T Cells, Predators and Finances all Walk the Levy Walk


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Lévy Distributions by Krishnavedala

A recent report from the University of Pennsylvania said that a study on mice revealed that the immune system’s T cells “hunt” parasites in a “Lévy walk” pattern.

This brings up a few questions for most of us, such as: Do T cells really walk? Who was Lévy, and …  Just what is a “Lévy walk, anyway?”

A Simple Description of a Lévy Walk

The researchers described the T cells’ “Lévy Walk” as having many short movements in one location, then a long move to another area where short movements are dominant. Many predators use such a pattern when hunting for prey that hides well.

A shark, for example, may hunt rather carefully near one reef, then swim much further afield to begin a new search elsewhere.

According to this research, a T cell seeks hidden parasites, such as Toxoplasma gondii, using a similar pattern of movement.

Introducing the Mathematics of a Lévy Walk

Usually known as a “Lévy Flight,” the Lévy Walk is a specific type of random movement based on the statistical Lévy distribution. This is similar to, but distinct from, the familiar “normal distribution” in the shape of a “bell curve.” In mathematics, a “random walk” describes a random series of discrete movements or steps. Unless a particular random walk is under discussion, it is generally assumed that each “step” is the same length, and each step is in a random direction. Brownian motion is an example from physics of a random walk in three dimensions, caused by random thermally induced collisions among molecules of a liquid. A Lévy Flight is a random walk with steps of different lengths. Most steps are short, and since the direction can change with each step, the path usually stays in a confined area. After a random number of short steps, a long step leads to a new area for short steps to “explore.”

Levy Flight showing Brownian Motion: Image by PAR

This image of Brownian motion shows an example of 1000 steps of an approximation to a Brownian-motion-type of Lévy flight in two dimensions. The origin of the motion is at [0, 0], the angular direction is uniformly distributed, and the step size is distributed according to a Lévy (i.e. stable) distribution with α=2 (“alpha = 2“) and β=0 (“beta = zero“); i.e. a (Normal distribution).” The image of distributions varies the parameter ‘c’, known as a “scaling parameter” in probability theory. As ‘c’ increases, the distribution spreads farther to the right and has a lower peak. Note that these distributions are graphed as smooth curves, but a “random walk” presents a haphazard zigzag path because a new direction is chosen based on the probability distribution.

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