## Measuring games in the Petri dish

For the next couple of months, Jeffrey Peacock is visiting Moffitt. He’s a 4th year medical student at the University of Central Florida with a background in microbiology and genetic engineering of bacteria and yeast. Together with Andriy Marusyk and Jacob Scott, he will move to human cells and run some in vitro experiments with non-small cell lung cancer — you can read more about this on Connecting the Dots. Robert Vander Velde is also in the process of designing some experiments of his own. Both Jeff and Robert are interested in evolutionary game theory, so this is great opportunity for me to put my ideas on operationalization of replicator dynamics into practice.

In this post, I want to outline the basic process for measuring a game from in vitro experiments. Games in the Petri-dish. It won’t be as action packed as Agar.io — that’s an actual MMO cells-in-Petri-dish game; play here — but hopefully it will be more grounded in reality. I will introduce the gain function, show how to measure it, and stress the importance of quantifying the error on this measurement. Since this is part of the theoretical preliminaries for my collaborations, we don’t have our own data to share yet, so I will provide an illustrative cartoon with data from Archetti et al. (2015). Finally, I will show what sort of data would rule-out the theoretician’s favourite matrix games and discuss the ego-centric representation of two-strategy matrix games. The hope is that we can use this work to go from heuristic guesses at what sort of games microbes or cancer cells might play to actually measuring those games.

## Abusing numbers and the importance of type checking

What would you say if I told you that I could count to infinity on my hands? Infinity is large, and I have a typical number of fingers. Surely, I must be joking. Well, let me guide you through my process. Since you can’t see me right now, you will have to imagine my hands. When I hold out the thumb on my left hand, that’s one, and when I hold up the thumb and the index finger, that’s two. Actually, we should be more rigorous, since you are imagining my fingers, it actually isn’t one and two, but i and 2i. This is why they call them imaginary numbers.

Let’s continue the process of extending my (imaginary) fingers from the leftmost digits towards the right. When I hold out my whole left hand and the pinky, ring, and middle fingers on my right hand, I have reached 8i.

But this doesn’t look like what I promised. For the final step, we need to remember the geometric interpretation of complex numbers. Multiplying by i is the same thing as rotating counter-clockwise by 90 degrees in the plane. So, let’s rotate our number by 90 degrees and arrive at $\infty$.

I just counted to infinity on my hands.

Of course, I can’t stop at a joke. I need to overanalyze it. There is something for scientists to learn from the error that makes this joke. The disregard for the type of objects and jumping between two different — and usually incompatible — ways of interpreting the same symbol is something that scientists, both modelers and experimentalists, have to worry about it.

If you want an actually funny joke of this type then I recommend the image of a ‘rigorous proof’ above that was tweeted by Moshe Vardi. My writen version was inspired by a variant on this theme mentioned on Reddit by jagr2808.

I will focus this post on the use of types from my experience with stoichiometry in physics. Units in physics allow us to perform sanity checks after long derivations, imagine idealized experiments, and can even suggest refinements of theory. These are all features that evolutionary game theory, and mathematical biology more broadly, could benefit from. And something to keep in mind as clinicians, biologists, and modelers join forces this week during the 5th annual IMO Workshop at the Moffitt Cancer Center.

## Operationalizing the local environment for replicator dynamics

Recently, Jake Taylor-King arrived in Tampa and last week we were brainstorming some projects to work on together. In the process, I dug up an old idea I’ve been playing with as my understanding of the Ohtsuki-Nowak transform matured. The basic goal is to work towards an operational account of spatial structure without having to commit ourselves to a specific model of space. I will take replicator dynamics and work backwards from them, making sure that each term we use can be directly measured in a single system or abducted from the other measurements. The hope is that if we start making such measurements then we might see some empirical regularities which will allow us to link experimental and theoretical models more closely without having to make too many arbitrary assumptions. In this post, I will sketch the basic framework and then give an example of how some of the spatial features can be measured from a sample histology.

