Mechanism of thermal renaturation and hybridization of nucleic acids: Kramers process and universality in Watson-Crick base pairing

Renaturation and hybridization reactions lead to the pairing of complementary single-stranded nucleic acids. We present here a theoretical investigation of the mechanism of these reactions in vitro under thermal conditions (dilute solutions of single-stranded chains, in the presence of molar concentrations of monovalent salts and at elevated temperatures). The mechanism follows a Kramers' process, whereby the complementary chains overcome a potential barrier through Brownian motion. The barrier originates from a single rate-limiting nucleation event in which the first complementary base pairs are formed. The reaction then proceeds through a fast growth of the double helix. For the DNA of bacteriophages T7, T4 and $\phi$X174 as well as for Escherichia coli DNA, the bimolecular rate $k_2$ of the reaction increases as a power law of the average degree of polymerization $$ of the reacting single- strands: $k_2 \prop ^\alpha$. This relationship holds for $100 \leq \leq 50 000$ with an experimentally determined exponent $\alpha = 0.51 \pm 0.01$. The length dependence results from a thermodynamic excluded-volume effect. The reacting single-stranded chains are predicted to be in universal good solvent conditions, and the scaling law is determined by the relevant equilibrium monomer contact probability. The value theoretically predicted for the exponent is $\alpha = 1-\nu \theta_2$, where $\nu$ is Flory's swelling exponent ($nu approx 0.588$) and $\theta_2$ is a critical exponent introduced by des Cloizeaux ($\theta_2 \approx 0.82$), yielding $\alpha = 0.52 \pm 0.01$, in agreement with the experimental results.