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UNWINDING THE DOUBLE HELIX: USING DIFFERENTIAL MECHANICS TO PROBE CONFORMATIONAL CHANGES IN 184 DNA Strand separation in living organisms frequently arises through interactive processes, in which local superhelical destabilization of the B-form acts in concert with other factors. Biological systems may exploit marginal decreases in the stability of the B-form that occur at discrete sites in superhelical molecules. For example, consider an enzyme that functions by recognizing a particular sequence and inducing separation there. It might be energetically able to induce the transition only if the B-form already is marginally destabilized at that site. This suggests that superhelical helix destabilization also can regulate biological processes through mechanisms that need not involve preexisting separations. For this reason it is important also to develop methods to predict sites where superhelicity marginally destabilizes the duplex. STATEMENT OF THE PROBLEM This chapter develops methods to predict the strand separation and helix destabilization experienced by a specified DNA sequence when superhelically stressed. We will focus specifically on predictions regarding several plasmids (that is, circular DNA molecules) that have been engineered to include the E. coli replication origin or variants thereof. This is done because experimental information is available regarding superhelical strand separation in these molecules. In principle, the analysis of conformational equilibria is quite direct. Because every base pair can separate, there are many possible states of strand separation available to a topologically constrained DNA molecule. By basic statistical mechanics, a population of identical molecules at equilibrium will be distributed among its accessible states according to Boltzmann's law. If these states are indexed by i, and if the free energy of state i is Gi, then the equilibrium probability pi of a molecule being in state i, which is the fractional occupancy of that state in a population at equilibrium, equals (7.3) Here Z is the so-called partition function, given by
UNWINDING THE DOUBLE HELIX: USING DIFFERENTIAL MECHANICS TO PROBE CONFORMATIONAL CHANGES IN 185 DNA (7.4) where R is the gas constant and T is the absolute temperature. Thus the fractional occupancies of individual states at equilibrium decrease exponentially as their free energies increase. If a parameter ζ has value ζi in state i, then its population average, that is, its expected value at equilibrium, is (7.5) This expression can be used to evaluate any equilibrium property of interest, once the governing partition function is known. The application of this approach to the rigorous analysis of conformational equilibria of superhelical DNA molecules is complicated by three factors. First, the number of the states involved is extremely large. Every base pair can be separated or unseparated, so specification of a state of a molecule containing N base pairs involves making N binary decisions. This yields a total of 2N distinct states of strand separation. This precludes the use of exact methods, in which all states are enumerated, to analyze molecules of biological interest, as these commonly have lengths exceeding 1,000 base pairs. Most DNAs have sites whose local sequences permit transitions to other conformations in addition to separation, further increasing the number of conformational states. Second, because the free energy needed to transform a base pair to an alternative conformation depends on the identity of the base pair involved, the analysis of equilibria must examine the specific sequence of bases in the molecule. This precludes several possible strategies for performing approximate analyses, including combinatorial methods that assume transition energetics to be the same for all base pairs, or that average the base composition of blocks. Third, and most importantly, the global and topological character of the superhelical constraint means that the conformations of all base pairs in the molecule are coupled together. Separation of a particular base pair alters its helicity, which changes the distribution of Tw, and hence of α, throughout the domain. This in turn affects the probability of transition of every other base pair. Whether transition occurs at a particular site depends not just on its local sequence, but also on how effectively this transition competes with all other
