Details

In this project, you are going to write a program which draws the trajectory of an E. coli bacterium looking for food.

• The food is located at position, $(0,300)$.
• The bacterium starts out at position, $(x_0,y_0) = (0,0)$.
• The initial heading of the bacterium, $\theta_0$, is random.
• When not tumbling, the bacterium moves in a straight line with speed one pixel per second.
• The distance between the bacterium and the food can be computed using the following formula:

  
  

  $$d_t = \sqrt{x_t^2 - (y_t - 300)^2}$$
  
  

  where $d_t$ is the distance of the bacterium from the food, $x_t$ and $y_t$ are the $x$ and $y$ coordinates of the bacterium's position (all at time, $t$).

• The probability that the bacterium will tumble at any given time can be computed using the following formula:

  
  

  $$p_t = \frac{1}{1 + e^{-100 (d_t - d_{t-1})}}$$
  
  

  where $d_{t-1}$ and $d_t$ are the distances of the bacterium from the food at time, $t-1$ and $t$. Notice that $p_t$ is nearly one when $(d_t - d_{t-1}) > 0$ (the bacterium is going in the wrong direction) and nearly zero when $(d_t - d_{t-1}) < 0$ (the bacterium is going in the right direction).

• The heading of the bacterium at time $\theta_t$ remains the same with probability, $1 - p_t$, and changes by $\breve{\phi}$, a random number between 0 and 359, with probability, $p_t$:

  
  

  $$\theta_t = \left\{ \begin{array}{cll} \theta_{t-1} & {\rm wi... ...1} + \breve{\phi} & {\rm with\;prob.} & p_t \end{array}\right.$$
  
  

  where $\breve{\phi}$ is a random number between 0 and 359.

• The position of the bacterium at time $t$ can be computed from the position at time $t-1$ and the heading using the following equations:
  $$\begin{eqnarray*} x_t & = & x_{t-1} - \sin \theta_{t-1}\\ y_t & = & y_{t-1} + \cos \theta_{t-1} \end{eqnarray*}$$
  
  
  
  
  

  where $\theta_{t-1}$ is the heading of the bacterium at time, $t-1$.

• The bacterium stops when it is less than one pixel from the food, i.e., $d_{t-1} < 1.0$.