Movement of Flagellated Bacteria Fei Yuan Terry Soo Supervisor: Prof. Thomas Hillen Mathematics Biology Summer School, UA May 12, 2004

Know something about f lagellated bacteria before we start... Flagellated bacteria swim in a manner that depends on the size and shape of the body of the cell and the number and distribution of their flagella. When these flagella turn counterclockwise, they form a synchronous bundle that pushes the body steadily forward: the cell is said to “run” When they turn clockwise, the bundle comes apart and the flagella turn independently, moving the cell this way and that in a highly erratic manner: the cell is said to “tumble” These modes alternate, and the cell executes a three- dimensional random walk.

Our objectives Describe the movement of an individual bacterium in 2-D and 3-D space using the model of random walk; Add a stimulus into the system and study the movement of a bacterium;

What have we done so far? The simulation of 2-D unbiased random walk The simulation of 2-D biased random walk The simulation of 3-D unbiased random walk The simulation of 3-D biased random walk

Cartesian coordinates or polar coordinates? Cartesian coordinates Polar coordinates

2-D unbiased random walk We define theta to be the direction that a bacteria moves each step Theta ~ Uniform(0, 2*Pi) Step size = 1

2-D directional biased random walk First approach Second approach

First approach We tried in the 2-D space... Calucate the gradient Grad(s) as (Sx, Sy), (Sx, Sy) = || Grad(s) || * (cos (theta), sin(theta) ) Probability density function of phi is ( cos ( phi – theta ) ) / K K = normalization constant Calculate the actual angle that the bacteria moves by inversing the CDF of phi and plugging in a random number U(0, 1)

Second approach We tried in the 2-D space... Consider attraction, say to a point mass or charge, that is attraction goes as 1/r^2 Use a N(u,s) distribution where u = the angle of approach s is related to r.

New questions arise in the 3-D world...

Solution??? Say X is Uniform on the unit sphere and write X = (theta, phi) We want to compute the distribution functions for theta and phi Theta is as before: Uniform(0, 2*Pi) However Phi is not uniform(0, Pi) For the half sphere, it is sin(x)(1- cos(x))

3-D unbiased random walk

3-D biased random walk

Have more fun??!! Let a bacteria to chase another?

More work in the future Study the movement of a whole population of bacteria Consider the life cycle of the population during the movement Consider the species of bacteria Plot the mean squared displacement as a function of time

The end Thank you! Any question?