## Operationalizing replicator dynamics and partitioning fitness functions

As you know, dear regular reader, I have a rather uneasy relationship with reductionism, especially when doing mathematical modeling in biology. In mathematical oncology, for example, it seems that there is a hope that through our models we can bring a more rigorous mechanistic understanding of cancer, but at the same time there is the joke that given almost any microscopic mechanism there is an experimental paper in the oncology literature supporting it and another to contradict it. With such a tenuous and shaky web of beliefs justifying (or just hinting towards) our nearly arbitrary microdynamical assumptions, it seems unreasonable to ground our models in reductionist stories. At such a time of ontological crisis, I have an instinct to turn — much like many physicists did during a similar crisis at the start of the 20th century in their discipline — to operationalism. Let us build a convincing mathematical theory of cancer in the petri dish with as few considerations of things we can’t reliably measure and then see where to go from there. To give another analogy to physics in the late 1800s, let us work towards a thermodynamics of cancer and worry about its many possible statistical mechanics later.

This is especially important in applications of evolutionary game theory where assumptions abound. These assumptions aren’t just about modeling details like the treatments of space and stochasticity or approximations to them but about if there is even a game taking place or what would constitute a game-like interaction. However, to work toward an operationalist theory of games, we need experiments that beg for EGT explanations. There is a recent history of these sort of experiments in viruses and microbes (Lenski & Velicer, 2001; Crespi, 2001; Velicer, 2003; West et al., 2007; Ribeck & Lenski, 2014), slime molds (Strassmann & Queller, 2011) and yeast (Gore et al., 2009; Sanchez & Gore, 2013), but the start of these experiments in oncology by Archetti et al. (2015) is current events[1]. In the weeks since that paper, I’ve had a very useful reading group and fruitful discussions with Robert Vander Velde and Julian Xue about the experimental aspects of this work. This Monday, I spent most of the afternoon discussing similar experiments with Robert Noble who is visiting Moffitt from Montpellier this week.

In this post, I want to unlock some of this discussion from the confines of private emails and coffee chats. In particular, I will share my theorist’s cartoon understanding of the experiments in Archetti et al. (2015) and how they can help us build an operationalist approach to EGT but how they are not (yet) sufficient to demonstrate the authors’ central claim that neuroendocrine pancreatic cancer dynamics involve a public good.

## Evolutionary game theory without interactions

When I am working on evolutionary game theory, I usually treat the models I build as heuristics to guide intuitions and push the imagination. But working on something as practical as cancer, and being in a department with many physics-trained colleagues puts pressure on me to think of moving more towards insilications or abductions. Now, Philip Gerlee and Philipp Altrock are even pushing me in that direction with their post on TheEGG. So this entry might seem a bit uncharacteristic, I will describe an experiment — at least as a theorist like me imagines them.

Consider the following idealized protocol that is loosely inspired by Archetti et al. (2015) and the E. coli Long-term evolution experiment (Lenski et al., 1991; Wiser et al., 2013; Ribeck & Lenski, 2014). We will (E1) take a new petri dish or plate; (E2) fill it with a fixed mix of nutritional medium like fetal bovine serum; (E3) put a known number N of two different cell types A and B on the medium (on the first plate we will also know the proportion of A and B in the mixture); (E4) let them grow for a fixed amount of time T which will be on the order of a cell cycle (or two); (E5) scrape the cells off the medium; and (E6) return to step (E1) while selecting N cells at random from the ones we got in step (E5) to seed step (E3). Usually, you would use this procedure to see how A-cells and B-cells compete with each other, as Archetti et al. (2015). However, what would it look like if the cells don’t compete with each other? What if they produce no signalling molecules — in fact, if they excrete nothing into the environment, to avoid cross-feeding interactions — and don’t touch each other? What if they just sit there independently eating their very plentiful nutrient broth?[1]

Would you expect to see evolutionary game dynamics between A and B? Obviously, since I am asking, I expect some people to answer ‘no’ and then be surprised when I derive some math to show that the answer can be ‘yes’. So, dear reader, humour me by being surprised.

## Memes, compound strategies, and factoring the replicator equation

When you work with evolutionary game theory for a while, you end up accumulating an arsenal of cute tools and tricks. A lot of them are obvious once you’ve seen them, but you usually wouldn’t bother looking for them if you hadn’t know they existed. In particular, you become very good friends with the replicator equation. A trick that I find useful at times — and that has come up recently in my on-going project with Robert Vander Veldge, David Basanta, and Jacob Scott — is nesting replicator dynamics (or the dual notion of factoring the replicator equation). I wanted to share a relatively general version of this trick with you, and provide an interpretation of it that is of interest to people — like me — who care about the interaction of evolution in learning. In particular, we will consider a world of evolving agents where each agent is complex enough to learn through reinforcement and pass its knowledge to its offspring. We will see that in this setting, the dynamics of the basic ideas — or memes — that the agents consider can be studied in a world of selfish memes independent of the agents that host them.

