This is part of my “journal club for credit” series. You can see the other computational neuroscience papers in this post.

## Unit: Diffusion

Organized by Ben Regner

- Standard Diffusion
- Anomalous Diffusion
- Life at Low Reynold’s Number

## Papers

Life at Low Reynold’s Number. By Purcell in 1977.

## Other Useful References

## Introduction

This is one of my favorite papers. The presentation style is extremely fun and readable without sacrificing any scientific integrity. I think it serves as a great introduction to fluid mechanics at low Reynold’s number. I don’t have too many comments since I think the paper explains it the best, but I will provide a few supplementary details for a more in depth exploration of the ideas from the paper.

And just to get you excited about fluid dynamics, I present an example of laminar flow:

## Basics of Fluid Mechanics

The fundamental equation of fluid mechanics is Navier-Stokes. The relevant version for this paper is the incompressible flow equations with pressure but no other external fields:

where is the velocity vector, is position, is density, is pressure, and is the kinematic viscosity. This equation can be made non-dimensional by the introduction of a characteristic velocity , length , and introducing the dynamic viscosity . This gives the following dimensionless variables:

Substituting in these characteristic length scales and doing some algebra, one arrives at the simplified equations:

with only one dimensionless constant, the Reynold’s number, defined as:

As explained in the paper, Reynold’s number is one of the essential constants describing a flow. High Reynold’s number leads to turbulent (chaotic) flow, while low Reynold’s number leads to laminar (smooth) flow. For extemely small Reynold’s number, Navier-Stokes simplifies to:

which is also just called Stoke’s equation.

At the end of the paper, Purcell describes another dimensionless number which he calls and in a footnote identifies as the Sherwood number. However, Ben Regner pointed out, that Purcell’s would actually be called the Peclet number today.

## Basics of Ecoli Chemotaxis

Chemotaxis and cellular sensing really deserves its own series of papers. But in the meantime, I recommend the following resources

- Chemotaxis on Wikipedia
- Howard Berg’s videos on individual Ecoli
- Howard Berg’s videos on swarms of Ecoli
- Berg and Purcell, Physics of chemoreception, 1977.

## Video Proof of Purcell’s Scallop Theorem

Reversible kicking does fine in water (high Reynold’s number)…

… but the same motion has issues in corn syrup (low Reynold’s number).

Here is a solution similar to what Ecoli and other bacteria employ.

## Fundamental Questions

- Purcell does an amazing job, so I have nothing to add.

## Advanced Questions

- What are some other strategies that are employed in biology to get around the issue of mobility at low Reynold’s number? Hint: I already linked to a video of one strategy. There are at least two other strategies, but to find these you will need to think about the assumptions leading to the basic Navier-Stokes equations.