Consider a spherical cow…

An inspirational spherical cow in my office.

My wife and I were on a refreshing vacation in Canada, walking around the cute town of Banff in the crisp morning air, and it was time for some ice cream. On past days, we had seen crowds gathered later in the afternoon at this ice cream shop, Cows, but it was shockingly empty at the time we entered. It was a good thing we did, though; not only did I get some delicious ice cream (good at any time and in any weather, I believe), but I found a very cute, roughly spherical cow. It now sits atop my computer in my office, encouraging me as I work on modeling cell biophysics.

My excitement over this purchase (and its inspiration role in my research) may require some further explanation. I recognize that mammals are not usually spherical, aside from in some very entertaining cartoon videos on YouTube. But in the physics and physics-adjacent community, spherical cows have become emblematic of a general approach to problem-solving.

Best I can tell from my extensive Wikipedia research [1], the origin of the “spherical cow” metaphor is a marginally amusing joke told at the slight expense of physicists. In brief, a farmer wishes to increase dairy production on his farm, and in a moment of extreme impracticality, phones up the university and hires a group of theoretical physicists, who tirelessly slave away on their chalk boards and eventually come back with a solution. The catch is their solution applies only when the cows are approximated as spherical bodies in a vacuum.

I can’t quite recall when I first came across this “spherical cow” idea (perhaps sometime during my years as an undergraduate), but I’ve become increasingly acquainted with this metaphor the many hours I’ve spent reading and listening to one of my favorite theoretical physicists, Sean Carroll. The idea comes up quite often in his Mindscape podcast, and features prominently in his newest book, "The Biggest Ideas in the Universe, Part 1”. In his words, the essential workflow behind this spherical cow approach is: “idealize a difficult problem down to a simple one by ignoring as many complications as you can. Get an answer to the simple problem. Then put the complications back in and calculate how they affect the answer to the simple problem.”

The crucial point behind this “spherical cow” business is that a sphere is much, much simpler than the actual geometry of a cow. To specify the geometry of the sphere, we only need to supply one number - the radius. Depending on what we are trying to model, we might consider other possible geometries with added complexity; perhaps a cylinder (requiring both a radius and a length) or a rectangular prism (requiring us to specify three lengths of sides). If we were to try to capture the cow in more detail (say, to the nearest centimeter), we would need to provide a large list of surface coordinates; if we’re lucky, we could get by treating it as bilaterally symmetric, cutting the amount of coordinates required in half.

The remarkable thing about this simple approach to modeling is that it works. Depending on what we hope to model, we might choose a different level of simplification (a different level of “coarse-graining”, some might call it). If we want to follow the trajectory of a cow jumping off a small ledge (don’t try that at home), we could probably just get by treating it as a point mass, or a sphere if we must. But if we want to model the biomechanics of cow walking, we likely require much greater detail. As succinctly put by that one guy (Albert Einstein), “Everything should be made as simple as possible, but no simpler”.

This thinking is not usually applied to cows, so let me give a couple concrete examples from physics. For starters, we can consider how Isaac Newton was able to show that planets orbit the sun in elliptical paths. At first glance, this may seem a monumental problem; certainly now we recognize the sun and planets are both extremely complicated physical entities. However, it turns out that if you treat each as point masses (the simplest possible geometry, effectively 0-dimensional) attracted according to an inverse-square law for gravity, the elliptical paths come out naturally. Newton essentially needed to invent calculus to solve this problem, but the basic idea is surprisingly simple and astoundingly accurate!

Perhaps the most powerful example of spherical cow thinking is something Sean Carroll arrives to at the end of his new book - black holes. Not long after Einstein wrote up his equations of general relativity, a solution was found that corresponded to a spherical massive body surrounded by empty space (the Schwarzschild solution). This solution now acted like the inverse square law in the Newtonian case mentioned above, but in the updated framework in which gravity corresponds to the warping of space-time (confusing, yes). Remarkably, this spherically symmetric solution was found to predict the existence of black holes - massive bodies which warp space-time to such an extreme level that nothing (not even light) can escape them. In other words, simply considering a spherical body in empty space (considerably simplified from what we might encounter in real-life) led to one of the most astounding predictions of 20th century physics!

Consider an axially symmetric cell… [2]

Now, lest you think we can’t really apply spherical cow thinking to biology, I’ll give a final example from my own research. My most recent papers have been addressing the mechanisms by which immune cells deform and spread over surfaces [2, 3]. Admittedly, this is a terribly complicated process, involving convoluted signaling cascades, rearrangements of structural proteins within the cell, production and use of chemical energy (ATP), and much more. However, rather than try to recapitulate the cell behavior in all its glory, we decided to treat the cell as a axially symmetric fluid body surrounded by an elastic boundary. Almost like a very small water balloon filled with a very viscous fluid (over 10,000 times the viscosity of water). If you recall anything about actual cells, this may sound somewhat ridiculously simplified. But it’s all a matter of the question being asked; if we wanted to uncover the effects of cell spreading on deformations of the nucleus, our model wouldn’t tell us a single thing (our model cell has no nucleus). But if we are asking questions about whole-cell deformations (which we were), this model could tell us quite a bit. Indeed, it showed us that immune cell spreading cannot realistically be driven by just passively adhering to a surface, but it requires the cell to exert forces from within (read more in my previous blog post).

Simulation of calcium dynamics for the realistic geometry of a small section of a neuron [4]. Unfortunately, far from spherical.

Hopefully this gives you some idea of how “spherical cow thinking” is quite useful in all different types of physics. This isn’t to say that the complexities of the real-world don’t give rise to interesting questions as well. For instance, in some of my work now, I’m using novel computational tools to examine how realistic cell geometries affect the dynamics of cell signaling events (see an example of simulations using these complicated geometries from previous work in the Rangamani Lab [2]). But of course, these models are still immensely simplified from the underlying molecular dynamics (and quantum mechanics) required to capture a more-or-less complete picture. It all comes down to the questions you wish to ask and the tools you have available.

References:

[1] Spherical cow. Wikipedia (2022).

[2] Francis, E. A. & Heinrich, V. Integrative experimental/computational approach establishes active cellular protrusion as the primary driving force of phagocytic spreading by immune cells. PLoS Comput Biol 18, e1009937 (2022).

[3] Francis, E. A., Xiao, H., Teng, L. H. & Heinrich, V. Mechanisms of frustrated phagocytic spreading of human neutrophils on antibody-coated surfaces. Biophysical Journal 121, 4714–4728 (2022).

[4] Lee, C. T. et al. 3D mesh processing using GAMer 2 to enable reaction-diffusion simulations in realistic cellular geometries. PLoS Comput Biol 16, e1007756 (2020).