Measuring relative stability of strand displacement intermediates

5.2 Appendix to Chapter 2

5.2.5 Measuring relative stability of strand displacement intermediates

strate by just one base more often than long toeholds. So, decreasingk_uni/k_bidisproportionately reduces the displacement rate of short toeholds and increases the orders of magnitude acceleration due to toehold length predicted by Multistrand (Figure5.7).

20 30 40 50 60 70 80 90 0.11

0.12 0.13 0.14 0.15

Temperature (º C)

Absorbance (AU)

Raw absorbance data

X05:Y05

X00:Y00 X01:Y01 X02:Y02

Hairpin    transitions

Figure 5.8:Raw absorbance data (at 260 nm), while annealing, at a concentration of 200 nM. Measurements were taken every0.1^◦Cbetween20^◦Cand90^◦C. The lower temperature transition is the (bimolecular) formation of the complex, while the higher temperature transition is the (unimolecular) formation of the hairpin. Data acquired by annealing and melting are essentially superimposable.

Temperature-dependent absorbance experiment protocols.Temperature-dependent absorbance experiments were performed using a Model 14 UV-Vis spectrophotometer, equipped with a water bath temperature controller, from AVIV Biomedical, Lakewood, NJ. UV absorbance at 260 nm be- tween 20^◦Cand 90^◦Cwas measured with a 1 nm bandwidth using 1.6 mL quartz cuvettes. The temperature step was set at 0.1^◦C/min with a 0.1^◦Cdead-band and an equilibration time of 0.25 minutes. All cuvettes were thoroughly cleaned before each experiment: each cuvette was washed 15 times in distilled water, once in 70% ethanol, another five times in distilled water, and finally once more in 70% ethanol.

One temperature-dependent absorbance experiment consisted of: (i) heating from 20 ^◦Cto 90^◦C, before taking any measurements; (ii) annealing from 90^◦Cto 20^◦Cwhile measuring ab- sorbance every 0.1^◦C; (iii) holding for 2h at 20^◦C; (iv) melting from 20^◦Cto 90^◦Cwhile mea- suring absorbance every 0.1^◦C. All heating and annealing steps in an experiment were at the temperature step mentioned above (0.1^◦C/min). An example of raw temperature-dependent ab- sorbance data at 200 nM obtained while annealing (step ii) is provided in Figure5.8.

For each complex, one sample was prepared at each of four different concentrations. For each of those samples, two runs of the temperature-dependent absorbance experiment described above were performed.

Concentration (nM) Upper normalization range (^◦C)

100 [61, 63]

150 [63, 65]

200 [64, 66]

300 [65, 67]

400 [65, 67]

500 [65, 67]

Table 5.4:Melt fraction for each complex is calculated from smoothed absorbance data by normalizing the absorbance in the[20, 35]^◦Crange to 0 and the absorbance in the concentration-dependent upper normal- ization range, specified in this Table, to 1. Our results are robust to this choice; this was verified by repeating the analysis with[65,67]^◦Cas the upper normalization range across all concentrations.

Two state model. We analyze the temperature-dependent absorbance data using a two-state model [191]: each molecule is assumed to be either in the fully bound state (Xi:Yj) or the fully dissociated state (Xi + Yj).

The raw absorbance data was smoothed by a moving average of 30 points (corresponding to a temperature interval of 3^◦C). The “melt fraction” or fraction of complex dissociated at tempera- tureT (f(T)) was calculated by normalizing the average absorbance of the bound state (between [20^◦C, 35^◦C]) to 0 and that of the dissociated state (between a concentration-dependent upper normalization range - see Table5.4) to 1. Note that the upper normalization range at a given con- centration is the same for all complexes. Our results are robust to the choice of upper normalization range; this was verified by repeating the analysis with[65,67]^◦Cas the upper normalization range across all concentrations.

Given the initial concentrationcof the complexXi:Yj, the melt fractionf(T)at temperatureT in the two-state model can be calculated from(∆H^◦,∆S^◦)as follows. Consider the reactionXi + YjXi:Yj, at temperatureT. Let us assume that the initial concentrationcofXi:Yjdissociates toXiandYjat concentrationxeach. Then,Xi:Yj is at concentrationc−x. We know that the equilibrium constantK_eq(T)is related tof(T)as

K_eq(T) =c−x

x² =1−f(T)

cf(T)² (5.69)

Solving the quadratic equation forf(T)≥0, we get f(T) =−1 +p

1 + 4cK_eq(T)

2cK_eq(T) (5.70)

SinceK_eq(T) = exp(−^∆G_RT^◦(T)), we may predict the entire temperature-dependent melt fraction curve by varying T appropriately.

!0.235 !0.230 !0.225 !0.220 !0.215

!86

!84

!82

!80

0.00 0.02 0.04 0.06

1.0

0.0

ï0 ï ï ï ï ï

Posterior Probability

ï ï ï ï ï ï 0

Posterior Probability

ï ï ï 0

Posterior Probability

A

C D

B

6Gº₅₅

6Sº 6Hº

Figure 5.9:Example posterior probability distributions obtained by Bayesian analysis over (A)(∆H^◦,∆S^◦) space and marginals over (B)∆G^◦at55^◦C, (C)∆H^◦and (D)∆S^◦for complexX10:Y10. All∆G^◦₅₅and∆H^◦ values are in kcal/mol while∆S^◦values are in kcal/K/mol. Note that the99%confidence interval is much more narrow for∆G^◦₅₅compared to∆H^◦and∆S^◦.

