(a) The definitions that we use for the signs associated with the crossing of two curves. Schematic representations of (b) the three topological configurations of DNA hairpins studied in this paper and (c) a control system with no topological or geometric constraints. The different topological configurations in (b) are: (i) Topologically unlinked, linking number Lk = 0. (ii) Topologically favoured, Lk = −1. (iii) Topologically frustrated Lk = +1.
Typical structures assumed by (a) a single hairpin and (b) the topologically unlinked (Lk = 0) kissing complex. Note that in (a) it almost looks as if the stem is longer than 10 base pairs, because the stacking tends to propagate beyond the end of the stem at this temperature. For (b) the chosen structure has 14 interstrand base pairs. To its right is a topological sketch of the configuration illustrating that the zero linking number is achieved by balancing positive and negative crossings. In panels (c)–(e) we show example structures for partially formed complexes with a total of 2, 6, and 10 base pairs formed between the loops, respectively. In our visualisation of the DNA structures, each backbone site is represented by a sphere and each base by an ellipsoid connected to the backbone site.
Free energy profile of two complementary hairpins that are topologically unlinked (i.e., Lk = 0) at T = 23 °C at a single strand concentration of 0.336 mM (squares). The free energy profile for hybridization of the control system (Fig. 1(c)) is also plotted (circles). The control system can also form 20 intermolecular base pairs but without any geometric or topological constraints. In the inset, the full profiles are plotted, showing the large (>30 k B T) free energy difference between the most stable states of the kissing complex and the control system. For comparison, the free energy profiles have been set to have the same value when the number of base pairs is 1.
Bonding probability as a function of nucleotide position in the loop for different N tot, the total number of correct base pairs in the kissing complex. Note that the probability is not completely symmetric around the centre of the loop sequence because the propensity to form non-native base pairs depends on which native base pairs are formed.
(a) Typical structure and topological sketch of the kissing complex with Lk = −1 and (b) free energy profile associated with the formation with the kissing complex, compared to that for the control system in Fig. 1(c). To aid this comparison, the two free energy profiles were set to have the same value at 1 base pair.
(a) Typical structure and topological sketch of the kissing complex with Lk = +1 and (b) free energy profile associated with the formation of the kissing complex, compared to that for Lk = −1.
Effect of the backbone excluded volume on the free energy profiles for a kissing complex with Lk = 0. The original value of σbb is 0.596 nm.
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