Analytical solution of equations governing molecular-motor-assisted transport of intracellular particles

Using the method of separation of variables, a generalized Fourier series solution is obtained for a one-dimensional transient problem describing the motor-assisted transport of intracellular organelles in a cell arm (a dendrite or axon), which is oriented in the x-direction. The one-dimensional transport of organelles occurs between two boundaries, the base of the dendrite and its tip. A fraction of space within the dendrite is occupied by filaments; the remaining space allows diffusion of unbound particles in the x-direction. The filaments are divided into outward filaments (molecular motors running along these filaments transport particles in the direction of increasing x) and inward filaments (molecular motors running along these filaments transport particles in the direction of decreasing x). Unbound (free) particles may bind to filaments; particles attached to filaments may detach from them; the binding/detachment kinetic processes are specified by the first rate constants.