Looking for species in cancer but finding strategies and players

Sometime before 6 August 2014, David Basanta and Tamir Epstein were discussing the increasing focus of mathematical oncology on tumour heterogeneity. An obstacle for this focus is a good definitions of heterogeneity. One path around this obstacle is to take definitions from other fields like ecology — maybe species diversity. But this path is not straightforward: we usually — with some notable and interesting examples — view cancer cells as primarily asexual and the species concept is for sexual organisms. Hence, the specific question that concerned David and Tamir: is there a concept of species that applies to cancer?

I want to consider a couple of candidate answers to this question. None of these answers will be a satisfactory definition for species in cancer. But I think the exercise is useful for understanding evolutionary game theory. With the first attempt to define species, we’ll end up using the game assay to operationalize strategies. With the second attempt, we’ll use the struggle for existence to define players. Both will be sketches that I will need to completely more carefully if there is interest.

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Darwin as an early algorithmic biologist

In his autobiography, Darwin remarked on mathematics as an extra sense that helped mathematicians see truths that were inaccessible to him. He wrote:

Darwin Turing HeadbandDuring the three years which I spent at Cambridge… I attempted mathematics… but got on very slowly. The work was repugnant to me, chiefly from my not being able to see any meaning in the early steps in algebra. This impatience was very foolish, and in after years I have deeply regretted that I did not proceed far enough at least to understand something of the great leading principles of mathematics, for [people] thus endowed seem to have an extra sense. But I do not believe that I should ever have succeeded beyond a very low grade. … in my last year I worked with some earnestness for my final degree of B.A., and brushed up … a little Algebra and Euclid, which later gave me much pleasure, as it did at school.

Today, this remark has become a banner to rally mathematicians interested in biology. We use it to convince ourselves that by knowing mathematics, we have something to contribute to biology. In fact, the early mathematical biologist were able to personify the practical power of this extra sense in Gregor Mendel. From R.A. Fisher onward — including today — mathematicians have presented Mendel as one of their own. It is standard to attributed Mendel’s salvation of natural selection to his combinatorial insight into the laws of inheritance — to his alternative to Darwin’s non-mathematical blending inheritance.

But I don’t think we need wait for the rediscovery of Mendel to see fundamental mathematical insights shaping evolution. I think that Darwin did have mathematical vision, but just lacked the algorithmic lenses to focus it. In this post I want to give examples of how some of Darwin’s classic ideas can be read as anticipating important aspects of algorithmic biology. In particular, seeing the importance of asymptotic analysis and the role of algorithms in nature.
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Proximal vs ultimate constraints on evolution

For a mathematician — like John D. Cook, for example — objectives and constraints are duals of each other. But sometimes the objectives are easier to see than the constraints. This is certainly the case for evolution. Here, most students would point you to fitness as the objective to be maximized. And at least at a heuristic level — under a sufficiently nuanced definition of fitness — biologists would agree. So let’s take the objective as known.

This leaves us with the harder to see constraints.

Ever since the microscope, biologists have been expert at studying the hard to see. So, of course — as an editor at Proceedings of the Royal Society: B reminded me — they have looked at constraints on evolution. In particular, departures from an expected evolutionary equilibrium is where biologists see constraints on evolution. An evolutionary constraint is anything that prevents a population from being at a fitness peak.

Winding path in a hard semi-smooth landscape

In this post, I want to follow a bit of a winding path. First, I’ll appeal to Mayr’s ultimate-proximate distinction as a motivation for why biologists care about evolutionary constraints. Second, I’ll introduce the constraints on evolution that have been already studied, and argue that these are mostly proximal constraints. Third, I’ll introduce the notion of ultimate constraints and interpret my work on the computational complexity of evolutionary equilibria as an ultimate constraint. Finally, I’ll point at a particularly important consequence of the computational constraint of evolution: the possibility of open-ended evolution.

In a way, this post can be read as an overview of the change in focus between Kaznatcheev (2013) and (2018).
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A month in papers: mostly philosophy of biology

I’ve seen a number of people that have aimed for reading one paper a day for every day of the year. Unfortunately, I am not great at new years resolutions, and would never be able to keep pace for all 365 days. Instead, in April I tried a one paper a day challenge for the month. I still came up short, only finishing 24 of 30 papers. But I guess that is enough for one paper per weekday.

