Random walk in spherical coordinates

[arandall]

Hi,

I'm modeling receptors moving along a cell surface that interact with proteins inside of a cell. I figured it would be easier to model the receptors in spherical coordinates, however I'm unsure of how to model a random walk. In cartesian coordinates, I basically model a step as:

x = x + sqrt(6*D*timeStep)*randn
y = y + sqrt(6*D*timeStep)*randn
z = z + sqrt(6*D*timeStep)*randn

Where D is my diffusion constant. How can I do this just using theta and phi? Modeling random walk in spherical coordinates will be really nice, because I can fix r such that the receptors can't leave the membrane of the cell, and just focus on how it moves in 2D with respect to the membrane.

 

[zhermes]

Because the receptors are so much smaller than the size of the cell, it should be fine if you treat theta and phi just like x-y; i.e. pretend its a 2D random walk in Cartesian coordinates.

 

[mathman]

To choose points on the surface of a sphere uniformly, the two angles should be chosen as follows (I'll call them latitude and longitude):

Longitude (θ) - θ uniform between 0 and 2π.
Latitude (φ) - sinφ uniform between -1 and 1.