Testing Landscape Theory for Biomolecular Processes with Single Molecule Fluorescence Spectroscopy

Although Kramers’ theory for diffusive barrier crossing on a 1D free energy profile plays a central role in landscape theory for complex biomolecular processes, it has not yet been rigorously tested by experiment. Here we test this 1D diffusion scenario with single molecule fluorescence measurements of DNA hairpin folding. We find an upper bound of $2.5\mu \mathrm{s}$ for the average transition path time, consistent with the predictions by theory with parameters determined from optical tweezer measurements.

Figure 1

(a) A one-dimensional free energy surface (blue curve) for folding with $x$ as the reaction coordinate (from Ref. [2]). The transition path is the rare segment of the folding trajectory during which there is a successful crossing of the barrier between the unfolded and folded states and is defined as a trajectory which crosses $x_0$ and reaches $x_1$ on the other side of the barrier without recrossing $x_0$ (violet curve). (b) Trajectory of an experimental observable, such as the FRET efficiency in single molecule fluorescence experiments or molecular extension in constant-force optical tweezer experiments, illustrating that in experiments the transition path appears as a nearly instantaneous jump.

Figure 2

Schematic diagram of the single-stranded DNA construct [$5^{\prime }\text{−}\mathrm{AACC}({\mathrm{T}}_{21})\mathrm{GGTT}-3^{\prime }$] in an unfolded and a hairpin-folded conformation with fluorescent dye labels and immobilized on a polyethylene glycol coated coverslip via a biotin-streptavidin-biotin linkage. $k_F$ and $k_U$ are the folding and unfolding rate coefficients. The 2 thymines that form base pairs in the folded hairpin are distinguished by a darker purple color.

Figure 3

Representative single molecule FRET data for an immobilized DNA hairpin at high power density ($20\mathrm{kW}/{\mathrm{cm}}^2$ and an average count rate of $870\mathrm{photons}/\mathrm{ms}$). (a) FRET efficiency trajectory. FRET efficiency was calculated for photons collected in $50\mu \mathrm{s}$ bins (b) Trajectories of donor (green) and acceptor (red) fluorescence ($50\mu \mathrm{s}$ bins). (c) Photon trajectory. Each circle represents a detected photon. Red circles (upper row) are acceptor photons. Green circles (lower row) are donor photons. The dashed line indicates a transition predicted by the Viterbi algorithm [38, 39]. (d) FRET efficiency histogram (power density of $20\mathrm{kW}/{\mathrm{cm}}^2$; $870\mathrm{detected}\mathrm{photons}/\mathrm{ms}$) calculated from photons in $50\mu \mathrm{s}$ bins [40]. The continuous red curves are Gaussian fits to the 2 peaks of the histogram using the width expected from shot noise. Intermediate values of the FRET efficiency contribute to the histogram, because $\sim 15\%$ of the bins used to construct the histogram contain a transition between the folded and unfolded states.

Figure 4

Kinetic model and maximum likelihood analysis of photon trajectories. (a) The noisy curve is a schematic of the instantaneous FRET for a transition path from the unfolded to folded states. The step curve is the simplest discretization of the path with average FRET efficiencies $E_U$ for the unfolded state, $E_F$ for the folded state, and an intermediate value $E_S=(E_U+E_F)/2$ for the virtual step state. (b) Comparison of the values of the likelihood function from the three state analysis, $L({\tau }_S)$, and a two-state analysis with an instantaneous transition, $L(0)$, calculated for 780 transitions from immobilized molecules. The result of the calculation is that no lifetime greater than $2.5\mu \mathrm{s}$ is detected by the measurement, so the average transition path time is shorter than $2.5\mu \mathrm{s}$. Acceptor blinking can interfere with the transition path time analysis [42], but was effectively eliminated by not including transitions in which there are nearby strings of 8 or more donor photons [31].

Figure 5

Transition path time as a function of the number of base pairs in the stem of the folded structure. The green circles with error bars were calculated from Eq. (2) by Neupane et al. [18] from force spectroscopic determinations of $\mathrm{\Delta }{G}^{*}$, $D^{*}$, and $k_b$ for DNA hairpins with stem lengths of 20 and 30 base pairs and a DNA aptamer with a stem of 9 base pairs. The red square with an arrow attached is the upper bound of $2.5\mu \mathrm{s}$ determined in this work for the 4-base-pair stem of the DNA hairpin using single molecule FRET. The broken curve is from Neupane et al., which assumes that $t_{\mathrm{TP}}\sim N$, while the continuous curve is $t_{\mathrm{TP}}\sim N^{\alpha }$ with $\alpha =1.6$ from Frederickx et al. [28]. Inset is the same points and curves on a linear scale.