As I went along, I posted tweet-length summaries in a long thread. In this post, I want to expand on and share what I read in April. And in the future, I think I’ll transform the month-goals into week goals of five papers per week. Just to avoid colossal twitter threads. I tried that last week, for example. But I don’t think I’ll end up making those into posts. Although, as happened in April, they might inspire thematic posts.

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Fusion and sex in protocells & the start of evolution

In 1864, five years after reading Darwin’s On the Origin of Species, Pyotr Kropotkin — the anarchist prince of mutual aid — was leading a geographic survey expedition aboard a dog-sleigh — a distinctly Siberian variant of the HMS Beagle. In the harsh Manchurian climate, Kropotkin did not see competition ‘red in tooth and claw’, but a flourishing of cooperation as animals banded together to survive their environment. From this, he built a theory of mutual aid as a driving factor of evolution. Among his countless observations, he noted that no matter how selfish an animal was, it still had to come together with others of its species, at least to reproduce. In this, he saw both sex and cooperation as primary evolutionary forces.

Now, Martin A. Nowak has taken up the challenge of putting cooperation as a central driver of evolution. With his colleagues, he has tracked the problem from myriad angles, and it is not surprising that recently he has turned to sex. In a paper released at the start of this month, Sam Sinai, Jason Olejarz, Iulia A. Neagu, & Nowak (2016) argue that sex is primary. We need sex just to kick start the evolution of a primordial cell.

In this post, I want to sketch Sinai et al.’s (2016) main argument, discuss prior work on the primacy of sex, a similar model by Wilf & Ewens, the puzzle over emergence of higher levels of organization, and the difference between the protocell fusion studied by Sinai et al. (2016) and sex as it is normally understood. My goal is to introduce this fascinating new field that Sinai et al. (2016) are opening to you, dear reader; to provide them with some feedback on their preprint; and, to sketch some preliminary ideas for future extensions of their work.

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Multiplicative versus additive fitness and the limit of weak selection

Previously, I have discussed the importance of understanding how fitness is defined in a given model. So far, I’ve focused on how mathematically equivalent formulations can have different ontological commitments. In this post, I want to touch briefly on another concern: two different types of mathematical definitions of fitness. In particular, I will discuss additive fitness versus multiplicative fitness.[1] You often see the former in continuous time replicator dynamics and the latter in discrete time models.

In some ways, these versions are equivalent: there is a natural bijection between them through the exponential map or by taking the limit of infinitesimally small time-steps. A special case of more general Lie theory. But in practice, they are used differently in models. Implicitly changing which definition one uses throughout a model — without running back and forth through the isomorphism — can lead to silly mistakes. Thankfully, there is usually a quick fix for this in the limit of weak selection.

I suspect that this post is common knowledge. However, I didn’t have a quick reference to give to Pranav Warman, so I am writing this.
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Eukaryotes without Mitochondria and Aristotle’s Ladder of Life

In 348/7 BC, fearing anti-Macedonian sentiment or disappointed with the control of Plato’s Academy passing to Speusippus, Aristotle left Athens for Asian Minor across the Aegean sea. Based on his five years[1] studying of the natural history of Lesbos, he wrote the pioneering work of zoology: The History of Animals. In it, he set out to catalog the what of biology before searching for the answers of why. He initiated a tradition of naturalists that continues to this day.

Aristotle classified his observations of the natural world into a hierarchical ladder of life: humans on top, above the other blooded animals, bloodless animals, and plants. Although we’ve excised Aristotle’s insistence on static species, this ladder remains for many. They consider species as more complex than their ancestors, and between the species a presence of a hierarchy of complexity with humans — as always — on top. A common example of this is the rationality fetish that views Bayesian learning as a fixed point of evolution, or ranks species based on intelligence or levels-of-consciousness. This is then coupled with an insistence on progress, and gives them the what to be explained: the arc of evolution is long, but it bends towards complexity.

