II. Design and Implementation of Stacking Bonds
II.2. Design, Results and Discussion
II.2.4. Thermodynamic measurements
II.2.4.2. First energy model: assuming loop-loop interactions are neutral
For the first model, we simply divided the binding energy by the number of blunt end stacks per bond (2p) to arrive at ∆Gst. For short DNA complexes, the free energy of hybridization is linear in the number of base pairs; thus we assumed that the total stacking bond energy would be linear in p, and ∆Gst would be roughly constant. Surprisingly, stacking bond energy appeared quite sublinear in p and ∆Gst increased from -2.6 kcal/mol for p=2, to -1.8 kcal/mol for p=4, to -1.4
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kcal/mol for p=6. While this range of values encompasses that measured for ‘GC’ blunt-end stacking elsewhere44, such sublinearity is predicted to decrease the performance of stacking bonds:it would result in a smaller energy difference between correct and incorrect bonds than is predicted by a linear energy model, and cause a correspondingly higher rate of incorrect bonds. It also suggests that stacking sequences with the same p-i but higher i/p might give higher error rates;
this is consistent with our observations for (7, 4), (8, 5) and (9, 6) sequences (see section “Why use 7 active patches with a mismatch constraint of 4?”).
Sublinear binding energies have been reported before in DNA tiling systems using sticky ends73. Here, we hypothesize that sublinearity might derive from deformation of the edge caused by residual local twist (twist correction sets only the global average twist), potential curvature induced because all breaks in the phosphate backbone lie on the same side of the origami, or a combination of both. If a few nearby patches bind, they would not have to bend or twist much; thus strain will contribute little to the stacking bond energy, and a large |∆Gst| that closely reflects the free solution stacking energy will be observed. Our data for p=2 indeed match free solution values fairly well (see below). In contrast, if numerous patches bind, strain will make a large contribution to the stacking bond energy, and |∆Gst| will be underestimated. Our hypothesis further suggests that the distribution of ‘1’s in a stacking sequence might affect bond energy, so we tested an additional sequence (Table II-6 and Figure II-15) for p=2 and two additional sequences for p=4, with more spread-out active patches (e.g., ‘100100001001’). The data support our earlier assumption that bonds with identical p are roughly isoenergetic: for p=2 no significant difference was measured; for p=4 small (up to 0.2 kcal/mol) but statistically significant differences were measured. The trend for p=4 is that spreading out active patches weakens stacking bonds, in agreement with the deformation hypothesis.
System& binary code [origami] (nM) ∆Gst (kcal/mol hx) N (origami count)
2patch-‐(6,7) 000001100000 0.424 -‐2.5889 362
2patch-‐(5,8) 000010010000 0.848* -‐2.6738 276
4patch-‐(5,6,7,8) 000011110000 0.424 -‐1.7644 178
4patch-‐(3,5,8,10) 001010010100 0.424 -‐1.6593 566
4patch-‐(1,4,9,12) 100100001001 0.424 -‐1.5578 360
6patch-‐(4,5,6,7,8,9) 000111111000 0.212# -‐1.4223 442
Table II-6. Free energy of the stacking bond per helix for various systems.
& Numbers in parentheses indicate the locations of active stacking patches (the 1’s in the binary sequences).
*A higher concentration was used because it was hard to find dimers for this system.
# A lower concentration was used because it was hard to find monomers for this system.
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Figure II-15. Free energy of the stacking bond per helix for various systems. Energy values per helix vary depending on the number of patches, indicating a nonlinear relationship between the stacking energy and the number of helices. The overall trend (decreasing |∆Gst| as the number of patches increases) suggests that patches farther away from the middle of the edge must bend more (to counter some remnant global deformation) to bind; this hypothesis is consistent with the trend within the 4-patch systems. The binding energies for the 2-patch systems did not show a statistically significant difference (the error bars partially overlap). Error bars indicate standard error, obtained by bootstrapping the count data and propagating errors through the equations.
II.2.4.2.1. Comparison with free energy of stacking in literature
Because we hypothesize that non-stacking factors are all destabilizing, we suggest that the average energy obtained for the 2-patch systems, –2.63 kcal/mol (1× TAE with 12.5 mM Mg2+, 22°C), is most reflective of a pure stacking interaction. One literature value44 for the energy of GC/CG stacking, measured under a condition closest to ours, is -2.17 kcal/mol (1× TBE solution at 37°C). The same group later reported temperature-dependent data under the same experimental setup62. While buffer conditions between our and their experiments differ, we did our best to make
ΔGst(kcal/molhx)
# patches
100100001001 001010010100 000011110000
000010010000 000001100000
000111111000
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the measurements comparable by correcting the literature value using the temperature-dependent data. Figure II-16 shows a plot that we reproduced based on experimental data given in the literature62: from their Figure 3a and Supplementary Table 2. Data were taken for five different temperatures (32°C, 37°C, 42°C, 47°C, and 52°C). Assuming that the temperature dependence of the enthalpy and the entropy of blunt-end stacking is negligible for the given temperature range, it is appropriate to make a linear fit to ∆Gst as a function of temperature. A regression line and its equation (R2 = 0.8943) are shown in Figure II-16. Linear extrapolation to the y-axis (T=22°C) gives an energy of -2.42 kcal/mol at 22°C, which is very close to the value we obtained.Figure II-16. Temperature dependence of the stacking free energy (data taken from ref. 62). A linear fit and its extrapolation gives a stacking free energy of -2.42 kcal/mol at 22°C, which is very close to the value we obtained, -2.63 kcal/mol, at the same temperature.