QBIOX: Distinguishing mathematical from verbal models in biology

There is a network at Oxford know as QBIOX that aims to connect researchers in the quantitative biosciences. They try to foster collaborations across the university and organize symposia where people from various departments can share their quantitative approaches to biology. Yesterday was my second or third time attending, and I wanted to share a brief overview of the three talks by Philip Maini, Edward Morrissey, and Heather Harrington. In the process, we’ll get to look at slime molds, colon crypts, neural crests, and glycolysis. And see modeling approaches ranging from ODEs to hybrid automata to STAN to algebraic systems biology. All of this will be in contrast to verbal theories.

Philip Maini started the evening off — and set the theme for my post — with a direct question as the title of his talk.

Does mathematics have anything to do with biology?

By the law of headlines — as Peter Jeavons pointed out — the answer is no. And Philip Maini would agree, at least if biology was logical. But it is not. It is bio-logical.

Puns aside, what Philip Maini meant by this is that biology has a lot of complex effects that often cannot be captured by simple verbal theories. This is where mathematical modelling comes in. Mathematical models sharpen our intuitions. They can show us where verbal theories fall short or compare systems via measurable parameters in a way that verbal theories simply cannot.

But mathematical models — even the simplest and most elegant one — are usually more complicated than the simple verbal theories that they challenge. And since Maini has a strong preference for the simplest models, he usually roots for the verbal theory. After all, if the verbal theory is sufficient then he doesn’t need to do any modelling and he can just stay home and watch football.

Unfortunately, verbal theories don’t always survive their encounters with evidence. Maini highlighted this effect with three cases from three different decades of his work. Three cases where a mathematical model overturned a verbal theory. Three cases where Maini couldn’t stay home and watch football.

In the 1990s, slime molds were all the rage as experimental systems. This was mostly because they had a fascinating life cycle, especially in their cooperative response to environmental austerity. They could be used to study signal transduction, chemotaxis, and differentiation. Höfer, Sherratt & Maini (1995b) set out to model this system, with a focus on the role of cell density. In the process, they confronted the verbal theory underlying the chemotactic wave paradox (see also Hofer et al., 1995a). As individual amoeba aggregate into a slime mold, they move up a chemogradient. However, the pulses that create this gradient are symmetric bursts — imagine them as an equilateral triangle pulses that travel past the cell — and so the standard verbal theory would expect the cell to actually move away from the source. The slime mold would never form. This is because the individual amoeba would be exposed to the increasing gradient for a shorter time (because it would move toward the source at that time, shortening exposure time) than to the decreasing gradient (because it would move away from the source at that time, following the wave). This was in clear qualitative contradiction to the observed evidence of amoeba moving together and forming a slime mold. Hence, the verbal theorists argued that a non-standard biochemistry was needed. But Hofer et al. (1995a,b) were able to show that if population density was taken into account, the paradox disappears and the standard biochemical pathways of chemotaxis allowed the individual amoeba to aggregate into a slime mold. A mathematical model resolved a paradox inherent in a non-rigorously formulated verbal theory.

In the 2000s, Maini shifted his focus to cancer. In particular, to the role of glycolysis — something that Robert Vander Velde and I have discuss extensively on TheEGG. In this case, verbal theory (for example, see Gatenby & Gillies, 2004) expects sweeps of progressively more aggressive clones. But formalizing this theory into a hybrid automaton model — as Gatenby et al. (2007) did — shows that although this is possible, a much more common outcome is an even more dangerous heterogeneous tumour. Something that was observed experimentally after the model’s suggestion. A mathematical model pushed a verbal theory into more surprising predictions.

In the 2010s, Maini unified both a focus on development and (a proxy for) tumour invasiveness by looking at neural crest cells. As with the amoeba, Maini found the most interesting questions about neural crest to be on chemotaxis. In this case, there are no pulses, but a more steady field of VEGF. As such, verbal theories predict that neural crest cells will steadily migration to their destination. This is consistent with reality. However, when McLennan et al. (2012) built a hybrid automaton model to implement this theory, they saw an obstacle: the cells at the front consume most of the VEGF and then the later cells get stuck without migrating far. In this case, a different biochemistry was actually needed that verbal theory couldn’t see. There need to be two types of cells: leaders that follow VEGF and followers that follow leaders. This was later confirmed experimentally. A mathematical model showed the incompleteness of a verbal theory and suggested new things to search for experimentally.

So although verbal theories can do a lot, Maini was able to point out cases when mathematical models can offer us more.

Edward Morrissey’s talk concentrated on what verbal theories cannot do and mathematical models can: compare systems via measurable parameters. In this case, he was comparing mice and men. Or at least the colonic crypts of a well-controlled mouse model against those of human patients. As slime molds were a popular model organism in the 90s, I feel like crypts are prominent experimental system today. At least among the oncologists that I regularly encounter. It is a system where popular topics like stem cells, somatic evolution, and heterogeneity can come together. Morrissey was looking at all three in his talk on the fixation and spread of somatic mutations in human colonic epithelium.

