Loop formation in polynucleotide chains. I. Theory of hairpin loop closure

A theoretical model to determine the probability of loop formation, based on an elaborated form of the Jacobson‐Stockmayer theory of cyclization equilibria, has been developed and used on RNA chains of homogeneous puckering and lengths up to 2 7 residues. The probability Q x ( q , γ0, ε0) of occurrence of hairpin loops of a particular chain x is given by Q x ( q , γ0, ε0) = [ W ( q ) δ r ][2Γ q (γ0)δγ] [E q ,γ0 (ε0) δε] where W ( q ) is the three‐dimensional density distribution function of end‐to‐end vectors r centered about the ideal loop closure position q ; Γ q (γ0) is an angular correlation factor that restricts the orientation of terminal bonds (i.e., bond x and bond 1) in the loop to a specified value Δθ = cos −1 (γ0) when r adopts a value within a range δ r from position q ; and E q ,γ0 (ε0) is the probability that ε falls within δε of ε0 when r and γ assume the values denoted by the respective subscripts. The parameter ε is related to the angle between a hypothetical bond ( x + 1) and bond 1. In these calculations, E q γ0 (ε0) was set equal to 1. For randomly coiling models previously developed to reproduce polynucleotide unperturbed dimensions, loop closure probability is maximized with chains of length 2 2 residues. Larger hairpin loops of 2 4 –2 5 residues, also favored in C3′‐ endo random coils of this type, are potential models of tRNA unfolding. W ( q ) is a stronger determinant of loop closure than Γ q (γ0). The angular correlation effect is most noticeable at chain lengths 2 2 and 2 3 where 2Γ q (γ0) deviates from unity.