## 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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## Multiple realizability of replicator dynamics

Abstraction is my favorite part of mathematics. I find a certain beauty in seeing structures without their implementations, or structures that are preserved across various implementations. And although it seems possible to reason through analogy without (explicit) abstraction, I would not enjoy being restricted in such a way. In biology and medicine, however, I often find that one can get caught up in the concrete and particular. This makes it harder to remember that certain macro-dynamical properties can be abstracted and made independent of particular micro-dynamical implementations. In this post, I want to focus on a particular pet-peeve of mine: accounts of the replicator equation.

I will start with a brief philosophical detour through multiple realizability, and discuss the popular analogy of temperature. Then I will move on to the phenomenological definition of the replicator equation, and a few realizations. A particular target will be the statement I’ve been hearing too often recently: replicator dynamics are only true for a very large but fixed-size well-mixed population.

## Choosing units of size for populations of cells

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.

## Abusing numbers and the importance of type checking

What would you say if I told you that I could count to infinity on my hands? Infinity is large, and I have a typical number of fingers. Surely, I must be joking. Well, let me guide you through my process. Since you can’t see me right now, you will have to imagine my hands. When I hold out the thumb on my left hand, that’s one, and when I hold up the thumb and the index finger, that’s two. Actually, we should be more rigorous, since you are imagining my fingers, it actually isn’t one and two, but i and 2i. This is why they call them imaginary numbers.

Let’s continue the process of extending my (imaginary) fingers from the leftmost digits towards the right. When I hold out my whole left hand and the pinky, ring, and middle fingers on my right hand, I have reached 8i.

But this doesn’t look like what I promised. For the final step, we need to remember the geometric interpretation of complex numbers. Multiplying by i is the same thing as rotating counter-clockwise by 90 degrees in the plane. So, let’s rotate our number by 90 degrees and arrive at $\infty$.

I just counted to infinity on my hands.

Of course, I can’t stop at a joke. I need to overanalyze it. There is something for scientists to learn from the error that makes this joke. The disregard for the type of objects and jumping between two different — and usually incompatible — ways of interpreting the same symbol is something that scientists, both modelers and experimentalists, have to worry about it.

If you want an actually funny joke of this type then I recommend the image of a ‘rigorous proof’ above that was tweeted by Moshe Vardi. My writen version was inspired by a variant on this theme mentioned on Reddit by jagr2808.

I will focus this post on the use of types from my experience with stoichiometry in physics. Units in physics allow us to perform sanity checks after long derivations, imagine idealized experiments, and can even suggest refinements of theory. These are all features that evolutionary game theory, and mathematical biology more broadly, could benefit from. And something to keep in mind as clinicians, biologists, and modelers join forces this week during the 5th annual IMO Workshop at the Moffitt Cancer Center.

## Operationalizing replicator dynamics and partitioning fitness functions

As you know, dear regular reader, I have a rather uneasy relationship with reductionism, especially when doing mathematical modeling in biology. In mathematical oncology, for example, it seems that there is a hope that through our models we can bring a more rigorous mechanistic understanding of cancer, but at the same time there is the joke that given almost any microscopic mechanism there is an experimental paper in the oncology literature supporting it and another to contradict it. With such a tenuous and shaky web of beliefs justifying (or just hinting towards) our nearly arbitrary microdynamical assumptions, it seems unreasonable to ground our models in reductionist stories. At such a time of ontological crisis, I have an instinct to turn — much like many physicists did during a similar crisis at the start of the 20th century in their discipline — to operationalism. Let us build a convincing mathematical theory of cancer in the petri dish with as few considerations of things we can’t reliably measure and then see where to go from there. To give another analogy to physics in the late 1800s, let us work towards a thermodynamics of cancer and worry about its many possible statistical mechanics later.

This is especially important in applications of evolutionary game theory where assumptions abound. These assumptions aren’t just about modeling details like the treatments of space and stochasticity or approximations to them but about if there is even a game taking place or what would constitute a game-like interaction. However, to work toward an operationalist theory of games, we need experiments that beg for EGT explanations. There is a recent history of these sort of experiments in viruses and microbes (Lenski & Velicer, 2001; Crespi, 2001; Velicer, 2003; West et al., 2007; Ribeck & Lenski, 2014), slime molds (Strassmann & Queller, 2011) and yeast (Gore et al., 2009; Sanchez & Gore, 2013), but the start of these experiments in oncology by Archetti et al. (2015) is current events[1]. In the weeks since that paper, I’ve had a very useful reading group and fruitful discussions with Robert Vander Velde and Julian Xue about the experimental aspects of this work. This Monday, I spent most of the afternoon discussing similar experiments with Robert Noble who is visiting Moffitt from Montpellier this week.

