Exact method for numerically analyzing a model of local denaturation in superhelically stressed DNA

Local denaturation, the separation at specific sites of the two strands comprising the DNA double helix, is one of the most fundamental processes in biology, required to allow the base sequence to be read both in DNA transcription and in replication. In living organisms this process can be mediated by enzymes which regulate the amount of superhelical stress imposed on the DNA. We present a numerically exact technique for analyzing a model of denaturation in superhelically stressed DNA. This approach is capable of predicting the locations and extents of transition in circular superhelical DNA molecules of kilobase lengths and specified base pair sequences. It can also be used for closed loops of DNA which are typically found in vivo to be kilobases long. The analytic method consists of an integration over the DNA twist degrees of freedom followed by the introduction of auxiliary variables to decouple the remaining degrees of freedom, which allows the use of the transfer matrix method. The algorithm implementing our technique requires ${O(N}^2)$ operations and $O\left(N\right)\mathrm{}$ memory to analyze a DNA domain containing N base pairs. However, to analyze kilobase length DNA molecules it must be implemented in high precision floating point arithmetic. An accelerated algorithm is constructed by imposing an upper bound M on the number of base pairs that can simultaneously denature in a state. This accelerated algorithm requires $O\left(\mathrm{MN}\right)\mathrm{}$ operations, and has an analytically bounded error. Sample calculations show that it achieves high accuracy (greater than 15 decimal digits) with relatively small values of M $\mathrm{}(M<\mathrm{}0.05N)\mathrm{}$ for kilobase length molecules under physiologically relevant conditions. Calculations are performed on the superhelical $\mathrm{pBR}322$ DNA sequence to test the accuracy of the method. With no free parameters in the model, the locations and extents of local denaturation predicted by this analysis are in quantitatively precise agreement with in vitro experimental measurements. Calculations performed on the fructose-1,6-bisphosphatase gene sequence from yeast show that this approach can also accurately treat in vivo denaturation.