For each complex, we infer (∆H^◦,∆S^◦) (and hence ∆G^◦₂₅, ∆G^◦₅₅) by fitting the predicted melt fraction curves to smoothed and normalized absorbance data across different concentra- tions. By comparing the free energies of different complexes, we can infer the contribution of the poly-Toverhangs. We do this in two ways: a Bayesian analysis and a descriptive “leave-one- concentration-out” fit.

Bayesian analysis. We essentially discretize the(∆H^◦,∆S^◦)space into a grid and calculate the likelihood that our experimental data for each complex (all data traces at four concentrations) arose from each candidate pair in the discretization, assuming an independent Gaussian noise model. Normalizing the likelihood yields the posterior distribution for∆H^◦, ∆S^◦and∆G^◦(e.g.

Figure5.9). We calculate posterior means and99%confidence intervals, under the assumptions of the two-state model and our Bayesian framework.

In other words, given candidate values of the standard enthalpy and entropy of formation, (∆H₀^◦,∆S₀^◦), for a particular complex, smoothed and normalized absorbance dataDT_i,c_j at a cer- tain temperatureTi and concentrationcj is assumed to be related to the predicted melt fraction fT_i,c_j as follows:

DTi,cj =fTi,cj+ξTi,cj (5.71) where

ξT_i,c_j ∼ N(0, σ²_Ti,cj)

is independent additive Gaussian noise. That is,ξT_i,c_j andξT_l,c_k are assumed to be independent ifTi 6= Tl orcj 6= ck. σ_T²

i,c_j is calculated as the sample variance of smoothed and normalized absorbance data points at concentrationcj in a neighborhood aroundTi (three points on either side ofTi).

With these assumptions, the likelihood of observing the data given the estimate(∆H₀^◦,∆S^◦₀)is simply

L(∆H₀^◦,∆S₀^◦) = Y

T_i,c_j

φ_fTi,cj,σ²

Ti,cj(DTi,cj) (5.72)

whereφµ,σ² is the probability density function of the Gaussian distribution with meanµand vari- anceσ².

Starting with a uniform prior over the (∆H^◦,∆S^◦) grid, the posterior probability distribution

0 1 2 5 10 Length of tails at the junction

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

0 1 2 5 10 15 20

Number of strand displacement steps completed

6 G$ 55 (kcal/mol)

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

A B

Figure 5.10: ^∆G◦of formation (at55^◦C) of complexes in thestrand displacement snapshotstudy (A) or the local overhangstudy (B). Error bars in black indicate Bayesian posterior means and99%confidence intervals.

Error bars in red 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.

is proportional to the likelihood (this standard result from Bayesian statistics is justified later). So, normalizing the likelihood of observing our data, we can calculate the posterior distribution:

P(∆H₀^◦,∆S₀^◦) = L(∆H₀^◦,∆S₀^◦) X

∆H_i^◦,∆S_j^◦

L(∆H_i^◦,∆S_j^◦) (5.73)

In practice, we first perform a coarse discretization of(∆H^◦,∆S^◦)space in order to identify the region containing non-zero values of the posterior probability; we then perform a fine discretiza- tion of that region and evaluate the posterior probability over it.

Once we have the posterior probability over(∆H^◦,∆S^◦)space, we find the smallest region containing99%of the probability, and then evaluate marginal posterior probability distributions for∆H^◦,∆S^◦,∆G^◦₂₅ and ∆G^◦₅₅ (Figure5.9). The 99% confidence intervals are relatively much narrower for∆G^◦₅₅ than∆H^◦or∆S^◦. This shows that our data permits accurate comparison of the stability of our complexes through∆G^◦, but cannot easily separate the enthalpic and entropic contributions. Also note that error bars and 99% confidence intervals are much narrower for∆G^◦₅₅ (Figure5.10) compared to∆G^◦₂₅(Figure2.8in the main text). This is to be expected because the former temperature is closer to the experimental melting temperature of our complexes. With the assumptions in the two-state model and our Bayesian framework, we report posterior means and 99% confidence intervals for quantities of interest (Table5.6, Figure5.10).

Relationship between posterior probability and likelihood. We now recall that with a uni- form prior, the posterior probability distribution is proportional to the likelihood. For a more detailed introduction, see Gelman et al. [220]. Supposeθ is a vector of parameters we want to infer, and that we have dataDwhich is informative aboutθ. Then, we know

P(θ,D) =P(θ)×P(D|θ) =P(D)×P(θ|D) (5.74) Therefore the posterior distributionP(θ|D)is obtained by

P(θ|D) = P(θ)×P(D|θ)

P(D) (5.75)

Here,P(θ)is constant because we start with a uniform prior. P(D) =P

θP(θ)×P(D|θ)is also independent ofθ.P(D|θ)is nothing but the likelihood. Hence, with a uniform prior, the posterior distribution is proportional to the likelihood.

Leave-one-concentration-out analysis. This is a simple and descriptive way of analyzing the data, which essentially serves as a sanity check. Data from each complex is analyzed separately to infer the free energy of formation of that complex. We measured temperature-dependent ab- sorbance data at four concentrations. Here, we sequentially leave out data from one concentration at a time, thus generating four datasets, each containing data from three concentrations. For each dataset, we perform a simultaneous nonlinear least squares fit (using the Levenberg-Marquardt algorithm, implemented by a built-in MATLAB function) of the predicted melt fraction curves to the smoothed and normalized absorbance data across all three concentrations present in the dataset. This procedure generates four estimates of(∆H^◦,∆S^◦)of formation for each complex, one for each leave-one-concentration-out dataset. We then calculate∆G^◦₂₅and∆G^◦₅₅for each of those four estimates and report the mean and standard deviation, for each complex (Table5.5).