## Stem cells, branching processes and stochasticity in cancer

October 25, 2014 6 Comments

When you were born, you probably had 270 bones in your body. Unless you’ve experienced some very drastic traumas, and assuming that you are fully grown, then you probably have 206 bones now. Much like the number and types of internal organs, we can call this question of science solved. Unfortunately, it isn’t always helpful to think of you as made of bones and other organs. For medical purposes, it is often better to think of you as made of cells. It becomes natural to ask how many cells you are made of, and then maybe classify them into cell types. Of course, you wouldn’t expect this number to be as static as the number of bones or organs, as individual cells constantly die and are replaced, but you’d expect the approximate number to be relatively constant. Thus number is surprisingly difficult to measure, and our best current estimate is around (Bianconi et al., 2013).

Both 206 and are just numbers, but to a modeler they suggest a very important distinction over which tools we should use. Suppose that my bones and cells randomly popped in and out of existence without about equal probability (thus keeping the average number constant). In that case I wouldn’t expect to see exactly 206 bones, or exactly 37200000000000 cells; if I do a quick back-of-the-envelope calculation then I’d expect to see somewhere between 191 and 220 bones, and between 37199994000000 and 37200006000000. Unsurprisingly, the variance in the number of bones is only around 29 bones, while the number of cells varies by around 12 million. However, in terms of the percentage, I have 14% variance for the bones and only 0.00003% variance in the cell count. This means that in terms of dynamic models, I would be perfectly happy to model the cell population by their average, since the stochastic fluctuations are irrelevant, but — for the bones — a 14% fluctuation is noticeable so I would need to worry about the individual bones (and we do; we even give them names!) instead of approximating them by an average. The small numbers would be a case of when results can depend heavily on if one picks a discrete or continuous model.

In ecology, evolution, and cancer, we are often dealing with huge populations closer to the number of cells than the number of bones. In this case, it is common practice to keep track of the averages and not worry too much about the stochastic fluctuations. A standard example of this is replicator dynamics — a deterministic differential equation governing the dynamics of average population sizes. However, this is not always a reasonable assumption. Some special cell-types, like stem cells, are often found in very low quantities in any given tissue but are of central importance to cancer progression. When we are modeling such low quantities — just like in the cartoon example of disappearing bones — it becomes to explicitly track the stochastic effects — although we don’t have to necessarily name each stem cell. In these cases we switch to using modeling techniques like branching processes. I want to use this post to highlight the many great examples of branching processes based models that we saw at the MBI Workshop on the Ecology and Evolution of Cancer.

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## Space and stochasticity in evolutionary games

January 28, 2015 by Artem Kaznatcheev 17 Comments

Two of my goals for TheEGG this year are to expand the line up of contributors and to extend the blog into a publicly accessible venue for active debate about preliminary, in-progress, and published projects; a window into the everyday challenges and miracles of research. Toward the first goal, we have new contributions from Jill Gallaher late last year and Alexander Yartsev this year with more posts taking shape as drafts from Alex, Marcel Montrey, Dan Nichol, Sergio Graziosi, Milo Johnson, and others. For the second goal, we have an exciting debate unfolding that was started when my overview of Archetti (2013,2014) prompted an objection from Philip Gerlee in the comments and Philipp Altrock on twitter. Subsequently, Philip and Philipp combined their objections into a guest post that begat an exciting comment thread with thoughtful discussion between David Basanta, Robert Vander Velde, Marc Harper, and Philip. Last Thursday, I wrote about how my on-going project with Robert, David, and Jacob Scott is expanding on Archetti’s work and was surprised to learn that Philip has responded on twitter with the same criticism as before. I was a little flabbergast by this because I thought that I had already addressed Philip’s critique in my original comment response and that he was reiterating the same exact text in his guest post simply for completeness and record, not because he thought it was still a fool-proof objection.

My biggest concern now is the possibility that Philip and I are talking past each other instead of engaging in a mutually beneficial dialogue. As such, I will use this post to restate (my understand of the relevant parts of) Philip and Philipp’s argument and extend it further, providing a massive bibliography for readers interested in delving deeper into this. In a future post, I will offer a more careful statement of my response. Hopefully Philip or other readers will clarify any misunderstandings or misrepresentations in my summary or extension. Since this discussion started in the context of mathematical oncology, I will occasionally reference cancer, but the primary point at issue is one that should be of interest to all evolutionary game theorists (maybe even most mathematical modelers): the model complexity versus simplicity tension that arises from the stochastic to deterministic transition and the discrete to continuous transition.

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Filed under Commentary, Models, Preliminary, Reviews, Technical Tagged with current events, discrete vs continuous, mathematical oncology, metamodeling, replicator dynamics, spatial structure