Calculating the Secrets of Life: Contributions of the Mathematical Sciences to Molecular Biology (1995)
Chapter: THE ENERGETICS OF A STATE
Page 186
alternatives. Thus, separations at particular sites can be analyzed only in the context of the entire molecule. Divide-and-conquer strategies, in which the sequence is partitioned into blocks that are individually analyzed, are thus not feasible. Superhelical transitions must be analyzed as global events, including simultaneous competitions among all possible transitions. This renders the accurate analysis of superhelical transitions extremely difficult.
It is not feasible to perform exact analyses of all states for the kilobase-length, topologically constrained molecules of biological importance, because the number of states grows exponentially. On the other hand, it is not enough to look only at the lowest energy states. Confining attention to the minimum-energy state provides a very poor depiction of transition behavior. Although any individual high-energy state is exponentially less populated, there are so many high-energy states that cumulatively they can dominate the minimum-energy state. The development of accurate methods to treat superhelical strand separation requires an intermediate approach (Benham, 1990). First, enough low-energy states must be treated exactly to provide an accurate depiction of the transition. Then the cumulative influence of the neglected, high-energy states must be estimated. Wherever possible, computed parameter values must be refined by the insertion of correction terms that account for the approximate influence of the neglected states. This is the strategy we adopt below.
The Energetics of a State
A superhelical linking difference a imposed on a DNA molecule can be accommodated by three types of deformation, each of which requires free energy. First, strand separations can occur. Second, the single strands in the separated regions can twist around each other, thereby absorbing some of the linking difference. Third, the portion of a not accommodated by these alterations imposes superhelical deformations on the balance of the molecule.
Each of these deformations requires free energy that can be described by some simple formulas. Opening each new region of strand separation requires a free energy a, while separating each individual base pair within a region takes free energy bATor bGC, depending on the