In this post, I want to unlock some of this discussion from the confines of private emails and coffee chats. In particular, I will share my theorist’s cartoon understanding of the experiments in Archetti et al. (2015) and how they can help us build an operationalist approach to EGT but how they are not (yet) sufficient to demonstrate the authors’ central claim that neuroendocrine pancreatic cancer dynamics involve a public good.
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## Space and stochasticity in evolutionary games

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.

## Truthiness of irrelevant detail in explanations from neuroscience to mathematical models

Truthiness is the truth that comes from the gut, not books. Truthiness is preferring propositions that one wishes to be true over those known to be true. Truthiness is a wonderful commentary on the state of politics and media by a fictional character determined to be the best at feeling the news at us. Truthiness is a one word summary of emotivism.

Truthiness is a lot of things, but all of them feel far from the hard objective truths of science.

Right?

Maybe an ideal non-existent non-human Platonic capital-S Science, but at least science as practiced — if not all conceivable versions of it — is very much intertwined with politics and media. Both internal to the scientific community: how will I secure the next grant? who should I cite to please my reviewers? how will I sell this to get others reading? And external: how can we secure more funding for science? how can we better incorporate science into schools? how can we influence policy decisions? I do not want to suggest that this tangle is (all) bad, but just that it exists and is prevalent. Thus, critiques of politics and media are relevant to a scientific symposium in much the same way as they are relevant to a late-night comedy show.

I want to discuss an aspect of truthiness in science: making an explanation feel more scientific or more convincing through irrelevant detail. The two domains I will touch on is neuroscience and mathematical modeling. The first because in neuroscience I’ve been acquainted with the literature on irrelevant detail in explanations and because neuroscientific explanations have a profound effect on how we perceive mental health. The second because it is the sort of misrepresentation I fear of committing the most in my own work. I also think the second domain should matter more to the working scientist; while irrelevant neurological detail is mostly misleading to the neuroscience-naive general public, irrelevant mathematical detail can be misleading, I feel, to the mathematically-naive scientists — a non-negligible demographic.

## Is cancer really a game?

A couple of weeks ago a post here on TheEGG, which was about evolutionary game theory (EGT) and cancer, sparked a debate on Twitter between proponents and opponents of the idea of using EGT to study cancer. Mainly due to the limitations inherent to Twitter the dialogue fizzled. Instead, here we are expanding ideas in this guest blog post, and eagerly await comments from the others in the debate. The post is written by Philip Gerlee and Philipp Altrock, with some editing from Artem. We will situate the discussion by giving a brief summary of evolutionary game theory, and then offer commentary and two main critiques: how spatial structure is handled, and how to make game theoretic models correspond to reality.
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## Models and metaphors we live by

George Lakoff and Mark Johnson’s Metaphors we live by is a classic, that has had a huge influence on parts of linguistics and cognitive science, and some influence — although less so, in my opinion — on philosophy. It is structured around the thought that “[m]etaphor is one of our most important tools for trying to comprehend partially what cannot be comprehended totally”.

The authors spend the first part of the book giving a very convincing argument that “even our deepest and most abiding concepts — time, events, causation, morality, and mind itself — are understood and reasoned about via multiple metaphors.” These conceptual metaphors structure our reality, and are fundamentally grounded in our sensory-motor experience. For them, metaphors are not just aspects of speech but windows into our mind and conceptual system:

Our ordinary conceptual system, in terms of which we both think and act, is fundamentally metaphorical in nature. … Our concepts structure what we perceive, how we get around the world, and how we relate to others. Our conceptual system thus plays a central role in defining our everyday realities. … Since communication is based on the same conceptual system that we use in thinking and actiong, language is an important source of evidence for what that system is like.

I found the book incredibly insightful, and in large agreement with many of my recent thoughts on the philosophies of mind and science. After taking a few flights to finish the book, I wanted to take a moment to provide a mini-review. The hope is to convincing you to make the time for reading this short volume.
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## Limits of prediction: stochasticity, chaos, and computation

Some of my favorite conversations are about prediction and its limits. For some, this is purely a practical topic, but for me it is a deeply philosophical discussion. Understanding the limits of prediction can inform the philosophies of science and mind, and even questions of free-will. As such, I wanted to share with you a World Science Festival video that THEREALDLB recently posted on /r/math. This is a selected five minute clip called “What Can’t We Predict With Math?” from a longer one and a half hour discussion called “Your Life By The Numbers: ‘Go Figure'” between Steven Strogatz, Seth Lloyd, Andrew Lo, and James Fowler. My post can be read without watching the panel discussion or even the clip, but watching the clip does make my writing slightly less incoherent.

I want to give you a summary of the clip that focuses on some specific points, bring in some of discussions from elsewhere in the panel, and add some of my commentary. My intention is to be relevant to metamodeling and the philosophy of science, but I will touch on the philosophy of mind and free-will in the last two paragraphs. This is not meant as a comprehensive overview of the limits of prediction, but just some points to get you as excited as I am about this conversation.