Transcript Mathematics Biology Summer School Project Movement of

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 threedimensional 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 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?