Movement of Flagellated Bacteria

Fei Yuan Terry Soo

Supervisor:Prof. Thomas Hillen

Mathematics Biology Summer School, UA

• May 12, 2004

Know something about flagellated 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 ) + 1.2 ) / 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 sphereand 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?