Simulate 2D Bronian motion
The script random_walk_w_confinement_fixed_wells.mat simulates a 2D Brownian motion, also called a random walk. Following Einsteins equation, displacements follow a Gaussian PDF with standard deviation given by k = sqrt(2 * D * dt) with D being the diffusion coefficient and dt the time step.
The script further offers the possibility to add areas of confinement to the trajectory. These are simply regions with a different diffusion coefficient D_conf (see below). Such trajectories can serve to test and calibrate means to identify confinements such as the confinement index. Look here for more details.
The input parameters are:
- D = 0.041; % diffusion coefficient, um2/s
- D_conf = D/100; % diffusion coefficient in confined areas, ?m2/s
- dwell_time = 100; % time the particle stays in the confined area
- start = [0 0]; % starting coordinates
- num_steps = 400; % number of time steps
- dt = 0.5; % interval between time steps, seconds
- dx = 1; % pixel size, um
- segment = 30; % Sm, segment length in frames see (Simson, Biophys J, 1995) for details
The script first calculates a random walk according to the defined parameters and displays the result including the two set confinement regions (red circles). Note that the confinement is set according to time, not space, i.e that particle will enter after a defined number of time steps and after leaving can not re-enter. To produce such a more realistic trajectory would require a modified script.
The following cell provides the option to save the trajectory as an animated plot such as the examples shown here.
The next part calculates the confinement index according to (Simson, Biophys J, 1995). The algorithm scans the trajectory searching for periods in which the molecule remains in a region considerably longer than expected for (free) Brownian motion. An input parameter is thus the diffusion coefficient as well as the radius of the confined region.
