Purification and labeling of nucleic acidsTerminally functionalized homopolymeric oligonucleotides with a 5ʹ-end dithiol and a 3ʹ-end primary amine for labeling (\({{{{{{\rm{dA}}}}}}}_{19}\), \({{{{{{\rm{dC}}}}}}}_{19}\), \({{{{{{\rm{dT}}}}}}}_{19}\), \({{{{{{\rm{dT}}}}}}}_{19}^{{{{{{\rm{ab}}}}}}}\), \({{{{{{\rm{dT}}}}}}}_{38}\), \({{{{{{\rm{rA}}}}}}}_{19}\), \({{{{{{\rm{rC}}}}}}}_{19}\) and \({{{{{{\rm{rU}}}}}}}_{19}\); Supplementary Table 1) were synthesized and purified by high pressure liquid chromatography (HPLC) by Integrated DNA Technologies. Prior to labeling, the oligonucleotides were dissolved in 10 mM sodium phosphate buffer pH 7, and filtered and concentrated (Amicon Ultra-0.5 mL, MWCO 3 KDa) to remove free primary amines that interfere with downstream reactions. After this step, each oligonucleotide was site-specifically labeled at the 5ʹ-end with thiol-reactive Alexa Fluor 594 maleimide, and at the 3ʹ-end with amine-reactive Alexa Fluor 488 succinimidyl ester according to the following procedure. The synthetically incorporated thiol groups at the 5ʹ-ends were reduced with 100 mM tris(2-carboxyethyl)phosphine (TCEP) at oligonucleotide concentrations of ~10 µM. After ~1 h, the buffer of the samples was exchanged to 10 mM sodium phosphate pH 7, and the samples were concentrated (Amicon Ultra-0.5 mL Centrifugal Filters MWCO 3 KDa) to ~10 µM. The acceptor dye (dissolved in 5 µL dimethyl sulphoxide (DMSO) and vortexed) was added to the sample at a ratio of 10:1 (dye:oligonucleotide) and incubated for 60 min. For the reaction of the amine-reactive donor dye with the 3ʹ-end of the oligonucleotides, the pH of the acceptor labeling reaction mixtures was increased to pH 8 by addition of 1 M sodium phosphate buffer pH 8. The donor dye (dissolved in 5 µl DMSO and vortexed) was added to the corresponding reaction mix in a tenfold excess over oligonucleotide and incubated for 60 minutes. Unreacted dye was removed with a desalting spin column (Zeba, Pierce, MWCO 7 kDa), and the labeled constructs were purified on a reversed-phase column (Dr. Maisch ReprosilPur 200 C18-AQ, 5 µm) using HPLC (Agilent 1100 series). Samples were lyophilized overnight, then dissolved in H2O, and stored at −80 °C until use.Production of ZMWsUsing borosilicate glass coverslips coated with a 100 nm aluminum layer (Deposition Research Laboratory, St. Charles, MO), ZMWs with a diameter of 120 nm were milled into the aluminum layer at a 90° angle using a gallium focused ion beam (FIB-SEM Zeiss Auriga 40 CrossBeam) with a voltage of 30 kV and a 10 pA beam current at room temperature. Before the experiments, the ZMWs were cleaned with double-distilled water and ≥ 99.7% ethanol to remove dust. They were then exposed to a 5 min air plasma treatment, followed by a 12 h incubation at room temperature in nitrogen atmosphere with 1 mg/mL of silane-modified PEG 1000 dissolved in ethanol and 1% acetic acid. After incubation, the ZMWs were washed with ethanol and 1% Tween 20 to remove excess PEG-silane, followed by a final rinse with ethanol and water before air drying.38,54.Single-molecule spectroscopySingle-molecule fluorescence experiments with and without ZMWs were performed on a four-channel MicroTime 200 confocal instrument (PicoQuant) equipped with either an Olympus UplanApo 60x/1.20 Water objective for measurements without ZMW or an Olympus UplanSapo 100x/1.4 Oil objective for measurements with ZMWs. Alexa 488 was excited with a diode laser (LDH-D-C-485, PicoQuant) at an average power of 100 µW (measured at the back aperture of the objective). The laser was operated in continuous-wave mode for nsFCS experiments and in pulsed mode with interleaved excitation (PIE)55 for fluorescence lifetime measurements. The wavelength range used for acceptor excitation was selected with two band pass filters (z582/15 and z580/23, Chroma) from the emission of a supercontinuum laser (EXW-12 SuperK Extreme, NKT Photonics) operating at a pulse repetition rate of 20 MHz (45 µW average laser power after the band pass filters). The SYNC output of the SuperK Extreme was used to trigger interleaved pulses from the 488-nm diode laser. Sample fluorescence was collected by the microscope objective, separated from scattered light with a triple band pass filter (r405/488/594, Chroma) and focused on a 100-µm pinhole. After the pinhole, fluorescence emission was separated into two channels, either with a polarizing beam splitter for fluorescence lifetime measurements, or with a 50/50 beam