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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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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Multiplicative versus additive fitness and the limit of weak selection

Previously, I have discussed the importance of understanding how fitness is defined in a given model. So far, I’ve focused on how mathematically equivalent formulations can have different ontological commitments. In this post, I want to touch briefly on another concern: two different types of mathematical definitions of fitness. In particular, I will discuss additive fitness versus multiplicative fitness.[1] You often see the former in continuous time replicator dynamics and the latter in discrete time models.

In some ways, these versions are equivalent: there is a natural bijection between them through the exponential map or by taking the limit of infinitesimally small time-steps. A special case of more general Lie theory. But in practice, they are used differently in models. Implicitly changing which definition one uses throughout a model — without running back and forth through the isomorphism — can lead to silly mistakes. Thankfully, there is usually a quick fix for this in the limit of weak selection.

I suspect that this post is common knowledge. However, I didn’t have a quick reference to give to Pranav Warman, so I am writing this.
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Evolutionary dynamics of acid and VEGF production in tumours

Today was my presentation day at ECMTB/SMB 2016. I spoke in David Basanta’s mini-symposium on the games that cancer cells play and postered during the poster session. The mini-symposium started with a brief intro from David, and had 25 minute talks from Jacob Scott, myself, Alexander Anderson, and John Nagy. David, Jake, Sandy, and John are some of the top mathematical oncologists and really drew a crowd, so I felt privileged at the opportunity to address that crowd. It was also just fun to see lots of familiar faces in the same place.

A crowded room by the end of Sandy's presentation.

A crowded room by the end of Sandy’s presentation.

My talk was focused on two projects. The first part was the advertised “Evolutionary dynamics of acid and VEGF production in tumours” that I’ve been working on with Robert Vander Velde, Jake, and David. The second part — and my poster later in the day — was the additional “(+ measuring games in non-small cell lung cancer)” based on work with Jeffrey Peacock, Andriy Marusyk, and Jake. You can download my slides here (also the poster), but they are probably hard to make sense of without a presentation. I had intended to have a preprint out on this prior to today, but it will follow next week instead. Since there are already many blog posts about the double goods project on TheEGG, in this post I will organize them into a single annotated linkdex.

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Hamiltonian systems and closed orbits in replicator dynamics of cancer

Last month, I classified the possible dynamic regimes of our model of acidity and vasculature as linear goods in cancer. In one of those dynamic regimes, there is an internal fixed point and I claimed closed orbits around that point. However, I did not justify or illustrate this claim. In this post, I will sketch how to prove that those orbits are indeed closed, and show some examples. In the process, we’ll see how to transform our replicator dynamics into a Hamiltonian system and use standard tricks from classical mechanics to our advantage. As before, my tricks will draw heavily from Hauert et al. (2002) analysis of the optional public good game. Studying this classic paper closely is useful for us because of an analogy that Robert Vander Velde found between the linear version of our double goods model for the Warburg effect and the optional public good game.

The post will mostly be about the mathematics. However, at the end, I will consider an example of how these sort of cyclic dynamics can matter for treatment. In particular, I will consider what happens if we target aerobic glycolysis with a drug like lonidamine but stop the treatment too early.

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Multiple realizability of replicator dynamics

Abstraction is my favorite part of mathematics. I find a certain beauty in seeing structures without their implementations, or structures that are preserved across various implementations. And although it seems possible to reason through analogy without (explicit) abstraction, I would not enjoy being restricted in such a way. In biology and medicine, however, I often find that one can get caught up in the concrete and particular. This makes it harder to remember that certain macro-dynamical properties can be abstracted and made independent of particular micro-dynamical implementations. In this post, I want to focus on a particular pet-peeve of mine: accounts of the replicator equation.

I will start with a brief philosophical detour through multiple realizability, and discuss the popular analogy of temperature. Then I will move on to the phenomenological definition of the replicator equation, and a few realizations. A particular target will be the statement I’ve been hearing too often recently: replicator dynamics are only true for a very large but fixed-size well-mixed population.

