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.

Just as uniformity can be expanded to new realms, sometimes we attempt testing for breaks in symmetry and violations of uniformity. For example, time-symmetry of physical laws is occasionally questioned and experimenters have attempted careful quantification of the limits on potential drift in the fundamental constants of physics (Fischer et al., 2004). Unsurprisingly, Fisher et al. (2004) showed that if there is recent drift in the constants then it is a very small effect; or as they write: “All these limits are consistent with zero”.

One of the first hurdles to finding violations of symmetry assumptions is noticing that the assumption is being made and then formulating that assumption explicitly. For example, although Anaximander and Aristotle used the principle of sufficient reason to impose homogeneity on their theories, modern philosophers usually attribute the explicit formulation of the principle to Spinoza and Leibniz — about two millennia later. But we don’t need to focus on just grand principles and big cosmology. Although useful to philosophers, some of this thought can seem idle to the working scientist. However, it is important to look out for implicit symmetries in everyday experiments. It is by noticing these small symmetries that we can show violations of them and move science forward. Doing so usually requires to read what isn’t written, to notice things that seem so irrelevant, clear, or common sense that they aren’t even reported in the experimental methods.

For example, the sex of the person that physically carried out a mouse study is not usually explicitly reported. Not the sex of the mouse — which usually isn’t reported either, and usually male — but the actual graduate student or lab tech that did the work for which all the paper authors claim credit. The sex of the experimenter is not reported because it just seems like an irrelevant detail. But this is just an implicit use of a symmetry principle: the inter-subjective uniformity of physical laws.

It doesn’t matter who did the experiment.

Except when it does.

A few years ago, researchers at the Department of Psychology of McGill University put this to the test. Sorge et al. (2014) found that mice were stressed out by male experimenters, and this reaction can skew research findings. In particular, the presence of men produced a stress response in mice and rats. This response was equivalent to the stress caused by restraining the rodents for a quarter of an hour in a tube or forcing them to swim for 3 minutes. More importantly, this stress response made the rodents less sensitive to pain.

The presence of women did not produce this stress response.

The men were also not strictly necessary: their smelly t-shirts would do. The effect was caused by the pheromones that men secrete from their armpits in higher concentrations than women. This is because all mammals share similar chemosignals.

But sex of experimenter is not the only unreported aspect of experiments, there’s also experimenter’s hair colour, day of the week, proximity to thesis deadline or grant renewal, etc. How do we pick which of the infinitely many unwritten parts of a methods section to focus on? In looking for violations of an implicit symmetry of experiments, it is particularly useful if there is a theory that is already available for dealing with the symmetry-break. As such, I like to view my recent work with Jeff Peacock, David Basanta, Andriy Marusyk, and Jacob Scott on measuring the games that cancer plays as an example of exploiting a broken symmetry.

In a typical cancer study, if the experimenters want to compare the growth rate of a drug sensitive and resistant cell-line, they usually do so in monotypical culture. They plate drug sensitive cells with or without drug (or in difference concentrations of drug), culture them and measure the growth rate. Then, they repeat with the drug resistant cells. The result is a number of growth-rates for the two populations in different concentration of drug. From this, the experimenters can estimate important factors like the cost of resistance or other quantities of interest.

But from the perspective of evolutionary game theory, this experimental set up did not compare apples to apples. Even at the same drug concentration, the micro-environments of the two monotypic experiments are not the same. A cell in the sensitive monoculture has a micro-environment full of sensitive cells and with no resistant, while a cell in the resistant monoculture has a micro-environment with no sensitive cells and full of resistant cells. By treating these two conditions as comparable, a traditional cancer biologist makes a symmetry assumption: it doesn’t matter for your growth rate if your neighbours are sensitive or resistant. This similar to assuming that growth rate is a cell-autonomous process.

Cell-autonomous growth rates are ones where the benefits/costs to growth rate are intrinsic to the cell: the presence of other cells are an irrelevant feature of the micro-environment. Thus, the growth rates from monotypic cultures provide all the necessary information. Non-cell-autonomous effects allow fitness to depend on a cell’s micro-environmental context, including the frequency of other cell types. A game theorist would not treat every initial seeding proportion of sensitive to resistant cells as different experimental conditions. A game theorist would demand a competitive fitness assay that is seldom use in cancer biology — but is the gold standard for studying resistance in bacteria. Although the bacteria studies don’t actually vary the seeding proportion, they compare experiments with an arbitrary fixed proportion. For a non-cell-autonomous process, growth rates need to be measured in competitive fitness assays over a range of seeding frequencies.

By breaking the assumption of cell-autonomous symmetry for non-small cell lung cancer in Kaznatcheev et al. (2017), we were able to show — among other things — that certain common beliefs about the cost of resistance might rest on this assumption. The classic model of resistance posits that the resistant cells receives a benefit in drug but is neutral, or even carries an inherent cost, in the abscence of treatment. For example, experimentalists frequently regard resistance granting mutations as selectively neutral in the absence of drug. The modeling community often goes further by considering explicit costs like up-regulating drug efflux pumps, investing in other defensive strategies, or lowering growth rate by switching to sub-optimal growth pathways.

If we looked at only monotypic fitness assays then our cell’s resistance to Alectinib would be consistent with the zero-cost classic model of the experimentalist. However, mixed cultures painted a different picture. One that is more consistent with a negative cost of resistance or at least one that switches from negative to positive dependent on the frequency of sensitive cells. For details, you can read the paper. The main point for me in the context of this post is that looking for symmetry breaking in scientific methodology matters at all levels, from grand cosmological theories to the petri dish.

What are some other implicit symmetries that we are assuming? How could we try breaking them? What theories could make use of such breaks?


Fischer, M., Kolachevsky, N., Zimmermann, M., Holzwarth, R., Udem, T., Hänsch, T. W., … & Marion, H. (2004). New limits on the drift of fundamental constants from laboratory measurements. Physical Review Letters, 92(23): 230802.

Kaznatcheev, A., Peacock, J., Basanta, D., Marusyk, A., & Scott, J. G. (2017). Fibroblasts and alectinib switch the evolutionary games that non-small cell lung cancer plays. bioRxiv, 179259.

Sorge, R. E., Martin, L. J., Isbester, K. A., Sotocinal, S. G., Rosen, S., Tuttle, A. H., … & Leger, P. (2014). Olfactory exposure to males, including men, causes stress and related analgesia in rodents. Nature Methods, 11(6): 629.


About Artem Kaznatcheev
From the Department of Computer Science at Oxford University and Department of Translational Hematology & Oncology Research at Cleveland Clinic, I marvel at the world through algorithmic lenses. My mind is drawn to evolutionary dynamics, theoretical computer science, mathematical oncology, computational learning theory, and philosophy of science. Previously I was at the Department of Integrated Mathematical Oncology at Moffitt Cancer Center, and the School of Computer Science and Department of Psychology at McGill University. In a past life, I worried about quantum queries at the Institute for Quantum Computing and Department of Combinatorics & Optimization at University of Waterloo and as a visitor to the Centre for Quantum Technologies at National University of Singapore. Meander with me on Google+ and Twitter.

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