As I’ve discussed before for gastric crypts — colonic crypts are similar — the bottom of the crypt is populated by stem cells. All the other cells in the crypt are eventually flushed out into the colon. Thus, a mutation can only be maintained in a crypt if it arises in the stem cell niche. Morrissey focused on nearly-neutral mutations that could be stained in the tissues. With such a marker, a given crypt could either be completely unstained, if no mutations are present; partially stained, if only some of the stem cells are mutated; or fully stained, if all the stem cells are mutated. So Morrissey wrote some computer vision code to count crypts in histologies. For a given histology, he could count the total number of crypts, as well as the number of partially stained and also fully stained crypts. From looking at histologies at various times in an individuals lives, he could construct a plots of the two quantities: the fraction of crypts that are partially stained and the fraction of crypts that are fully stained.

From both verbal and mathematical models, we expect the total number of fully stained crypts to increase linearly with time and the the partially stained crypts to quickly saturate at a constant number. This is consistent with the data, The verbal theory was not ruled out, but it didn’t mean that Morrissey could watch football.

Unlike the verbal theory, the mathematical model allows us to transform the quantitative measurements of the partially stained crypt number saturation point and the slope of the totally stained crypt number into estimates for the number of active stem cells and the stem cell replacement rate. Kozar et al. (2013) first made these measurements in the mouse system. This was already an exciting result, since it showed that the number of active stem-cells (around 5) is lower than the total number (around 16). Although I wonder if this could also be explained by effective population size vs head-count population size. Either way, this was a surprise.

Recently Nicholson et al. (2018) repeated his process for human samples. From this, Morrissey could conclude that the humans’ stem cell replacement rate is very slow: 500 times slower than mouse. This is also a surprise. The popular wisdom is that humans should roughly the same replacement rate in the crypt.

These surprises would have been impossible from verbal theories. These surprises required the parameter inference that mathematical models make possible.

But it is not always possible to infer all the hidden parameters of a model. There are simply too many parameters and too little data. This is where Heather Harrington stepped in. She is focused on the comparisons of data and models with the use of computational algebra. Her goal — at least in my interpretation — is to use computational algebra to make the qualitative comparison of models to data more rigorous and automatic. Qualitative comparisons similar to the ones Maini used to rule our certain verbal theories with his mathematical models. Except, unlike Maini, Harrington doesn’t aim to rule of a single verbal theory, but narrow down large spaces of mathematical models. She calls this algebraic systems biology.

This appeals to me because I view the algorithmic biology that I am developing as a means to do rigorous qualitative analysis. And I am to rule out whole classes of models based on the shape of their qualitative predictions. In this way, Harrington and I are kindred spirits.

Unlike me, though, Harrington has a much more impressive record of success. She shared four of these successes as examples: the Notch-Wnt interactions of the intestinal crypt (Kay et al., 2017), multi-site phosphorylation (Jovanovic et al., 2015), analyzing neural networks from fMRI data, and the topology of biological images.

Most importantly, she shared the workflow of algebraic systems biology.

I think the above nicely formalizes the relationship between verbal theories, mathematical models, and experimental work. And I am glad that a network like QBIOX exists at Oxford to facilitate this process.

References

Gatenby, R. A., & Gillies, R. J. (2004). Why do cancers have high aerobic glycolysis?. Nature Reviews Cancer, 4(11), 891.

Gatenby, R. A., Smallbone, K., Maini, P. K., Rose, F., Averill, J., Nagle, R. B., … & Gillies, R. J. (2007). Cellular adaptations to hypoxia and acidosis during somatic evolution of breast cancer. British Journal of Cancer, 97(5), 646.

Höfer, T., Maini, P. K., Sherratt, J. A., Chaplain, M. A. J., & Murray, J. D. (1995a). Resolving the chemotactic wave paradox: A mathematical model for chemotaxis of Dictyostelium amoebae. Journal of Biological Systems, 3(04), 967-973.

Höfer, T., Sherratt, J. A., & Maini, P. K. (1995b). Dictyostelium discoideum: cellular self-organization in an excitable biological medium. Proc. R. Soc. Lond. B, 259(1356): 249-257.

Jovanovic, G., Sheng, X., Ale, A., Feliu, E., Harrington, H. A., Kirk, P., … & Stumpf, M. P. (2015). Phosphorelay of non-orthodox two component systems functions through a bi-molecular mechanism in vivo: The case of ArcB. Molecular BioSystems, 11(5), 1348-1359.

Kay, S. K., Harrington, H. A., Shepherd, S., Brennan, K., Dale, T., Osborne, J. M., … & Byrne, H. M. (2017). The role of the Hes1 crosstalk hub in Notch-Wnt interactions of the intestinal crypt. PLoS Computational Biology, 13(2), e1005400.

Kozar, S., Morrissey, E., Nicholson, A. M., van der Heijden, M., Zecchini, H. I., Kemp, R., … & Winton, D. J. (2013). Continuous clonal labeling reveals small numbers of functional stem cells in intestinal crypts and adenomas. Cell Stem Cell, 13(5), 626-633.

McLennan, R., Dyson, L., Prather, K. W., Morrison, J. A., Baker, R. E., Maini, P. K., & Kulesa, P. M. (2012). Multiscale mechanisms of cell migration during development: theory and experiment. Development, 139(16), 2935-2944.

Nicholson, A. M., Olpe, C., Hoyle, A., Thorsen, A. S., Rus, T., Colombé, M., … & Malhotra, S. (2018). Fixation and spread of somatic mutations in adult human colonic epithelium. Cell Stem Cell.

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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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