Diffusion in angles - Explore the surface area of a sphere

Simulate Diffusion on the surface of a hemisphere.

These programs simulate a model describing the motion of anchored motor proteins interacting with biopolymers. Explicitly this simulation is proposed to explain the motion of a fluorescent molecule tagged to the heavy chain of the cytoskeletal motor protein dynein when it is bound to the cell cortex.

It is assumed that the fluorescent label of a single dynein when bound to the cell cortex essentially travels about the surface area of a hemisphere. In the presence of a bio-polymer, for instance a microtubule, the total surface area that the fluorescent molecule can be explore shrinks; due to the propoensity of dynein to interact with the microtubule.

The programs simulate motion in spherical coordinates and map the trajectory to cartesian space to match the signal output collected using single molecule imaging techniques (I.E. single molecule imgaging collects the position of the fluorescent molecule in a 2D plane (x,y)).

The polar angle theta is generated by a Gaussian (normal) distribution. While the azimuthal angle (phi) is dictated by a Uniform distribution. Angles are selected at random for a fixed "radial" coordinate at each time step to generate a trajectory. The coordinates are mapped to cartesian coordinates, and the displacements between successive positions calculated.

Refer to https://www.biorxiv.org/content/10.1101/2022.09.11.507465v1 for more information.