The Noble Eightfold Path to Mathematical Biology

Twitter is not a place for nuance. It is a place for short, pithy statements. But if you follow the right people, those short statements can be very insightful. In these rare case, a tweet can be like a kōan: a starting place for thought and meditation. Today I want to reflect on such a thoughtful tweet from Rob Noble outlining his template for doing good work in mathematical biology. This reflection is inspired by the discussions we have on my recent post on mathtimidation by analytic solution vs curse of computing by simulation.

So, with slight modification and expansion from Rob’s original — and in keeping with the opening theme — let me present The Noble Eightfold Path to Mathematical Bilogy:

1. Right Intention: Identify a problem or mysterious effect in biology;
2. Right View: Study the existing mathematical and mental models for this or similar problems;
3. Right Effort: Create model based on the biology;
4. Right Conduct: Check that the output of the model matches data;
5. Right Speech: Humbly write up;
6. Right Mindfulness: Analyse why model works;
7. Right Livelihood: Based on 6, create simplest, most general useful model;
8. Right Samadhi: Rewrite focussing on 6 & 7.

The hardest, most valuable work begins at step 6.

The only problem is that people often stop at step 5, and sometimes skip step 2 and even step 3.

This suggests that the model is more prescriptive than descriptive. And aspiration for good scholarship in mathematical biology.

In the rest of the post, I want to reflect on if it is the right aspiration. And also add some detail to the steps.

Given how tempting it is to stop at step 5, Jorge Peña suggests that we start at step 6. And as long as we first do step 2 then I think skipping to 6 can be a good idea, especially for a theorist outside of biology entering into biology. From my experience, there are a lot of models and papers in biology that just focused on step 3, 4, and 5. This means that there is a whole world of different models that aren’t formally related to each other. As I’ve discussed before, I believe that techniques from theoretical computer science can be used to abstract over and unify these models. Rather that understanding why your own models works and abstracting it, it can be even more rewarding to understand why other people’s models work, finding a common explanation, and abstracting it out. I think that in the long term, this kind of work is of fundamental importance.

This highlights an important feature of the Noble Path. It doesn’t need to be followed by a single scientist or work, but can be followed by a field. If you’re seeing that a lot of people in your area stopped at step 5, just continue their work from step 6.

It is important to note that this post-5 abstraction process doesn’t need to be theoretical.

We can also consider empirical abstraction. Most commonly, steps 3 and 4 are done as two parallel tracks. The model and experiment are based on some common conceptual grounds, but the model dynamics are project forward from some parameters on one track, and the experiments are run separately. Then the results of both are compared — often qualitatively. The abstraction process can then aim to unify these two tracks. By abstracting the model, it can be changed from a model to a measurement. In that case, the experiment ends up directly defining or selecting the model.

However, sometimes there is not enough existing work, or a clear enough question and intuition to start at step 6. This is something that Patrick Ellsworth and Rob Noble mentioned in our earlier discussion. Sometimes you don’t realize that something is a problem in need of an explanation until you’ve gone through steps 3,4 and 5. Often these steps are required to build the basic tools with which to generate a field-endogenous question which can then be answered by a more abstract and general theory. In these cases, we might not be able to follow Jorge straight to step 5 because we don’t yet understand what the relevant ‘works’ is that needs a ‘why’.

But good work cannot stop at step 5. Or at least, we should aspire to not stop at 5.

This reminds me of a post where Philip Gerlee wrote (and Jacob Scott agreed in the comments) that expecting both novel mathematics and novel biology from an exploration in an interdisciplinary field like mathematical oncology is asking for too much. Science is about keeping as many parameters fixed as possible and trying to vary only one at a time. I am not sure if I agree.

I think that if we follow the Noble Eightfold Path then we can achieve both biology and mathematical novelty. In steps 3 to 5, we usually use standard approaches and thus are unlikely to yield any mathematical novel results. What keeps us going through these steps is the biologically novel results. However, when we turn to explaining why our model works — that is where we have the space for mathematically novel results. This doesn’t introduce new parameters, rather it helps us understand our existing parameters. Figure out which of the countless knobs of the original model actually matter. Good abstraction should make us more confident in our results.

Of course, this ideal of interdisciplinary should not be a requirement for any work that a mathematical biologist does. Rather, it should be a requirement for calling that work interdisciplinary. A mathematical biologist can do steps 3 to 5 and produce a work of biology. Or she can do steps 6 and 7 and produce a work of applied mathematics. But only when she follows the whole eightfold path, does she reach a truly interdisciplinary state — a contribution to both worlds. Of course, this is more difficult to do. But if we are going to praise interdisciplinary (as people tend to do) then it makes sense for it to be more difficult.