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Acidity and vascularization as linear goods in cancer

Last month, Robert Vander Velde discussed a striking similarity between the linear version of our model of two anti-correlated goods and the Hauert et al. (2002) optional public good game. Robert didn’t get a chance to go into the detailed math behind the scenes, so I wanted to do that today. The derivations here will be in the context of mathematical oncology, but will follow the earlier ecological work closely. There is only a small (and generally inconsequential) difference in the mathematics of the double anti-correlated goods and the optional public goods games. Keep your eye out for it, dear reader, and mention it in the comments if you catch it.[1]

In this post, I will remind you of the double goods game for acidity and vascularization, show you how to simplify the resulting fitness functions in the linear case — without using the approximations of the general case — and then classify the possible dynamics. From the classification of dynamics, I will speculate on how to treat the game to take us from one regime to another. In particular, we will see the importance of treating anemia, that buffer therapy can be effective, and not so much for bevacizumab.

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Don’t treat the player, treat the game: buffer therapy and bevacizumab

No matter how much I like modeling for the sake of modeling, or science for the sake of science, working in a hospital adds some constraints. At some point people look over at you measuring games in the Petri dish and ask “why are you doing this?” They expect an answer that involves something that benefits patients. That might mean prevention, early detection, or minimizing side-effects. But in most cases it means treatment: how does your work help us treat cancer? Here, I think, evolutionary game theory — and the Darwinian view of cancer more generally — offers a useful insight in the titular slogan: don’t treat the player, treat the game.

One of the most salient negative features of cancer is the tumour — the abnormal mass of cancer cells. It seems natural to concentrate on getting rid of these cells, or at least reducing their numbers. This is why tumour volume has become a popular surrogate endpoint for clinical trials. This is treating the player. Instead, evolutionary medicine would ask us to find the conditions that caused the system to evolve towards the state of having a large tumour and how we can change those conditions. Evolutionary therapy aims to change the environmental pressures on the tumour, such that the cancerous phenotypes are no longer favoured and are driven to extinction (or kept in check) by Darwinian forces. The goal is to change the game so that cancer proves to be a non-viable strategy.[1]

In this post I want to look at the pairwise game version of my joint work with Robert Vander Velde, David Basanta, and Jacob Scott on the Warburg effect (Warburg, 1956; Gatenby & Gillies, 2004) and acid-mediated tumour invasion (Gatenby, 1995; Gatenby & Gawlinski, 2003). Since in this work we are concerned with the effects of acidity and vascularization on cancer dynamics, I will concentrate on interventions that affect acidity (buffer therapy; for early empirical work, see Robey et al., 2009) or vascularization (angiogenesis inhibitor therapy like bevacizumab).

My goal isn’t to say something new about these therapies, but to use them as illustrations for the importance of changing between qualitatively different dynamic regimes. In particular, I will be dealing with the oncological equivalent of a spherical cow in frictionless vacuum. I have tried to add some caveats in the footnotes, but these could be multiplied indefinitely without reaching an acceptably complete picture.

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Lotka-Volterra, replicator dynamics, and stag hunting bacteria

Happy year of the monkey!

Last time in the Petri dish, I considered the replicator dynamics between type-A and type-B cells abstractly. In the comments, Arne Traulsen pointed me to Li et al. (2015):

We have attempted something similar in spirit with bacteria. Looking at frequencies alone, it looked like coordination. But taking into account growth led to different conclusions […] In that case, things were more subtle than anticipated…

So following their spirit, I will get more concrete in this post and replace type-A by Curvibacter sp. AEP13 and type-B by Duganella sp. C1.2 — two bacteria that help fresh water Hydra avoid fungal infection. And I will also show how to extend our replicator dynamics with growth and changing cell density.

Although I try to follow Arne’s work very closely, I had not read Li et al. (2015) before, so I scheduled it for a reading group this past Friday. I really enjoyed the experiments that they conducted, but I don’t agree with their interpretations that taking growth into account leads to a different conclusion. In this post, I will sketch how they measured their experimental system and then provide a replicator equation representation of the Lotka-Volterra model they use to interpret their results. From this, we’ll be able to conclude that C and D are playing the Stag Hunt — or coordination, or assurance, pick your favorite terminology — game.

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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 — 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.
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