On the probability of ring closure of lambda DNA

Abstract

The equilibrium constant, K, as a function of temperature (58 to 66°C) for the joining of the right half-molecules and the left half-molecules of λb2b5c DNA has been measured and compared to the previously measured equilibrium constant, Kc, for cyclization of the whole molecules. The observed values of ΔH, −108 (±10)kcal./mole and −88 (±10) kcal./mole (in 2·0 M -NaCl) for the joining of the halves and for cyclization, respectively, are the same within experimental error. Theoretically, the ratio of Kc to K is predicted to be the Jacobson—Stockmayer factor for the probability density of zero end-to-end distance in a random coil polymer, j = (3/2πbl)3/2, where l is the contour length and b is the hydrodynamic segment length. If b and l are taken from sedimentation velocity and electron microscope data, respectively, j=3·6×1011 molecules/cc; whereas the equilibrium measurements give j = 3·4 (±1·0) × 1011 molecules/cc. The equilibrium constant for the joining of the halves is found to be independent of concentration, as expected. The rate of joining of half-molecules has been measured, and follows the expected second-order kinetics. The kinetics of joining of halves and the kinetics of cyclization are related by the same factor, j. The kinetic data give j = 2·8 (±0·5) × 1011 molecules/cc. Thus the present investigation tends to confirm the assumption that the joining of the cohesive ends is an identical process for cyclization and for the association of half-molecules, and to confirm the model for a large DNA molecule as a random coil. With the available equilibrium data, it is possible to make precise predictions as to the equilibrium concentrations of linear and cyclic n-mers for all values of n. The appropriate formulae are presented. The cyclic monomer predominates at low temperature and low concentration of DNA, whereas linear n-mers of high n predominate at low temperature and high concentration. It is estimated that a small equilibrium concentration of a catenane exists under appropriate conditions. An estimate of the diffusion constant for relative diffusion of the ends of linear λ DNA of about 10−6 cm2 sec−1 is made, based on reported relaxation times for DNA in electric birefringence experiments. It is argued that segment diffusion rather than translational diffusion of the molecules as a whole is the main source of diffusional encounters between segments on different molecules.