Measuring relative stability of branch migration intermediates

2.2 Materials, Methods, and Results

2.2.3 Measuring relative stability of branch migration intermediates

Multistrand is particularly sensitive to the ratiok_uni/k_bi. Decreasingk_unislows down all unimolec- ular transitions - which brings the branch migration rate closer to experimentally inferred values, but makes fraying too slow. Two parameters,k_uniandk_bi, are simply not enough [153] to simul- taneously match the four distinct time scales involved: rates of hybridization, fraying, branch migration, and branch migration initiation. Therefore, even unrealistically low choices ofk_uni/k_bi are unable to match observed acceleration in strand displacement rates due to toehold length (see Supplementary Figure S7).

The SSK analysis confirms that understanding what the IEL’s∆G_sand∆G_prepresent requires examining features not present in the NN model. Multistrand models branch migration as a fray–and–replace process and interprets the IEL’s sawtooth transition state as one in which the substrate-incumbent base pair at the junction is frayed. This choice, when coupled with ak_uni calibrated to match fraying rates [168], results in a branch migration rate that is much faster than experimentally inferred step times [165,166].

Indeed, the thermodynamics of the branch migration junction, e.g. states i and j in Fig- ure2.5(A), is not well characterized in the standard nearest-neighbor secondary structure model, as it involves overhangs, dangles, and coaxial stacking. Reflecting the lack of consensus, tools like NUPACK [123], Vienna RNA [183] and Mfold [184] offer several ways of treating dangle contribu- tions; however, none of the three “dangle options" in the NUPACK energy model [120] improved Multistrand predictions (Supplementary Figure S6(A)).

prising hairpinXiand strandYj(Figure2.6). XiandYjhave poly-Toverhangs of lengthiand j respectively. Varying(i, j)from(20,0) to(0,20)with i+j = 20yields complexes which are

“frozen snapshots” of branch migration, with no expected branch migration possible. X20:Y00 mimics the binding of the invader by the toehold, whileX19:Y01represents the displacement of 1 base, and so on. X10:Y10represents the “half way stage” of branch migration, whileX00:Y20 captures nearly successful displacement. Measuring the relative stability of these frozen snapshots is expected to be indicative of the relative free energies of branch migration intermediates. Exper- iments involving these complexes will henceforth be referred to as thestrand displacement snapshot study. To investigate the consequences of short overhangs at the junction, we designed complexes Xi:Yi(for i = 0, 1, 2, 5 and 10). These experiments will be referred to as thelocal overhangstudy. All our complexes have the same base pairs at the junction and poly-Toverhangs, while branch mi- gration typically involves different bases at each step. We can therefore study the thermodynamic consequences of junction geometry, without the complication of sequence dependence.

Temperature dependent absorbance experiments.We measure the UV absorbance (at 260 nm) of each complex between 20^◦Cand 90^◦C, at four different concentrations. Since the absorbance of single-stranded DNA (ssDNA) is higher than that of double-stranded DNA (dsDNA), and the fraction of ssDNA is dependent on the temperature, a temperature-dependent absorbance curve is obtained at each concentration (Supplementary Figure S8). Each complexXi:Yjexhibits two tran- sitions: the bimolecular, lower temperature transition and the unimolecular, higher temperature transition due to the hairpin inXiclosing or opening. The unimolecular transition was identified both by it being independent of concentration and by control melts involving the hairpins only (data not shown). At the concentrations chosen, the bimolecular and unimolecular transitions are distinct.

For each complex, we infer the enthalpy (∆H^◦) and entropy (∆S^◦) of formation by fitting the smoothed and normalized temperature-dependent absorbance curves (Figure2.7) to a two-state

Domain Sequence Length

d CCTCATCATACTACG 15

e CTCCATGTCACTTC 14

Table 2.1:Sequences for domains from Figure2.6, listed 5’ - 3’.