## Approximating spatial structure with the Ohtsuki-Nowak transform

Can we describe reality? As a general philosophical question, I could spend all day discussing it and never arrive at a reasonable answer. However, if we restrict to the sort of models used in theoretical biology, especially to the heuristic models that dominate the field, then I think it is relatively reasonable to conclude that no, we cannot describe reality. We have to admit our current limits and rely on thinking of our errors in the dual notions of assumptions or approximations. I usually prefer the former and try to describe models in terms of the assumptions that if met would make them perfect (or at least good) descriptions. This view has seemed clearer and more elegant than vague talk of approximations. It is the language I used to describe the Ohtsuki-Nowak (2006) transform over a year ago. In the months since, however, I’ve started to realize that the assumptions-view is actually incompatible with much of my philosophy of modeling. To contrast my previous exposition (and to help me write up some reviewer responses), I want to go through a justification of the ON-transform as a first-order approximation of spatial structure.

## Evolution as a risk-averse investor

I don’t know about you, but most of my money is in my savings account and not in more volatile assets like property, bonds, or stocks. This is a consequence of either laziness to explore my options, or — the more comforting alternative — extreme risk-aversion. Although it would be nice to have a few thousand dollars more to my name, it would be devastating to have a few thousand dollars less. As such if I was given a lottery where I had a 50% chance of loosing $990 or a 50% chance of winning$1000 then I would probably choose not to play, even though there is an expected gain of $10; I am risk averse, the extra variance of the bet versus the certainty of maintaining my current holdings is not worth$10 for me. I most cases, so are most investors, although the degree of expected profit to variance trade-off differs between agents.

Daniel Bernoulli (8 February 1700 – 17 March 1782) was one of the mathematicians in the famous Bernoulli family of Basal, Switzerland, and contemporary and friend of Euler and Goldbach. He is probably most famous for Bernoulli’s principle in hydrodynamics that his hyper-competitive father Johann publishing in a book he pre-dated by ten years to try and claim credit. One of Daniel’s most productive times was working alongside Euler and Goldbach in the golden days (1724-1732) of the St. Petersburg Academy. It was in Russia that he developed his solution to the St. Petersburg paradox by introducing risk-aversion, and made his contribution to probability, finance, and — as we will see — evolution.

## Black swans and Orr-Gillespie theory of evolutionary adaptation

The internet loves fat tails, it is why awesome things like wikipedia, reddit, and countless kinds of StackExchanges exist. Finance — on the other hand — hates fat tails, it is why VaR and financial crises exist. A notable exception is Nassim Taleb who became financially independent by hedging against the 1987 financial crisis, and made a multi-million dollar fortune on the recent crisis; to most he is known for his 2007 best-selling book The Black Swan. Taleb’s success has stemmed from his focus on highly unlikely events, or samples drawn from far on the tail of a distribution. When such rare samples have a large effect then we have a Black Swan event. These are obviously important in finance, but Taleb also stresses its importance to the progress of science, and here I will sketch a connection to the progress of evolution.

## Epistasis and empirical fitness landscapes

Biologists tend to focus on nuances — to the point that Rutherford considered the field as stamp-collecting — and very local properties of systems, leading at times to rather reductionist views. These approaches are useful for connecting to experiment, but can be shown to underspecify conceptual models that need a more holistic approach. In the case of fitness landscapes, the metric that biologists study is epistasis — the amount of between locus interactions — and is usually considered for the interaction of just two loci at a time; although Beerenwinkel et al. (2007a,b) have recently introduced a geometric theory of gene interaction for considering epistasis across any number of loci. In contrast, more holistic measures can be as simple as the number of peaks in the landscape, or the computational or as complicated as the global combinatorial features of interest to theoretical computer scientists. In this post I discuss connections between the two and provide a brief overview of the empirical work on fitness landscapes.

Epistasis in fitness graphs of two loci. Arrows point from lower fitness to higher fitness, and AB always has higher fitness than ab. From left to right no epistasis, sign epistasis, reverse sign epistasis.