Loop formation of a flexible polymer with two random reactive sites

The dynamics of loop formation is investigated for a flexible polymer owing to the reversible intrachain reaction. The polymer is made of N hard spheres tethered by inextensible bonds, and the two reactive sites with binding energy -εare randomly located. The coil-to-loop crossover is characterized by the probability curve which depicts the variation of the open-state probability with temperature. The midpoint temperature (β_m^-1) is related to the conformational entropy loss ΔS from coil to loop states. It is found that β_mε = ΔS/k_B = ln N ^a + G, and the loop can be classified into three types: (i) end-to-end, α ≃ 1.98; (ii) end-to-interior, α ≃ 2.16; and (iii) interior-to-interior, α ≃ 2.48. The constant G varies with locations of the reactive sites. The kinetic rate constants can be well depicted by the Arrhenius kinetics with free energy barriers in agreement with crossover thermodynamics. Although the end-to-end loop is most easily formed for long enough polymers, the interior-to-interior loop is preferred for short chains. This consequence indicates that biopolymers may utilize loop types and contour distance between reactive sites to control the probability of loop formation.