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### Abstract

A dynamic Monte Carlo simulation of the collapse transition of polymer chains is presented. The chains are represented as self-avoiding walks on the simple cubic lattice with a nearest-neighbor contact potential to model the effect of solvent quality. The knot state of the chains is determined using the knot group procedure presented in the accompanying paper. The equilibrium knot spectrum and the equilibrium rms radius of gyration as functions of the chain length and the contact potential are reported. The collapse transition was studied following quenches from good-to poor-solvent conditions. Our results confirm the prediction that the newly formed globule is not yet at equilibrium, since it has not yet achieved its equilibrium knot spectrum. For our model system, the relaxation of the knot spectrum is about an order of magnitude slower than that of the radius of gyration. The collapse transition is also studied for a model in which both ends of the chain remain in good-solvent conditions. Over the time scale of these simulations, knot formation is frustrated in this inhomogeneous model, verifying that the mechanism of knotting is the tunneling of chain ends in and out of the globule.

These computations were performed using the High Performance Computing Facility at Stevens Institute of Technology.

I. INTRODUCTION

II. COMPUTATIONAL DETAILS

A. Hamiltonians of the model chains

B. Chain rearrangement maneuvers

C. Definition of knot state

III. EQUILIBRIUM RESULTS

IV. MODELING THE COLLAPSE TRANSITION

V. SUMMARY AND CONCLUSIONS

## Figures

Schematics of the chain rearrangement maneuvers employed.

Schematics of the chain rearrangement maneuvers employed.

Snapshots of several chain conformations. (a) Good solvent, . (b) Theta solvent, . (c) Very poor solvent, collapsed chain, . (d) Inhomogeneous model, bulk of chain is collapsed, end segments under good-solvent conditions and therefore exposed, , .

Snapshots of several chain conformations. (a) Good solvent, . (b) Theta solvent, . (c) Very poor solvent, collapsed chain, . (d) Inhomogeneous model, bulk of chain is collapsed, end segments under good-solvent conditions and therefore exposed, , .

Size scaling for the indicated values of the contact potential. The theta point is assigned to be , since then . Fluctuations are strongest as we pass through the transition , and so there the results display some sampling error.

Size scaling for the indicated values of the contact potential. The theta point is assigned to be , since then . Fluctuations are strongest as we pass through the transition , and so there the results display some sampling error.

Variation of through the collapse transition for chains of length . The approximate position of the theta point is indicated.

Variation of through the collapse transition for chains of length . The approximate position of the theta point is indicated.

Variation of the mean knot complexity score through the collapse transition for chains of length . The approximate position of the theta point is indicated.

Variation of the mean knot complexity score through the collapse transition for chains of length . The approximate position of the theta point is indicated.

Probability that a chain is found without a knot as a function of chain length and at various values of the contact potential. Curves for are labeled with values. Curves for all display probabilities near 1 and appear near the top of the graph. The horizontal dashed curve corresponds to and determines, by interpolation or extrapolation, the value of .

Probability that a chain is found without a knot as a function of chain length and at various values of the contact potential. Curves for are labeled with values. Curves for all display probabilities near 1 and appear near the top of the graph. The horizontal dashed curve corresponds to and determines, by interpolation or extrapolation, the value of .

The value controls the crossover from unknotted to knotted behavior. Its variation as a function of the contact potential is shown. The approximate location of the theta point is indicated.

The value controls the crossover from unknotted to knotted behavior. Its variation as a function of the contact potential is shown. The approximate location of the theta point is indicated.

Snapshots of a collapsing chain at several moments during collapse. At , three “pearls” have condensed along the chain (enclosed in circles). One pearl grows at the expense of the others until the globule has formed at . The globule at is still not at equilibrium, we must wait much longer to see its knot state reach equilibrium.

Snapshots of a collapsing chain at several moments during collapse. At , three “pearls” have condensed along the chain (enclosed in circles). One pearl grows at the expense of the others until the globule has formed at . The globule at is still not at equilibrium, we must wait much longer to see its knot state reach equilibrium.

The relaxation functions and for both the homogeneous and the inhomogeneous models. Relaxation times for and for the homogeneous model; 100 and 960, respectively, are estimated from the slopes of the least-squares lines, shown dashed. The function remains very close to 0 over the course of the simulation, except for a minor fluctuation near , indicating that knots do not form in the inhomogeneous model.

The relaxation functions and for both the homogeneous and the inhomogeneous models. Relaxation times for and for the homogeneous model; 100 and 960, respectively, are estimated from the slopes of the least-squares lines, shown dashed. The function remains very close to 0 over the course of the simulation, except for a minor fluctuation near , indicating that knots do not form in the inhomogeneous model.

## Tables

Columns 2 and 3 give the unnormalized selection probabilities for each type of maneuver. To anneal a chain, all maneuvers are appropriate, while to model dynamics, we restrict ourselves to local maneuvers. Column 4 displays the change in chain length that accompanies each maneuver.

Columns 2 and 3 give the unnormalized selection probabilities for each type of maneuver. To anneal a chain, all maneuvers are appropriate, while to model dynamics, we restrict ourselves to local maneuvers. Column 4 displays the change in chain length that accompanies each maneuver.

Comparison of simulation vs exact enumeration results on short chains. is the mean chain length, is the radius of gyration, and is the probability of finding a chain of length .

Comparison of simulation vs exact enumeration results on short chains. is the mean chain length, is the radius of gyration, and is the probability of finding a chain of length .

Occurance probability of each of the indicated knots or knot classes in chains of length at the indicated values of the contact potential, . The last column gives the knot complexity score for the indicated knot. The knot class designation U.C. indicates “unrecognizably complex,” or a knot too complex for its knot state to be determined by the knot group technique.

Occurance probability of each of the indicated knots or knot classes in chains of length at the indicated values of the contact potential, . The last column gives the knot complexity score for the indicated knot. The knot class designation U.C. indicates “unrecognizably complex,” or a knot too complex for its knot state to be determined by the knot group technique.

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