Fig. 2

Binding dynamics of the one-box system with limited trap density. a One-box system with finite trap density. The quasi-twodimensional synthesis chip of height \(h\) and length \(l\) with \(h\ll l\) is filled with a solution of proteins (orange spheres) that bind to molecular traps (yellow cones) to form protein-trap complexes. b Number \(\rm{N}_\mathrm{b}(t^*)\) of bound proteins at \(t=t^*\) and c maximum of the trap-binding efficiency \(\rm{f}_\mathrm{b}^\mathrm{max}=\rm{f}_\mathrm{b}(t^*)\) for different ratios of synthesis rate \(\alpha _\mathrm{syn}\) and binding affinity \(\kappa _\mathrm{bin}\). d Number \(\rm{N}_\mathrm{b}(t)\) of proteins bound to traps normalized to the maximal value \(\rm{N}_\mathrm{b}^\mathrm{max}\) and e time-evolution of the trap-binding efficiency \(\rm{f}_\mathrm{b}(t)\) for different choices of synthesis rate \(\alpha _\mathrm{syn}\) and binding affinity \(\kappa _\mathrm{bin}\). Note that the maximum of the trap-binding efficiency \(\rm{f}_\mathrm{b}^\mathrm{max}\) is determined by the ratio of synthesis rate \(\alpha _\mathrm{syn}\) and binding affinity \(\kappa _\mathrm{bin}\), see also (c). For the light blue and magenta lines this ratio is one, thus \(\rm{f}_\mathrm{b}^\mathrm{max}\) is identical but reached at different time points. Background color indicates binding-dominated (orange) and synthesis-dominated (gray) regime for \(\alpha _\mathrm{syn}= {0.001\,\mathrm{\text {s}^{-1}}}\) and \(\kappa _\mathrm{bin}= {0.1\,\mathrm{\text {s}^{-1}}}\). Transition from the former to the latter occurs at \(t= t^*\), i.e., when the maximum \(\rm{f}_\mathrm{b}^\mathrm{max}\) of the trap-binding efficiency is reached