FIG. 4.

Effects of domain walls and finite size on density profiles. Left, determination of domain wall positions in the phase-plane trajectory and the density profile (inset). Red points show kMC simulation results. The boundary points are $({\sigma }_R,{\sigma }_L)=(-\frac{1}{2},\frac{1}{2})$ and $({\sigma }_R,{\sigma }_L)=(\frac{1}{2},-\frac{1}{2})$. The solution locally follows the flow, which cannot connect the boundary points without crossing the lines with σ_R = 0 and σ_L = 0 where the velocity is ill defined. Crossing these lines uses the matching condition to connect to another exact solution curve, introducing a domain wall. Inset shows the density profiles of ρ_R (purple, dashed) and ρ_L (brown, solid). The sharp transitions in ρ_R and ρ_L indicate the domain walls. There are three regions which are separated by two domain walls: regions near ends follow C₁, and the region at the center is described by C₂. Right, finite-size constraint. The blue points (longer curve) are kMC simulation results for N = 1000, and the green (shorter curve) N = 500. For a larger number of sites, the dimensionless motor speed is smaller, moving the solution closer to the LI. The switching rate is 0.44 s⁻¹(left), 0.1 s⁻¹(right), the bulk motor concentration c = 200 nM, and the motor speed 5 μm s⁻¹; other parameters are the reference values of Table I. Arrows indicate the vector field, which has the mathematical form of Eqs. (12) and (13).