We study the dynamics of the passage of a polymer through a membrane pore (translocation), focusing on the scaling properties with the number of monomers N. The natural coordinate for translocation is the number of monomers on one side of the hole at a given time. Commonly used models that assume Brownian dynamics for this variable predict a mean (unforced) passage time tau that scales as N2, even in the presence of an entropic barrier. In particular, however, the time it takes for a free polymer to diffuse a distance of the order of its radius by Rouse dynamics scales with an exponent larger than two, and this should provide a lower bound to the translocation time. To resolve this discrepancy, we perform numerical simulations with Rouse dynamics for both phantom (in space dimensions d=1 and 2), and self-avoiding (in d=2) chains. The results indicate that for large N, translocation times scale in the same manner as diffusion times, but with a larger prefactor that depends on the size of the hole. Such scaling implies anomalous dynamics for the translocation process. In particular, the fluctuations in the monomer number at the hole are predicted to be nondiffusive at short times, while the average pulling velocity of the polymer in the presence of a chemical-potential difference is predicted to depend on N.

We consider the passage of long polymers of length N through a hole in a membrane. If the process is slow, it is in principle possible to focus on the dynamics of the number of monomers s on one side of the membrane, assuming that the two segments are in equilibrium. The dynamics of s(t) in such a limit would be diffusive, with a mean translocation time scaling as N 2 in the absence of a force, and proportional to N when a force is applied. We demonstrate that the assumption of equilibrium must break down for sufficiently long polymers (more easily when forced), and provide lower bounds for the translocation time by comparison to unimpeded motion of the polymer. These lower bounds exceed the time scales calculated on the basis of equilibrium, and point to anomalous (sub-diffusive) character of translocation dynamics. This is explicitly verified by numerical simulations of the unforced translocation of a self-avoiding polymer. Forced translocation times are shown to strongly depend on the method by which the force is applied. In particular, pulling the polymer by the end leads to much longer times than when a chemical potential difference is applied across the membrane. The bounds in these cases grow as N 2 and N 1+ν , respectively, where ν is the exponent that relates the scaling of the radius of gyration to N . Our simulations demonstrate that the actual translocation times scale in the same manner as the bounds, although influenced by strong finite size effects which persist even for the longest polymers that we considered (N = 512).

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Kantor, Kardar, and Nelson Reply:Recently we reported some results on statistical mechanics of two-dimensional networks. 1 Because of entropy effects, a self-avoiding manifold assumes a typical radius R that grows with its linear size L as R~~L V . Monte Carlo simulations in three dimensions lead to v~0.8 in good agreement with a Flory estimate of j. For strips of width w and length L, we suggested a scaling form /£ -(wL 3 ) 1/5 which interpolates between R~~L Af5 for w~~ L and R~~L 3/5 for vv^CL. Note that j is the Flory estimate for a free polymer in 3D. The crumpled balls obtained by crushing strips of paper 2 ' 3 are nonequilibrium configurations that belong to an ensemble entirely different from the equilibrium ones mentioned earlier. Here we briefly remark on similarities and differences between these two ensembles, and present a simple explanation for the experimental results of the preceding Comment. 3 Consider a Z)-dimensional manifold (D = 1 for polymers, and Z)=2 for surfaces) crumpled under strong pressure in d dimensions. If the network collapses to a compact object whose fractal dimension equals the dimension d of space, the radius grows as R~~L Vc with v c =D/d (since mass ~~L D -R d ). A string of length L can easily be compacted in this fashion into a ball of radius R~~L Vl in d = 3. It is, however, much more difficult for surfaces to achieve this compaction without tearing. 4 This is why crumpled aluminum foils 1 and crushed pieces of paper 2 ' 3 result in a nontrivial exponent V2 « 0.8 > j. The interior of a crushed paper ball (seen by gently opening the ball) looks rather similar to pictures of free surfaces from Monte Carlo simulations, which may account for the similarity in the exponent v for the two systems.For the compacted strips of length L and width w studied by Gomes and Vasconcelos, 3 we would expect R~~w a L^ with an exponent that interpolates between compacted strings (R~~~L 1/3 ) for L»H>, and crushed surfaces (R~~L V2~~ ) for L -w. This leads to the conclusion a = y~~0.33 and P = V2-y--0.47. These ex-ponents are quite consistent with the results reported in the preceding Comment. 3 Indeed, that agreement seems to improve for samples including a larger range of L/w, and hence more likely to be compacted to a uniform density.

We examine the statistics of knots with numerical simulations of a simplified model of polyethylene. We can simulate polymers of up to 1000 monomers (each representing roughly three CH(2) groups), at a range of temperatures spanning coil (good solvent) and globule (bad solvent) phases. We quantify the abundance of knots in the globule phase and in confined polymers, and their rarity in the swollen phase. Since our polymers are open, we consider (and test) various operational definitions for knots, which are rigorously defined only for closed chains. We also associate a typical size with individual knots, which are found to be small (tight and localized) in the swollen phase but large (loose and spread out) in the dense phases.