Movement of Flagellated Bacteria

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

2. 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.

3. 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;

4. 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

5. Cartesian coordinates or polar coordinates? Cartesian coordinates Polar coordinates

6. 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

7. 2-D directional biased random walk First approach Second approach

8. 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)

9. 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.

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

11. 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))

12. 3-D unbiased random walk

13. 3-D biased random walk

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

15. 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

16. The end Thank you! Any question?