Choosing units of size for populations of cells
April 11, 2016
Recently, I have been interacting more and more closely with experiment. This has put me in the fortunate position of balancing the design and analysis of both theoretical and experimental models. It is tempting to think of theorists as people that come up with ideas to explain an existing body of facts, and of mathematical modelers as people that try to explain (or represent) an existing experiment. But in healthy collaboration, theory and experiment should walk hand it hand. If experiments pose our problems and our mathematical models are our tools then my insistence on pairing tools and problems (instead of ‘picking the best tool for the problem’)
means that we should be willing to deform both for better communication in the pair.
Evolutionary game theory — and many other mechanistic models in mathematical oncology and elsewhere — typically tracks population dynamics, and thus sets population size (or proportions within a population) as central variables. Most models think of the units of population as individual organisms; in this post, I’ll stick to the petri dish
and focus on cells as the individual organisms. We then try to figure out properties of these individual cells and their interactions based on prior experiments or our biological intuitions. Experimentalists also often reason in terms of individual cells, making them seem like a natural communication tool. Unfortunately, experiments and measurements themselves are usually not about cells. They are either of properties that are only meaningful at the population level — like fitness — or indirect proxies for counts of individual cells — like PSA or intensity of fluorescence. This often makes counts of individual cells into an inferred theoretical quantity and not a direct observable. And if we are going to introduce an extra theoretical term then parsimony begs for a justification.
But what is so special about the number of cells? In this post, I want to question the reasons to focus on individual cells (at the expense of other choices) as the basic atoms of our ontology.
So, let’s look at what we could mean by ‘size of population’. I started with the obvious definition of number of cells, and if all I ever did was in silico
simulations then it is the definition I would have stuck to. Especially for agent-based models, it is very tempting to have cells as your agents and building everything up around them. But trying to overcome the limitations of our experimental system — or more accurately, the limitation of my current computer vision skills — made me question the naturalness of this definition. 
At least in cell cultures, I realized that there are some reasons to doubt individual cells as the most natural choice of units.
Consider two populations that have the same number of cells and everything else is equal, except …
… the cells in the first population are metabolically twice as active as cells in the second population.
In this case, the more active cells can easily strain their environment more, as they use more resources to fuel themselves. If your limiting resource in the petri dish is growth medium then the more metabolically active cells will consume more of it. 
With slower metabolic activity, the cell becomes less of an effect not only on its own future, but also on other cells it interacts with — for example, by moving around less or releasing fewer cytokines and thus interacting with fewer other cells. In this case, the more natural set of physical units might be the power consumption in terms of watts-used.
… the cells in the first population are twice as big as cells in the second population.
From the point of view of games mediated by things like diffusive factors or cell-cell contact, the bigger cells will have more area to absorb/release factors or to contact other cells. In we are working in vitro
, larger cells also exhaust the limiting factor of free space quicker than small cells. 
On top of this, size feeds back into the first point, with larger cells usually doing more things metabolically and in terms of other activity. In this case, the more natural set of physical units might be area-covered.
Of course, we could try to express the above in terms of individual cells by converting back and forth between numbers of cells and watts-used or area-covered. Practically, this would mean finding a conversion factor which amounts to a measure of how much power or area a typical cell uses. But in doing so, we have swept some amount of heterogeneity under the rug — after all, each cell takes up a different amount of space or uses a different amount of energy, especially when facing new circumstances like chemotherapy — and it is not clear what useful thing we got in return.
Without individual cells to ground us, reductionist story telling becomes more difficult; something that can be both a plus or a minus:
On the one hand, it is hard to imagine how 10 watts-used by cancer interacts with 10 watts-used by fibroblasts, instead we are forced to make these measurements experimentally. Since these measurements are almost always at the level of populations, we don’t feel a need to make sense of them in reductionist terms of how a single watt-used interacts with another watt-used. You might have noticed that even the word ‘interaction’ felt awkward in the last two sentences. Watt or area use invite us to recognize the importance of both the size of the other population and the environment more generally. This makes it easier to notice evolutionary games with only indirect interactions through processes like heterogeneous feeding rates
On the other hand, these more abstract units hinder the imagination, and it can often become more difficult to explain the work or to design new experiments. The alternative units I considered in this post also obscure the discrete nature of cells
, a discreteness that isessential to life and can have significant side-effects on model conclusions (for discussion, see Durrett & Levin, 1994; Shnerb et al., 2000). 
It is also much easier to write that “we used fluorescent area as a proxy for population size” than to delve into the discussion of this post. 
And in the context of most work, such discussion is inconsequential. However, in the broader context, I think this is a discussion that we should have. Whichever way that conversation goes, we will emerge with a better understanding of the basic ontology on which we build our theories and experiments.
Notes and References
- Although I have bumped against these themes from purely theoretical considerations, I would have never bothered to worry more and write about them if it wasn’t for wearing many hats in a tight-knit collaboration. If there was three separate people to design the experiments, quantify the data, and build the theoretical models then I suspect they would be more likely to continue communicating in terms of individual cell counts. It was only due to difficulties in segmenting cells that I started to wonder what would be more useful to do: to work more on refining the computer vision algorithm to properly identify the individual cells, or to consider a theoretical model grounded in something other than individual cells?
- The extreme case of this is cells that are completely inactive or maybe even dead. This doesn’t come up as much in simulation, since I can just cleanly remove dead agents. In our experimental system, on the other hand, cancer cells often don’t lysis and a dead cell will maintain its membrane integrity and continue to look just like a live cell under phase-contrast microscopy. Further, many drugs — like Alectinib in our experimental system — are not cytotoxic but cytostatic, causing affected cells to shut (or slow) down their activity.
- In the case of cancer, the tumor corresponding to the more voluminous population would be much more burdensome to the patient. In fact, tumor burden is often measured and reported as volume in x-ray or other imaging. The number of cells in the tumor is then inferred from these volumetric measures by assuming (or measuring outside the body) the size of a typical cancer cell.
The importance of area has also come up in thinking aboutcancer in the bone. Osteoclasts and osteoblasts, take up drastically different amounts of area on the bone, and they are only of significant consequence to the model if they are in contact with the bone (else they are not remodeling it). Area-Mo-Bone becomes the important variable here. Focusing on area also makes fictitious player like free-space more sensical (for example, see my comments on Li et al., 2005
Although in the case of replicator dynamics, it is not clear how much this would matter since they already ignore the discrete nature of life through the use of ODEs. This would matter much more for branching process models
. Unfortunately, I am much less familiar with those.
- In fact, the reason this post exists is because a post I was writing about our recent experimental work had gotten out-of-hand in terms of lengths and tangents. This was one of those tangents that can now be replaced by a link.
Durrett, R., & Levin, S. (1994). The Importance of Being Discrete (and Spatial). Theoretical Population Biology
, 46: 363-394.
Li, X.-Y., Pietschke, C., Fraune, S., Altrock, P.M., Bosch, T.C., & Traulsen, A. (2015). Which games are growing bacterial populations playing? Journal of the Royal Society Interface
, 12 (108)
Shnerb, N.M., Louzoun, Y., Bettelheim, E., & Solomon, S. (2000). The importance of being discrete: life always wins on the surface. Proceedings of the National Academy of Sciences, 97
(19), 10322-4 PMID: 10962027