End-to-end distance trajectory of a polymer chain. Transition paths from A to B are the trajectory segments, where the end-to-end distance, R, enters the transition region at R A and reaches R B without first returning to R A . The transit time t AB is the time spent by a transition path within the transition region R B < R < R A . In contrast, trajectory segments that start at R A and later exit the transition region into A prior to reaching R B (duration t AA ) will contribute to the mean first passage time from A to B but not to the mean transit time.
Transit times vs first passage times. The mean transit time from A to B (filled symbols) is independent of the chain length N, in contrast to the strongly chain-length-dependent mean first passage from A to B (empty symbols). Here both times are plotted as a function of the spatial extent of the transition region R A − R B , with R B fixed at 2.5 equilibrium bond lengths.
Polymer model vs Free diffusion model vs 1D Smoluchowski model for polymer cyclization transit times. The mean transit times for cyclization of a polymer chain (filled circles) are independent of the chain length. For small values of R A − R B they agree with the free diffusion model (open circles), which ignores the polymer chain and assumes that the end monomers move freely. In contrast, the model of Eq. (2) (solid, dashed, and dotted lines) that assumes 1D diffusion in a one-dimensional potential of mean force is inconsistent with the polymer data and shows significant dependence on the chain length. As in Fig. 2, the distance R B is fixed here at 2.5 equilibrium bond lengths.
Cyclization of a polymer chain can be achieved via straightening of a chain segment containing n ≪ N monomers.
The effective number of monomers rearranging in a cyclization transition [estimated from Eqs. (5) and (6)]. The effective number of monomers n(R A , R B ) involved in the cyclization of a chain increases with the change in the end-to-end distance R A − R B required to close the loop. For small values of R A − R B loop closure is accomplished by moving only n(R A , R B ) = 2 end monomers. Here R B is fixed at 2.5 bond lengths. Dashed line shows Eq. (4), which predicts that n(R A , R B ) is proportional to the distance R A − R B traveled during a transition.
The distribution of transit times obtained from the free diffusion model, when rescaled according to Eq. (7) to account for the effective number of monomers involved, is nearly identical to the actual distribution obtained from simulations of polymer chains. (a): N = 160, R A = 10.39, and R B = 2.5. (b): N = 160, R A = 25.97, and R B = 2.5.
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