+

^e

T^j

Yj

A

Xi

Tⁱ d

d* e*

T⁶ T⁶

Tⁱ d

Xi:Yj

e T^j

d* e*

B

T⁶

T²⁰ d

X20:Y00

e

d* e*

(i  =  20,  j  =  00) T⁶

T¹⁰ d

X10:Y10

e T¹⁰

d* e*

(i  =  10,  j  =  10) T⁶

T²⁰ d

X00:Y20

e

d* e*

(i  =  00,  j  =  20)

Figure 2.6:(A) ComplexXi:Yjcomprises hairpinXiand strandYj, with poly-Toverhangs of lengthiandj respectively. Domainsdandeare designed to be orthogonal to each other and the overhangs (sequences in Table2.1). (B) Varying (i,j) from(20,00)to(00,20)withi+j= 20mimics the geometry branch migration intermediates (X20:Y00(start),X10:Y10(middle) andX00:Y20(end) respectively). No branch migration is intended in these complexes.

model [191]:

[X_i:Y_j]

[Xi][Yj] = e^{−(∆H◦−T∆S◦)/RT} (2.11) where∆H^◦and∆S^◦are assumed to be temperature independent. We perform this fitting using a Bayesian analysis and confirm our findings using a simpler descriptive “leave-one-concentration- out” approach. Details are provided in Supplementary Section S5; see Supplementary Figure S9 and Supplementary Tables S3, S4 and S5.

From∆H^◦and∆S^◦we can calculate the free energy of formation,∆G^◦(T) = ∆H^◦−T∆S^◦. By comparing the free energies of different complexes, we infer the contribution of the poly-T overhangs. Our two-state assumption means that∆G^◦(T)is assumed to be linear inT– deviations from linearity limit the accuracy of extrapolation from values around the melting temperature of the complexes. Values of∆G^◦at 55^◦C, which is closer to the melting temperature, are plotted in Supplementary Figure S10.

Second overhang causes de-stabilization due to a local effect. Unlike the predictions of NN thermodynamic models [118, 120] of DNA, the immobile complexes we designed are not all of equal free energy; a second overhang at the junction causes a thermodynamic penalty (Fig- ure2.8A). At 25^◦C, we infer a free energy penalty of∼2.0kcal/mol (3.4 RT) as branch migration

proceeds from 0 to 10 steps (X20:Y00vs.X10:Y10), with the majority (∼1.5kcal/mol) arising from the first step (X20:Y00vs.X19:Y01). An approximately symmetric decrease is inferred for steps 11 to 20 (X10:Y10vs.X00:Y20).

Since the de-stabilization due to an additional overhang plateaus so quickly, we suspect that the penalty is due to local effects at the junction, which is supported by thelocal overhangstudy (Figure2.8B). Two 1-base overhangs on either side of the junction (X01:Y01) result in a penalty of∼ 1.4kcal/mol relative to no-overhangs (X00:Y00). Lengthening the overhangs increases this penalty, but each additional base contributes progressively less, with an overall penalty of 3.0 kcal/mol and 3.2 kcal/mol respectively for 5-base (X05:Y05) and 10-base (X10:Y10) overhangs.

Our experiments suggest that current NN models of DNA do not capture the free energy land- scape of strand displacement accurately enough to capture the kinetics of branch migration. This explains in part the inability of SSK models like Multistrand to match experimentally observed toehold-mediated acceleration.

20 30 40 50 60 70 80 90

0 0.5 1 1.5 2

Temperature( ° C)

Melt Fraction

Smoothed and normalized absorbance

X00:Y00 X01:Y01 X02:Y02

X05:Y05

Bimolecular  transitions

Unimolecular transitions

Figure 2.7: Smoothed and normalized UV absorbance data while annealing (at 200 nM). The lower tem- perature transition is the (bimolecular) formation of the complex, while the higher temperature transition is the (unimolecular) formation of the hairpin. The mean absorbance between20^◦Cand35^◦Cis normal- ized to 0 and that between64^◦Cand66^◦C(indicated by vertical lines) to 1. The temperature range whose mean absorbance is normalized to 1 is concentration-dependent (Supplementary Table S3). Data acquired by annealing and melting are essentially superimposable. The dashed line indicates the halfway point of the bimolecular transition.

0 1 2 5 10 15 181920 Number of strand displacement steps completed

6 G$ 25 (kcal/mol)

Bayesian analysis Leave one concentration out NUPACK (Dangles = None) NUPACK (Dangles = Some)

0 1 2 5 10

Length of tails at the junction Bayesian analysis Leave one concentration out NUPACK (Dangles = None) NUPACK (Dangles = Some)

A B

Figure 2.8:^∆G◦25of formation for complexes in thestrand displacement snapshotstudy (A) and thelocal over- hangstudy (B). Black error bars indicate Bayesian posterior means and99%confidence intervals, while red error bars indicate means and standard deviations of leave-one-concentration-out least square fits. NUPACK predictions with dangles options “some” and “none” are provided for comparison.