Localization of a Polymer with Internal Constraints

We argue that Kantor and Kardar’s assertion that their simulation results contradict our criterion for the localization of a softly constrained ideal polymer is incorrect. Our criterion is inapplicable to the model used in these simulations. We argue that our criterion for localization of as polymer with internal constraints strongly suggests that the models with soft and hard constraints belong to different universality classes. The size measure used by Kantor and Kardar is also not adequate for studying polymer localization. We also show that, in spite of these contrasts, combining the simulation results with our criterion for ideal chains seems to uncover some intriguing physics. PACS numbers: 82.35.+t, 87.15.By, 61.41.+t Typeset using REVTEX 1 In the accompanying comment, Kantor and Kardar (KK) [1] discuss simulations of an ideal polymer chain of N monomers with M randomly chosen distance constraints. These simulations use an analogy between ideal polymers and resistor networks [2] to calculate the mean-squared end-to-end distance (r) of the polymer. The simulations together with a lower bound for r are used to suggest that the polymer remains extended unless the number of constraints exceeds some Mc which is of order N . They concluded that this result is at variance with our recent theory in which we argue that Mc ∼ N/ logN for an entirely different model (BT) [3]. Here we show that this assertion is too strong, and that their results for ideal polymers may shed an interesting light on ours. The most obvious contrast between our study and that of KK is that the two studies investigate different models. In the random resistor analogy a short corresponds to a Dirac delta function (a hard constraint). In our Letter we explicitly excluded such delta function constraints; see footnote 9 of BT. Although a polymer with soft constraints will be more expanded than one with hard constraints, soft constraints add a new length scale (inducing longer range interactions) to the system which is present in our localization condition (see equations 1 and 10 of BT). The size of a polymer with soft constraints depends on both the number of constraints and the size of the constraints. Thus, the two models may be in different universality classes, making the objections of KK circumspect. The above notwithstanding, expanding our localization condition (eq. 10 of BT) for large φ ≡ (b/c) and small δ ≡ 1− μ gives the localization condition δ < logL − log 2 logφ+ logL − 1 + O( 1 φ ) + O(δ). (1) The hard constraint case corresponds to the limit φ → ∞ with φ ≫ L ≫ 1, yielding a maximum value of δ → 0, self-consistent with our expansion and consistent (!) with the arguments of KK that the critical value of M ∼ N in this limit. The purpose of our Letter was to determine whether a polymer with soft internal constraints is better approximated by an expanded chain or by a collapsed chain. The nature of the transition between these two regimes could not be determined by our methods . A