Real Mathematics – Life vs. Maths #6


The word itself sounds like a taxi application for smart phones, and that is not a bad metaphor for its true meaning: Chemotaxis is the movement of cells (or organisms) towards a source of a chemical gradient. This chemical gradient could either be good for the organism or bad for it.

If there is a chemical gradient in an environment that is good for the organism, it would move towards that environment. If the chemical gradient is bad for the organism’s life, it would move away from that environment. These movements are called Chemotaxis.

According to the illustrations red matter is bad, blue is good for the organism.

Random Path

In the previous article I talked about one-dimensional biased random walks. Now let’s use all our knowledge to create a two-dimensional biased random walk.

There were four possible outcomes for two-dimensional random walk: Up, down, right and left. Each and every one of these outcomes has the same probability that is ¼. That means taking one step in any of those directions have the same probability. In order to create a two-dimensional biased random walk let us keep the probabilities same but change the number of steps taken for right and upward directions.

Up: ¼ probability, 2 steps.

Down: ¼ probability, 1 step.

Right: ¼ probability, 2 steps.

Left: ¼ probability, 1 step.

This change will give biased results. If one would take N random steps under these conditions, that person will likely end up at a point which is located up-right hand side of the origin:

To illustrate a two-dimensional biased random walk I rolled a 12-sided dice 10 times (1-2-3 upwards, 4-5-6 downwards, 7-8-9 right, 10-11-12 left).

As seen in the pictures a two-dimensional biased random walk and Chemotaxis are essentially same with one another. If there was a food source for an organism on the up-right hand side of the origin, that organism would make movements as shown in the picture.

Now let’s analyze these illustrations:

On left we are looking at a random walk of an organism without any chemical presence in the environment. On right blue dots are food sources for the organism, which is why the organism is moving randomly towards where those blue dots are dense. If the food source is denser on the up-right, organism will make random movements towards that area.

This is a real life example for the biased random walk. Let’s get more real though.

Dive and Tumble

E. coli is a type of bacteria that could be fatal in some cases. Sugar is one of its main nutritional sources. This is why E. coli bacteria choose to move randomly towards areas where there is sugar. This makes E. coli a biased random walker.


There are organelles that look like whips on E. coli bacterium. These organelles (or propellers) help E. coli to move in two ways. Whenever these propellers are lined in the clockwise direction, they make bacteria to move forward, like swimming or diving. And whenever the propellers rotate counter-clockwise direction the bacteria makes a tumble that helps to change direction. Hence we call the two fundamental movements of E. coli dive and tumble.

E. coli uses dive and tumble in order to survive, and these movements are in fact biased random walks.

E. coli bacteria will always make random movements toward the sugar if there is some in an environment.


Even one-cell organisms that have no brains use mathematics for their survival.

Then, how is it still possible to see the smartest known creatures (humans) claim that they don’t need mathematics in their lives?

Human beings must use mathematics in order to understand the events that are occurring around them. Because mathematics is the only language we use for to explain those events.

Now that you are aware of this fact, try to find the mathematics behind the things that are happening around you. (Mighty Google at work!) Don’t hesitate to contact me in case you have questions. I’ll be glad to assist you.

For Those Interested In Random Walks: Random Walks in Nature

Brownian Motion: It is the random movement of particles in a fluid that are colliding with other atoms/molecules inside the very same fluid.

Robert Brown is the first person whom observed this sort of movement. In 1827, he was working with pollens and he observed that they are constantly and randomly moving inside the water. But he was not able to explain the reason behind the movement. It was Albert Einstein whom explained this phenomenon in one of his 1905 papers. (Click here to see Brownian movement.)


One of the easiest and most observable real life examples for Brownian motion is the movements of dust particles. Try it with some light in a dark room; you will be mesmerized with their random dance. The reason behind this movement is that the dust particles are actually colliding with invisible air atom&/molecules. You could imagine dust particles as ping pong and air molecules as racket. As you can see Brownian motion can be observed both in liquid and in gas.

M. Serkan Kalaycıoğlu


Leave a Comment

Fill in your details below or click an icon to log in: Logo

You are commenting using your account. Log Out /  Change )

Twitter picture

You are commenting using your Twitter account. Log Out /  Change )

Facebook photo

You are commenting using your Facebook account. Log Out /  Change )

Connecting to %s