Fig. 3

Binding dynamics in a two-box system with diffusion rates \(\widetilde{D}={0.001\,\mathrm{\text {s}^{-1}}}\) (left) and \(\widetilde{D}={0.1\,\mathrm{\text {s}^{-1}}}\) (right), and synthesis rates \(\alpha _\mathrm{syn}\) and binding affinities \(\kappa _\mathrm{bin}\) as indicated. a The two-box system with box height \(h\), box length \(l\), and \(h<< 2\,l\). Proteins (orange spheres) can freely diffuse between both boxes and bind to traps (yellow cones). New proteins are synthesized only in the left box (blue). b Numbers \(\rm{N}_{i,\rm{b}}(t)\) of bound proteins in box \(i=1\) (dashed lines) and \(i=2\) (solid lines). Because synthesis is restricted to box 1, saturation of traps in box 2 is lagging behind. This lag is not observable in well-mixed systems with high diffusion rates. c In two-box systems, the trap-binding efficiency \(\rm{f}_\mathrm{b}(t)\) exhibits a pure binding-dominated (\(0 \le t < t^*_\mathrm{1}\)), a mixed (\(t^*_\mathrm{1} \le t < t^*_\mathrm{2}\)), and a pure synthesis-dominated regime (\(t \ge t^*_\mathrm{2}\)) as indicated by background shading. d) Individual trap binding efficiencies \(\rm{f}_\mathrm{i,b}(t)\equiv \rm{N}_{i,\rm{b}}(t)/\rm{N}_i(t)\) of box \(i=1\) (dashed lines) and \(i=2\) (solid lines). Transitions from the binding-dominated to the synthesis-dominated regimes in the boxes 1 and 2 occur at time points \(t^*_\mathrm{1}\) and \(t^*_\mathrm{2}\), respectively. Arrows point to the maxima of \(\rm{f}_\mathrm{1,b}(t)\) and \(\rm{f}_\mathrm{2,b}(t)\) at \(t = t^*_\mathrm{1}\) and \(t = t^*_\mathrm{2}\), respectively