By performing a global analysis using Bayesian inference, all nec- essary parameters can be determined even without in-detail prior knowledge; however, some of the parameters are correlated and thus there is not enough information in the recorded data to accurately determine these parameters individually, most notably the rates k12
andk21(figure 4.9 A).
Here, the uncertainties in the amplitude of the experimental corre- lation function caused by for example, poor statistics make it impossi-
4.2 Influence of dye selection on DNA hairpin dynamics 71 12 21 12 21 12 12 21(1/s) p ro b a b ili ty (a .u )
C
D
p ro b a b ili ty (a .u ) k + k /s-1 12 21 k /s-1 12 k + k / s -1 12 21 k / s -1 21 omega / omegaz r p ro b a b il ity / a .u p ro b a b il ity / a .u omega /µmr o m e g a /µ m zA B
Figure 4.9: Correlations of experimental parameters. A) Two-dimensional histogram of the equally weighted samples obtained for the rate constants
k12andk21for an Atto532–Atto647N sample containing 10mMNaCl mea-
sured at 60µWlaser power. The top and side show the corresponding one-
dimensional projection. The two parameters are strongly correlated. B) Green focal volume results showing a strong correlation betweenωr and
ωz. C) Two-dimensional histogram and one-dimensional projections of the sum of transition rates versus the structural parameter (ωz/ωr). In contrast to the individual rates, the sum of rates as well as the structural parameter can be determined accurately. D) Schematic representation of the fit-error determination. Errors are given as the distance of the best fit-ting sample (blue point) to the lower and upper boarders that mark 1σof the distribu-
tion (black lines). Since this best fitting sample is not required to show an equal distance to both boarders, the greater of the two distances was used as fit error for weighted averaging of the results.
72 New developments and applications
ble to resolve parameters relying on small changes in this amplitude (this is problematic also for E1 and E2, data not shown). Instead,
the sum of the ratesk12+k21 (rate constant of the exponential decay in Equations 4.8– 4.10) can be determined with high accuracy (fig- ure 4.9 C). Likewise, the structural parameter defined as the ratio of lengthωzto widthωr of the focal volume (see figure 4.9 B and C) can
be determined accurately whileωz andωrare highly correlated.
To summarize the accuracy of the parameter estimation, the fit error was always defined as the distance of the maximum posterior sample to the points in the distribution including 68 % of the prob- ability as shown in figure 4.9 D. Note that for this type of multi- dimensional analysis, oftentimes the best fitting sample (i.e. maxi- mum posterior) is not positioned exactly in the middle of that inter- val; in fact, it does not even need to lie inside it.
Both the choice of dye molecules and the salt buffer conditions (i.e. the salt concentration) have an effect on the equilibrium con- stant, as well as on the kinetics between open and closed states for DNA hairpins. For the particular hairpins used in this work, assum- ing a perfect two-state (open-closed) system, one expects transitions from the open state with∼10–20 % FRET (depending on the Förster distance of the respective dye pair) to the closed state with ∼100 % FRET.
The results obtained for the kinetics of hairpins labeled with dif- ferent dye pairs and measured at different salt concentrations are summarized in figure 4.10. The total rate shows almost no depen- dence on the dye pair at 10 mM salt. At 160 mM NaCl, only the Alexa532-Alexa647 dye combination has a significantly higher rate whereas at 320mMNaCl, more distinct influences of the dye molecules are observed:
Hairpins labeled with Atto647N show slow transitions but no differences between the donor dyes (6-Tamra, Alexa532, Atto532). Changing the red dye to Alexa647 leads, however, to a significant rate increase. In this case, Atto532 and 6-Tamra labeled hairpins show equal rates while hairpins with Alexa532 open and close much faster but at rates similar to the ones found for 160mMNaCl.
4.2 Influence of dye selection on DNA hairpin dynamics 73
Dye combination
0 1000 2000 3000 4000 5000 6000 7000 8000 9000 Alexa532 Atto647N Tamra-6 Atto647N Atto532 Atto647N Atto532 Alexa647 Tamra-6 Alexa647 Alexa532 Alexa647 10 mM 160 mM 320 mM 2 1 1 2k
+
k
Figure 4.10: Summary of the kinetic results obtained from the FCS exper- iments. Overview of the determined hairpin kinetic rates for a variety of dye combinations and salt concentrations; determined sum of kinetic rates
k12 andk21for 10mM(blue), 160mM(red) and 320mM(green) NaCl. Ex- periments were performed in triplicates and error bars were calculated for the fit-error-weighted averages. Samples not showing useful kinetic infor- mation are marked in white (dominating photophysical effects : Atto532 Atto647N 10mM, no kinetic signal detected : Tamra-6 Alexa647 10mM).
used. If the two state model is valid no systematic fluctuations should be observable. While the FCS data presented in figure 4.5 show the typical situation for hairpins containing Alexa647, which can be described using a simple two-state model, it was found that using Atto647N instead causes unexpectedly high residuals of the fits to the correlation functions (figure 4.11).The observed anti-correlations of the residuals from the fits to the donor auto-correlation and to the donor-FRET cross-correlation show an anti-correlation on a timescale of a few 100µs(figure 4.11 B), which has previously been described as an indicator for the presence of additional kinetics [125]. More than two states have been reported for a five-base-pair T21 DNA hairpin,
74 New developments and applications
Figure 4.11: Summary of FCS datasets showing systematic deviations from the simple two-state dynamic model. A) Summary of data with (Yes, light grey) and without (No, dark grey) significant remaining residuals after globally fitting to a two-state dynamic model as described in chap- ter 4.2.3, B) Exemplary overlay of GGxGG auto-correlation (green) and GRxGG cross-correlation (red) residuals (Atto532-Atto647N, 320mMNaCl)
showing anti-correlated behavior.
but until now only one transition had been observed by conventional FCS [49].