 Solution

The single-block DNA zipper partition function retains one contiguous paired block and ignores multiple separated bubbles. A block of $j\ge 1$ paired bases can start at any of $N-j+1$ positions. Its first pair has initiation weight $\kappa$, and the remaining $j-1$ pairs each have propagation weight $s$. ThusThe source's symbol $k$ in the displayed expressions denotes the same initiation constant as $\kappa$, not Boltzmann's constant. Association and other reference-state factors are absorbed into that statistical weight; the model treats the propagation weights as uniform despite the actual sequence's different bases.

Put $r=j-1$. Since $N-r$ counts how many terms of a finite geometric series contain $s^r$,ConsequentlyThe apparent singularity is removable: at $s=1$, the original sum gives $q=\kappa N(N+1)/2$, which is $45\kappa$ for these nine bases.

At $s\gg 1$, the longest block dominates:Conditional on at least one pair, the fully paired probability approaches one, with leading correction of order $2/s$. If a fully unpaired reference state of weight one is included, the full partition function is $Q=1+q$, so a large propagation constant alone does not guarantee predominant duplex if $\kappa s^{N-1}$ is still very small.

The given sequence contains five A–T and four G–C pairs when fully complementary. Real duplex stability also depends strongly on stacking, sequence context, salt and temperature; it cannot be calculated just by counting two versus three hydrogen bonds. All else comparable, sequences with greater duplex propagation stability resist bubble opening more strongly, while easier-to-open, often A–T-rich start-site regions facilitate initiation. The homogeneous-$s$ zipper demonstrates that principle but does not resolve a quantitative sequence-specific melting profile or polymerase's active stabilization of the open state.