Equilibrium Thermodynamics of Binding Reactions  

The equilibrium constant Keq for the reaction of the binding of an oligonucleotide to a site on a template strand is defined as:

where b is the concentration of the bound template-oligo complex, u is the concentration of unbound templates, and c is the concentration of unbound oligos, all at equilibrium and with all concentrations expressed as unit-free Molar concentrations (that is, as a dimensionless number representing the number of moles of substance per liter of solution). That is, if it were the case that a solution with 1 Molar bound complex, 1 Molar unbound template, and 1 Molar unbound oligos would be a system at equilibrium, then the Keq would be exactly 1 (no units). (There is nothing magical about the concentration 1 Molar, it is just a convenient and conventional baseline.)

The quantity of interest to us here is b/u, the ratio of bound to unbound templates. Renaming this ratio r and solving for it we get:

The value of c is not necessarily known a priori, but if we assume that the total concentration of templates (bound or unbound) is much less than the total concentration of oligos, then it follows that the concentration of free oligos will always be essentially equal to the total concentration of oligos, since only a small fraction of them could ever become bound. The total concentration of oligos is a quantity that we do know (because we control it when we set up the reaction). The program currently does make this assumption; however, the assumption could be relaxed and the exact r found fairly easily by solving a quadratic equation, an exercise we will not delve into here.

The other unknown in the equation for r is Keq. The Keq for any reaction can be expressed

where DeltaG0 is a quantity known as the standard free energy change for the reaction (the amount of work per unit substance that must be input to convert reactants to products when all substances are present at 1 Molar concentrations), R is Boltzmann's constant 0.001986 kcal/(mol K), and T is the temperature of the system. DeltaG0 itself is dependent on temperature, and is given by:

where DeltaH0 and DeltaS0 are (more or less) temperature independent constants for the reaction. DeltaH0 is the standard enthalpy change for the reaction, meaning the net energy input (work and heat) per unit substance required for converting reactants to products, and DeltaS0 is the standard entropy change, reflecting the increase in disorder involved in the conversion. The higher the DeltaS0 the less the amount of input energy that must be in the form of organized work rather than heat.

Estimates of DeltaH0 and DeltaS0 for DNA inter-strand binding reactions were derived using the procedure of Breslauer et al., which involves summing up contributions from all the base pairs along the strand. Actually, in Breslauer's technique it is not base pairs themselves but actually "stacks" of adjacent base pairs that are the basic elements of the calculation. This is because it has been found that the stability of a DNA duplex (pair of bound strands) can be modeled more accurately by taking these nearest-neighbor interactions into account. Figures for the enthalpies of each possible stack of neighboring Watson-Crick base-pairs were taken from Breslauer's paper. Entropy figures were taken from the more recent Quartin & Wetmur '89, as recommended in Tijssen '93. For mismatched base pairs, the quality of the data in the literature is poorer; we used DeltaH0 and DeltaS0 values extrapolated from those given in a limited study by Werntges et al.'86.