In the early months of TheEGG, Julian Xue turned to explaining the why behind the evolution of complexity with ideas like irreversible evolution as the steps up the ladder of life.[2] One of Julian’s strongest examples of such an irreversible step up has been the transition from prokaryotes to eukaryotes through the acquisition of membrane-bound organelles like mitochondria. But as an honest and dedicated scholar, Julian is always on the lookout for falsifications of his theories. This morning — with an optimistic “there goes my theory” — he shared the new Kamkowska et al. (2016) paper showing a surprising what to add to our natural history: a eukaryote without mitochondria. An apparent example of a eukaryote stepping down a rung in complexity by losing its membrane-bound ATP powerhouse.
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Mutation-bias driving the evolution of mutation rates

In classic game theory, we are often faced with multiple potential equilibria between which to select with no unequivocal way to choose between these alternatives. If you’ve ever heard Artem justify dynamic approaches, such as evolutionary game theory, then you’ve seen this equilibrium selection problem take center stage. Natural selection has an analogous ‘problem’ of many local fitness peaks. Is the selection between them simply an accidental historical process? Or is there a method to the madness that is independent of the the environment that defines the fitness landscape and that can produce long term evolutionary trends?

Two weeks ago, in my first post of this series, I talked about an idea Wallace Arthur (2004) calls “developmental bias”, where the variation of traits in a population can determine which fitness peak the population evolves to. The idea is that if variation is generated more frequently in a particular direction, then fitness peaks in that direction are more easily discovered. Arthur hypothesized that this mechanism can be responsible for long-term evolutionary trends.

A very similar idea was discovered and called “mutation bias” by Yampolsky & Stoltzfus (2001). The difference between mutation bias and developmental bias is that Yampolsky & Stoltzfus (2001) described the idea in the language of discrete genetics rather than trait-based phenotypic evolution. They also did not invoke developmental biology. The basic mechanism, however, was the same: if a population is confronted with multiple fitness peaks nearby, mutation bias will make particular peaks much more likely.

In this post, I will discuss the Yampolsky & Stoltzfus (2001) “mutation bias”, consider applications of it to the evolution of mutation rates by Gerrish et al. (2007), and discuss how mutation is like and unlike other biological traits.

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Variation for supply driven evolution

I’ve taken a very long hiatus (nearly 5 years!) from this blog. I suppose getting married and getting an MD are good excuses, but Artem has very kindly let me return. And I greatly appreciate this chance, because I’d like to summarize an idea I had been working on for a while. So far, only two publication has come out of it (Xue et al., 2015a,b), but it’s an idea that has me excited. So excited that I defended a thesis on it this Tuesday. For now, I call it supply-driven evolution, where I try to show how the generation of variation can determine long-term evolution.

Evolutionary theoreticians have long known that how variation is generated has a decisive role in evolutionary outcome. The reason is that natural selection can only choose among what has been generated, so focusing on natural selection will not produce a full understanding of evolution. But how does variation affect evolution, and can variation be the decisive factor in how evolution proceeds? I believe that the answer is “frequently, yes,” because it does not actually compete with natural selection. I’ll do a brief overview of the literature in the first few posts. By the end, I hope how this mechanism can explain some forms of irreversible evolution, stuff I had blogged about five years ago.

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Evolutionary non-commutativity suggests novel treatment strategies

In the Autumn of 2011 I received an email from Jacob Scott, now a good friend and better mentor, who was looking for an undergraduate to code an evolutionary simulation. Jake had just arrived in Oxford to start his DPhil in applied mathematics and by chance had dined at St Anne’s College with Peter Jeavons, then a tutor of mine, the evening before. Jake had outlined his ideas, Peter had supplied a number of email addresses, Jake sent an email and I uncharacteristically replied saying I’d give it a shot. These unlikely events would led me to where I am today — a DPhil candidate in the Oxford University Department of Computer Science. My project with Jake was a success and I was invited to speak at the 2012 meeting of the Society of Mathematical Biology in Knoxville, TN. Here I met one of Jake’s supervisors, Alexander Anderson, who invited me to visit the Department of Integrated Mathematical Oncology at the Moffitt Cancer Center and Research Institute for a workshop in December of that year. Here Dr. Anderson and I discussed one of the key issues with the work I will present in this post, issues that now form the basis of my DPhil with Dr. Anderson as one of two supervisors. Fittingly, the other is Peter Jeavons.

Jake was considering the problem of treating and avoiding drug resistance and in his short email provided his hypothesis as a single question: “Can we administer a sequence of drugs to steer the evolution of a disease population to a configuration from which resistance cannot emerge?”

In Nichol et al. (2015), we provide evidence for an affirmative answer to this question. I would like to use this post to introduce you to our result, and discuss some of the criticisms.

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