Random Walks Model Diffusion

The wanderlust of a randomly walking particle

You may feel like a single, coherent being, but you are just a skin-covered bag of trillions of cells that act largely independently. Cells are full of proteins, complex macromolecules that perform nearly every cellular function. If a protein could move in a straight line, then it would move at 20 kph or faster¹, meaning that the protein would cover a distance 1 billion times its length every second (analogous to a car traveling at 20 billion kph). However, the cytoplasm filling the cell is so densely packed with water molecules that the protein ping-pongs off them, frequently changing direction.

We will model the movements of a cellular particle such as a protein by a random walk in a two-dimensional plane. At each step, the particle moves a single unit of distance in a randomly chosen direction. The video below shows a randomly walking particle taking 1000 steps.

The distance that the particle wanders from its starting point may surprise you. And yet perhaps this particle is just an outlier, and the typical particle would be much more of a homebody.

From one particle to many

If we animate the action of many independent particles following random walks, then although some particles hug the starting point and some wind up far away, most particles steadily drift outward. The following video shows a simulation of multiple randomly walking particles that all begin their walk at the same point and that diffuse throughout their environment.

Although the movements of a single particle are random, we can draw conclusions about the average-case behavior of many particles can be predicted, as the following theorem indicates.

Random Walk Theorem: After n steps of unit length in a random walk, a particle will on average find itself a distance from its origin that is proportional to \(\sqrt{n}\).

Note: If you love mathematics and are interested in seeing a proof of this theorem, click here.

Our experience of the world confirms the Random Walk Theorem’s statement that randomly walking particles tend to drift away from their starting point. We understand, for example, that someone who has a respiratory virus can infect many others within an enclosed space as the viral particles expand outward. We also know that when a cake is baking in the oven, we will not need to wait long for delicious smells to waft from the kitchen.

If you are interested in seeing how to build the random walk simulation shown in the video above, then please visit the following software tutorial. This tutorial uses CellBlender, an add-on to the popular open graphics software program Blender, which allows us to create and visualize biological models. These models rely on particle-based reaction-diffusion simulations that are implemented by the program MCell. We will use this software for our work in biological modeling in this prologue as well as module 1.

Note: We have designed this course so that you can appreciate the key ideas behind the biological models that we build without following software tutorials. But we also provide these tutorials so that you can explore the modeling software that we have used to generate our conclusions.

Visit tutorial

Brownian motion: big numbers in small spaces

Later in this course, we will see that random walks power a simple but powerful approach that bacteria like E. coli use to explore their environment in the hunt for food. In the next lesson, we will see that randomly moving particles can produce high-level patterns if the particles interact when they collide.

Before continuing, we will point you to a beautiful animation illustrating just how far a single randomly moving particle can travel in a relatively small amount of time. This animation, which shows a simulation of the path taken by a glucose molecule as the result of Brownian motion, starts at 6:10 of the following excellent video developed by the late Joel Stiles.