Colon cancer, mathematical time travel, and questioning the sequential mutation model.

On Saturday, I arrived in Columbus, Ohio for the the MBI Workshop on the Ecology and Evolution of Cancer. Today, our second day started. The meeting is an exciting combination of biology-minded mathematicians and computer scientists, and math-friendly biologist and clinicians. As is typical of workshops, the speakers of the first day had an agenda of setting the scope. In this case, the common theme was to question and refine the established model as embodied by Hannah & Weinberg’s (2000) hallmarks of cancer outlined. For an accessible overview of these hallmarks, I recommend Buddhini Samarasinghe’s series of posts. I won’t provide a full overview of the standard model, but only focus on the aspects at issue for the workshop participants. In the case of the first two speakers, the standard picture in question was the sequential mutation model. In the textbook model of cancer, a tumour acquires the hallmark mutations one at a time, with each subsequent mutation sweeping to fixation. Trevor Graham and Darryl Shibata presented their work on colon cancer, emphasizing tumour heterogeneity, and suggesting that we might have to rewrite the sequential mutation page of our Cancer 101 textbooks to better discuss the punctuated model.

Trevor A. Graham started the day by reminding us what cancer is and why we get it. Cancer is as old as — and in some-way dual to — multicellularity; it is somatic cells selfishly breaking their covenant of helping germ cells reproduce to instead advance their individual fitness by reproducing themselves without restraint. The differences in somatic mutations are heritable variation, and selection is seen through processes like clonal expansion. Thus, carcinogenesis is fundamentally an evolutionary process (Nowell, 1976) with selection for cancer at the level of individual cells. Studying this process in humans is hard and Graham focused on four difficulties: (1) we can only sample cancers that have already established themselves in the body, (2) longitudinally-collected samples are hard to obtain, (3) perturbation experiments are unethical and thus mostly impossible, and (4) tracing clonal expansion through in vivo cell marking is nearly impossible.

Of course, Graham’s group is happy with a challenge, and they have developed techniques to overcome some of these limitations in the study of colon cancer. Colon cancer develops in intestinal crypts, which are wonderful test-tubes for cancer research both in their appearance and their ability to serve as small semi-independent grounds for experiment. The crypts provide a physical constraint to the cells that are quickly recycled as their filling. This filling on the inside edge of the crypt is a single layer that provides direct insight into stem cell dynamics. The crypts come with a natural tracer because of occasional loss of cytochrome c oxidase (CCO) activity due to a somatic mitochondrial DNA mutation (Taylor et al., 2003). By colouring the CCO deficient cells blue and imaging individual crypts, Graham’s group can look at the temporal evolution of the crypt stem cells (Baker et al, 2014).


These colour-processed maps serve as a time versus population proportion plot, allowing Graham to use mathematical modeling to travel back in time and examine the dynamic history. By modeling the changes in the number of blue cells from line to line as a random walk, Baker et al. (2014) estimate the selection strength as nearly neutral (an unbiased random walk) with a diffusion coefficient that differs between normal and neoplastic human colon. In earlier work with a faster molecular clock based on methylation, Hymphries et al. (2013) saw that on the level of the population of crypts that make up an adenoma, there was a large amount of heterogeneity without recent clonal sweeps of the whole population. Graham used these two results to question the standard view of cancer as a sequential series of mutations swept to dominate the whole tumour, instead suggesting that the weak selection allowed for a punctuated model where many independent mutations exist at intermediate levels in the population.

In the second talk of the day, Darryl Shibata continued this exploration of colon cancer. He focused more on the whole tumour, over the individual crypts, and introduced another method of mathematical time travel. A typical tumour might be on the order of centimeters and from the point of view of individual crypts, these are huge distances encompassing billions of cells. In terms of the subtle variety of this fully grown tumour, the complexity is immense, but in terms of information, the most significant genetic changes come from early in the tumour development. By exploiting the spatial structure, and sequencing from opposite sides of the tumour, Shibata’s group is able to uncover the early private mutations causing tumour heterogeneity (for a general overview of this, clinical considerations, and some more reference, see Shibata (2012)). Further, by focusing on the amount of overlap in the private mutations between distant sides of the tumour, Shibata can estimate the amount of motility in the early tumour. This allows him to explain why tumours with mutational patches are usually benign adenomas, while those with a poka dot pattern (a lot of mixing) are cancers that can metastasize to elsewhere in the body. The former are clearly non-motile, while the later are “born to be bad”, and contain motility from early in the development. This stands in contrast to the accepted view of motility as something that is acquired late in tumour progression, and suggests why we tend to find primarily non-killers when we screen. By the time the tumour reaches a size that is clinically detectable, it is already bad or benign. The benign ones linger for a long time until they are noticed in a screen. The bad one, however, go on to quickly metastasize and kill the patient before there is time for a screen to notice the cancer.

Although both speakers focused on colon cancer, and advocated for punctuated model of mutations, they differed in the mechanisms they suggested. For Graham it was a decrease in selection strength, while for Shibata it was higher genomic instability in the early tumour. Joel Brown briefly hinted at this point during his talk today, but to sketch it in more detail would require that I introduce the evolutionary ecology perspective on cancer, which will be a whole other post. Stay tuned!


Baker AM, Cereser B, Melton S, Fletcher AG, Rodriguez-Justo M, Tadrous PJ, Humphries A, Elia G, McDonald SA, Wright NA, Simons BD, Jansen M, & Graham TA (2014). Quantification of crypt and stem cell evolution in the normal and neoplastic human colon. Cell reports, 8 (4), 940-7 PMID: 25127143

Hanahan, D., & Weinberg, R. A. (2000). The hallmarks of cancer. Cell, 100(1), 57-70.

Humphries, A., Cereser, B., Gay, L. J., Miller, D. S., Das, B., Gutteridge, A., … & Graham, T. A. (2013). Lineage tracing reveals multipotent stem cells maintain human adenomas and the pattern of clonal expansion in tumor evolution. Proceedings of the National Academy of Sciences, 110(27), E2490-E2499.

Nowell, P. C. (1976). The clonal evolution of tumor cell populations. Science, 194(4260), 23-28.

Shibata, D. (2012). Heterogeneity and tumor history. Science, 336(6079): 304-305.

Taylor, R. W., Barron, M. J., Borthwick, G. M., Gospel, A., Chinnery, P. F., Samuels, D. C., … & Turnbull, D. M. (2003). Mitochondrial DNA mutations in human colonic crypt stem cells. Journal of Clinical Investigation, 112(9): 1351-1360.

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.

5 Responses to Colon cancer, mathematical time travel, and questioning the sequential mutation model.

  1. Pingback: Experimental and comparative oncology: zebrafish, dogs, elephants | Theory, Evolution, and Games Group

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