  In cells and in vitro assays the number of motor proteins involved in
biological transport processes is far from being unlimited. The cytoskeletal
binding sites are in contact with the same finite reservoir of motors (either
the cytosol or the flow chamber) and hence compete for recruiting the available
motors, potentially depleting the reservoir and affecting cytoskeletal
transport. In this work we provide a theoretical framework to study,
analytically and numerically, how motor density profiles and crowding along
cytoskeletal filaments depend on the competition of motors for their binding
sites. We propose two models in which finite processive motor proteins actively
advance along cytoskeletal filaments and are continuously exchanged with the
motor pool. We first look at homogeneous reservoirs and then examine the
effects of free motor diffusion in the surrounding medium. We consider as a
reference situation recent in vitro experimental setups of kinesin-8 motors
binding and moving along microtubule filaments in a flow chamber. We
investigate how the crowding of linear motor proteins moving on a filament can
be regulated by the balance between supply (concentration of motor proteins in
the flow chamber) and demand (total number of polymerised tubulin
heterodimers). We present analytical results for the density profiles of bound
motors, the reservoir depletion, and propose novel phase diagrams that present
the formation of jams of motor proteins on the filament as a function of two
tuneable experimental parameters: the motor protein concentration and the
concentration of tubulins polymerized into cytoskeletal filaments. Extensive
numerical simulations corroborate the analytical results for parameters in the
experimental range and also address the effects of diffusion of motor proteins
in the reservoir.
