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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Chemical games and the origin of life from prebiotic RNA

From bacteria to vertebrates, life — as we know it today — relies on complex molecular interactions, the intricacies of which science has not fully untangled. But for all its complexity, life always requires two essential abilities. Organisms need to preserve their genetic information and reproduce.

In our own cells, these tasks are assigned to specialized molecules. DNA, of course, is the memory store. The information it encodes is expressed into proteins via messenger RNAs.Transcription (the synthesis of mRNAs from DNA) and translation (the synthesis of proteins from mRNAs) are catalyzed by polymerases necessary to speed up the chemical reactions.

It is unlikely that life started that way, with such a refined division of labor. A popular theory for the origin of life, known as the RNA world, posits that life emerged from just one type of molecule: RNAs. Because RNA is made up of base-complementary nucleotides, it can be used as a template for its own reproduction, just like DNA. Since the 1980s, we also know that RNA can act as a self-catalyst. These two superpowers – information storage and self-catalysis – make it a good candidate for the title of the first spark of life on earth.

The RNA-world theory has yet to meet with empirical evidence, but laboratory experiments have shown that self-preserving and self-reproducing RNA systems can be created in vitro. Little is known, however, about the dynamics that governed pre- and early life. In a recent paper, Yeates et al. (2016) attempt to shed light on this problem by (1) examining how small sets of different RNA sequences can compete for survival and reproduction in the lab and (2) offering a game-theoretical interpretation of the results.

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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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Counting cancer cells with computer vision for time-lapse microscopy

Competing cellsSome people characterize TheEGG as a computer science blog. And although (theoretical) computer science almost always informs my thought, I feel like it has been a while since I have directly dealt with the programming aspects of computer science here. Today, I want to remedy that. In the process, I will share some Python code and discuss some new empirical data collected by Jeff Peacock and Andriy Marusyk.[1]

Together with David Basanta and Jacob Scott, the five of us are looking at the in vitro dynamics of resistance to Alectinib in non-small cell lung cancer. Alectinib is a new ALK-inhibitor developed by the Chugai Pharmaceutical Co. that was approved for clinical use in Japan in 2014, and in the USA at the end of 2015. Currently, it is intended for tough lung cancer cases that have failed to respond to crizotinib. Although we are primarily interested in how alectinib resistance develops and unfolds, we realize the importance of the tumour’s microenvironment, so one of our first goals — and the focus here — is to see how the Alectinib sensitive cancer cells interact with healthy fibroblasts. Since I’ve been wanting to learn basic computer vision skills and refresh my long lapsed Python knowledge, I decided to hack together some cell counting algorithms to analyze our microscopy data.[2]

In this post, I want to discuss some of our preliminary work although due to length constraints there won’t be any results of interest to clinical oncologist in this entry. Instead, I will introduce automated microscopy to computer science readers, so that they know another domain where their programming skills can come in useful; and discuss some basic computer vision so that non-computational biologists know how (some of) their cell counters (might) work on the inside. Thus, the post will be methods heavy and part tutorial, part background, with a tiny sprinkle of experimental images.[3] I am also eager for some feedback and tips from readers that are more familiar than I am with these methods. So, dear reader, leave your insights in the comments.

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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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Hadza hunter-gatherers, social networks, and models of cooperation

At the heart of the Great Lakes region of East Africa is Tanzania — a republic comprised of 30 mikoa, or provinces. Its border is marked off by the giant lakes Victoria, Tanganyika, and Malawi. But the lake that interests me the most is an internal one: 200 km from the border with Kenya at the junction of mikao Arusha, Manyara, Simiyu and Singed is Lake Eyasi. It is a temperamental lake that can dry up almost entirely — becoming crossable on foot — in some years and in others — like the El Nino years — flood its banks enough to attract hippos from the Serengeti.

For the Hadza, it is home.

The Hadza number around a thousand people, with around 300 living as traditional nomadic hunter-gatherers (Marlow, 2002; 2010). A life style that is believed to be a useful model of societies in our own evolutionary heritage. An empirical model of particular interest for the evolution of cooperation. But a model that requires much more effort to explore than running a few parameter settings on your computer. In the summer of 2010, Coren Apicella explored this model by traveling between Hadza camps throughout the Lake Eyasi region to gain insights into their social network and cooperative behavior.

Here is a video abstract where Coren describes her work:

The data she collected with her colleagues (Apicella et al., 2012) provides our best proxy for the social organization of early humans. In this post, I want to talk about the Hadza, the data set of their social network, and how it can inform other models of cooperation. In other words, I want to freeride on Apicella et al. (2012) and allow myself and other theorists to explore computational models informed by the empirical Hadza model without having to hike around Lake Eyasi for ourselves.

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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 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.
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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.
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