can this inform thought on the correspondence between nature and algorithms more generally?


I spend a lot of time writing about how we can use algorithmic thinking to understand aspects of biology. It is much less common for me to write about how we can use biology or nature to understand and inspire algorithms. In fact, I feel surprisingly strong skepticism towards the whole field of natural algorithms, even when I do write about it. I suspect that this stems from my belief that we cannot learn algorithms from nature. A belief that was shaken, but not overturned, when I saw the snake’s skeleton in Gaudi’s attic. In this post, I will try to substantiate the statement that we cannot learn algorithms from nature. My hope is that someone, or maybe just the act of writing, will convince me otherwise. I’ll sketch my own position on algorithms & nature, and strip the opposing we-learn-algorithms-from-nature position of some of its authority by pulling on a historic thread that traces this belief from Plato through Galileo to now. I’ll close with a discussion of some practical consequences of this metaphysical disagreement and try to make sense of Gaudi’s work from my perspective.

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