The polymer threading a membrane transition (PTM), which is a first-order thermodynamic phase transition for an isolated linear polymer in the limit of infinite molecular weight, is coupled to the other four phase transitions of the isolated polymer molecule. They are (1) the helix–random coil (HR) phase transition which can be diffuse (polypeptides), second-order (DNA) or first-order (collagen) depending on the number of strands, (2) the collapse (C) transition which is usually second-order but can be first-order for polymeric solvents, (3) adsorption onto a surface (SA) which is second-order, (4) a model of equilibrium polymerization (P) which is first-order. In each case an exact expression for the partition function of the coupled pair is given as a one-dimensional summation over products of the individual partition functions corresponding to sides 1 and 2. Using a procedure analogous to evaluation of the grand canonical ensemble the summation can be performed and the character of the transition elucidated in the limit of infinite molecular weight. Given that the solutions on either side are sufficiently diverse there are 15 possible translocation pair couplings. They are PTM–PTM, HR–HR, C–C, SA–SA, P–P, PTM–HR, PTM–C, PTM–SA, PTM–P, HR–C, HR–SA, HR–P, C–SA, C–P, SA–P. The PTM–P coupling is most interesting because one can create polymer in the PTM side even though the P side is in the depolymerization regime. For HR–HR there are eight possible translocation modes. For example, as we raise the temperature we can have H1→H2→R1→R2 in obvious notation. These exact model solutions provide a thermodynamic base for the study of the kinetics of significant technological problems such as the translocation of DNA through pores imbedded in membranes. They also throw light on the nature of polymer–membrane–pore interactions in living cells and viruses.

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