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.