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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Modeling influenza at ECMTB/SMB 2016

This week, I am at the University of Nottingham for the joint meeting of the Society of Mathematical Biology and the European Conference on Mathematical and Theoretical Biology — ECMTB/SMB 2016. It is a huge meeting, with over 800 delegates in attendance, 308 half-hour mini-symposium talks, 264 twenty-minute contributed talks, 190 posters, 7 prize talks, 7 plenary talks, and 1 public lecture. With seventeen to eighteen sessions running in parallel, it is impossible to see more than a tiny fraction of the content. And impossible for me to give you a comprehensive account of the event. However, I did want to share some moments from this week. If you are at ECMTB and want to share some of your highlights for TheEGG then let me know, and we can have you guest post.

I did not come to Nottingham alone. Above is a photo of all the current/recent Moffitteers that made their way to the meeting.

I did not come to Nottingham alone. Above is a photo of current/recent Moffitteers that made their way to the meeting this year.

On the train ride to Nottingham, I needed to hear some success stories of mathematical biology. One of the ones that Dan Nichol volunteered was the SIR-model for controlling the spread of infectious disease. This is a simple system of ODEs with three compartments corresponding to the infection status of individuals in the population: susceptible (S), infectious (I), recovered (R). It is given by the following equations

\begin{aligned}  \dot{S} & = - \beta I S \\  \dot{I} & = \beta I S - \gamma I \\  \dot{R} & = \gamma I,  \end{aligned}

where \beta and \gamma are usually taken to be constants dependent on the pathogen, and the total number of individuals N = S + I + R is an invariant of the dynamics.

As the replicator dynamics are to evolutionary game theory, the SIR-model is to epidemiology. And it was where Julia Gog opened the conference with her plenary on the challenges of modeling infectious disease. In this post, I will briefly touch on her extensions of the SIR-model and how she used it to look at the 2009 swine flu outbreak in the US.
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Computational kindness and the revelation principle

In EWD1300, Edsger W. Dijkstra wrote:

even if you have only 60 readers, it pays to spend an hour if by doing so you can save your average reader a minute.

He wrote this as the justification for the mathematical notations that he introduced and as an ode to the art of definition. But any writer should heed this aphorism.[1] Recently, I finished reading Algorithms to Live By by Brian Christian and Tom Griffiths.[2] In the conclusion of their book, they gave a unifying name to the sentiment that Dijkstra expresses above: computational kindness.

As computer scientists, we recognise that computation is costly. Processing time is a limited resource. Whenever we interact with others, we are sharing in a joint computational process, and we need to be mindful of when we are not carrying our part of the processing burden. Or worse yet, when we are needlessly increasing that burden and imposing it on our interlocutor. If you are computationally kind then you will be respectful of the cognitive problems that you force others to solve.

I think this is a great observation by Christian and Griffiths. In this post, I want to share with you some examples of how certain systems — at the level of the individual, small group, and society — are computationally kind. And how some are cruel. I will draw on examples from their book, and some of my own. They will include, language, bus stops, and the revelation principle in algorithmic game theory.
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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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Systemic change, effective altruism and philanthropy

Keep your coins. I want change.The topics of effective altruism and social (in)justice have weighed heavy on my mind for several years. I’ve even touched on the latter occasionally on TheEGG, but usually in specific domains closer to my expertise, such as in my post on the ethics of big data. Recently, I started reading more thoroughly about effective altruism. I had known about the movement[1] for some time, but had conflicting feelings towards it. My mind is still in disarray on the topic, but I thought I would share an analytic linkdex of some texts that have caught my attention. This is motivated by a hope to get some guidance from you, dear reader. Below are three videos, two articles, two book reviews and one paper alongside my summaries and comments. The methods range from philosophy to comedy and from critical theory to social psychology. I reach no conclusions.

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EGT Reading Group 56 – 60

Since my last update in February, the evolutionary game theory reading group has passed another milestone with 5 more meetings over the last 4 months. We looked at a broad range of topics, from life histories in cancer to the effects of heterogeneity and biodiversity. From the definitions of fitness to analyzing digital pathology. Part of this variety came from suggested papers by the group members. The paper for EGT 57 was suggested by Jill Gallaher, EGT 58 by Robert Vander Velde, and the second paper for EGT 60 came from a tip by Jacob Scott. We haven’t yet recovered our goal of regular weekly meetings, but we’ve more than halved the time it took for these five meetings compared to the previous ones.

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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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Eukaryotes without Mitochondria and Aristotle’s Ladder of Life

In 348/7 BC, fearing anti-Macedonian sentiment or disappointed with the control of Plato’s Academy passing to Speusippus, Aristotle left Athens for Asian Minor across the Aegean sea. Based on his five years[1] studying of the natural history of Lesbos, he wrote the pioneering work of zoology: The History of Animals. In it, he set out to catalog the what of biology before searching for the answers of why. He initiated a tradition of naturalists that continues to this day.

Aristotle classified his observations of the natural world into a hierarchical ladder of life: humans on top, above the other blooded animals, bloodless animals, and plants. Although we’ve excised Aristotle’s insistence on static species, this ladder remains for many. They consider species as more complex than their ancestors, and between the species a presence of a hierarchy of complexity with humans — as always — on top. A common example of this is the rationality fetish that views Bayesian learning as a fixed point of evolution, or ranks species based on intelligence or levels-of-consciousness. This is then coupled with an insistence on progress, and gives them the what to be explained: the arc of evolution is long, but it bends towards complexity.

In the early months of TheEGG, Julian Xue turned to explaining the why behind the evolution of complexity with ideas like irreversible evolution as the steps up the ladder of life.[2] One of Julian’s strongest examples of such an irreversible step up has been the transition from prokaryotes to eukaryotes through the acquisition of membrane-bound organelles like mitochondria. But as an honest and dedicated scholar, Julian is always on the lookout for falsifications of his theories. This morning — with an optimistic “there goes my theory” — he shared the new Kamkowska et al. (2016) paper showing a surprising what to add to our natural history: a eukaryote without mitochondria. An apparent example of a eukaryote stepping down a rung in complexity by losing its membrane-bound ATP powerhouse.
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Population dynamics from time-lapse microscopy

Half a month ago, I introduced you to automated time-lapse microscopy, but I showed the analysis of only a single static image. I didn’t take advantage of the rich time-series that the microscope provides for us. A richness that becomes clearest with video:

Above, you can see two types of non-small cell lung cancer cells growing in the presence of 512 nmol of Alectinib. The cells fluorescing green are parental cells that are susceptible to the drug, and the ones in red have an evolved resistance. In the 3 days of the video, you can see the cells growing and expanding. It is the size of these populations that we want to quantify.

In this post, I will remedy last week’s omission and share some empirical population dynamics. As before, I will include some of the Python code I built for these purposes. This time the code is specific to how our microscope exports its data, and so probably not as generalizable as one might want. But hopefully it will still give you some ideas on how to code analysis for your own experiments, dear reader. As always, the code is on my github.

Although the opening video considers two types of cancer cells competing, for the rest of the post I will consider last week’s system: coculturing Alectinib-sensitive (parental) non-small cell lung cancer and fibroblasts in varying concentrations of Alectinib. Finally, this will be another tools post so the only conclusions are of interest as sanity checks. Next week I will move on to more interesting observations using this sort of pipeline.
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