splitter for nsFCS measurements to avoid the effects of detector deadtimes and afterpulsing on the correlation functions5. Finally, the fluorescence photons were distributed by wavelength into four channels by dichroic mirrors (585DCXR, Chroma), additionally filtered by band pass filters (ET 525/50 M and HQ 650/100, Chroma), and focused onto one of four single-photon avalanche detectors (SPCM-AQRH-14-TR, Excelitas). The arrival times of the detected photons were recorded with a HydraHarp 400 counting module (PicoQuant, Berlin, Germany).All free-diffusion single-molecule experiments were conducted with labeled oligonucleotide concentrations between 100 and 250 pM without ZMWs or between 50 and 300 nM with ZMWs in 10 mM HEPES buffer pH 7.0 (adjusted with 35 mM NaOH), 0.01% Tween 20, 143 mM β-mercaptoethanol (BME), 150 mM NaCl, and for the viscosity dependence with appropriately chosen concentrations of glycerol (without ZMW) in 18-well plastic slides (ibidi) or in ZMWs at 22 °C.Single-molecule FRET data analysisData analysis was carried out using the Mathematica (Wolfram Research) package Fretica (https://github.com/SchulerLab). For the identification of photon bursts, the photon recordings were time-binned (1 ms binning for measurements without ZMWs, 0.2 ms for measurements with ZMWs). Photon numbers per bin were corrected for background, crosstalk, differences in detection efficiencies and quantum yields of the fluorophores, and for direct excitation of the acceptor56. Bins with more than 50 photons were identified as photon bursts. Ratiometric transfer efficiencies were obtained for each burst from E = nA/(nA + nD), where nA and nD are the corrected numbers of donor and acceptor photons in the photon burst, respectively. The E values were histogrammed. The subpopulation corresponding to the FRET-labeled species was fitted with a Gaussian peak function or analyzed by photon distribution analysis taking into account the experimentally observed burst size distribution57,58,59 (Supplementary Fig. 1). Bursts from experiments in PIE mode were further selected according to the fluorescence stoichiometry ratio60,61,62, S (0.2 < S < 0.8) (Fig. 2, Supplementary Fig. 3).End-to-end distance distributionsFor analyzing the single-molecule FRET data of the ssNA variants, we employed end-to-end distance distributions of analytical polymer models as well as the distance distributions obtained by the hierarchical chain growth (HCG) approach31 (see hierarchical chain growth). The three polymer models used and the corresponding end-to-end distance probability density functions were:Gaussian chain (GC)8:$${P}_{G{{{{{\rm{C}}}}}}}\left(r\right)=4\pi {r}^{2}{\left[\frac{3}{2\pi \left\langle {r}^{2}\right\rangle }\right]}^{\frac{3}{2}}{e}^{-\frac{3}{2}\frac{{r}^{2}}{\left\langle {r}^{2}\right\rangle }}$$
(2)
Worm-like chain (WLC)8,41:$${P}_{{{{{{\rm{WLC}}}}}}}\left(r\right)=\; \left\{\begin{array}{cc}C{\left(r/{l}_{c}\right)}^{2}{\left(1-{\left(r/{l}_{c}\right)}^{2}\right)}^{-9/2}{e}^{-{3l}_{c}/{\left[4{l}_{p}{\left(1-{\left(r/{l}_{c}\right)}^{2}\right)}\right]}},&r\le {l}_{c}\\ \hfill 0 \hfill,& r \, > \, {l}_{c}\end{array}\right.$$
(3)
where \({l}_{c}\) and \({l}_{p}\) are the contour- and persistence lengths of the chain, respectively (Supplementary Table 3). C is a normalization constant.Self-avoiding walk polymer (SAW-ν)63:$$\begin{array}{c}{P}_{{{{{{\rm{SAW}}}}}}-\nu }\left(r\right)=A\frac{4\pi }{R}{\left(\frac{r}{R}\right)}^{2+g}{e}^{-\alpha {\left(\frac{r}{R}\right)}^{\delta }},\\ {{{{{\rm{with}}}}}}\;{R=\left\langle {r}^{2}\right\rangle }^{\frac{1}{2}},\;\,g=\frac{\gamma -1}{\nu },\;\delta=\frac{1}{1-\nu },\;\gamma \, \approx \, 1.1615,\;{{{{{\rm{and}}}}}}\;\nu=\frac{{{{{\mathrm{ln}}}}}\left(\frac{R}{b}\right)}{{{{{\mathrm{ln}}}}}(n)},\end{array}$$
(4)
where \(b\) and \(n\) are the segment length and the number of segments of the polymer, respectively (Supplementary Table 3). \(A\) is a normalization constant.If the rotational correlation time, \({\tau }_{{rot}}\) (Supplementary Table 2), of the chromophores is short relative to the fluorescence lifetime, \({\tau }_{D}\), of the donor (such that orientational factor \({\kappa }^{2} \, \approx \, 2/3\))34, and the end-to-end distance dynamics of the polypeptide chain (with relaxation time \({\tau }_{r}\)) are slow relative to \({\tau }_{D}\), the experimentally determined mean transfer efficiency, \(\left\langle E\right\rangle\), can be related to the distance distribution, \(P(r)\), by64:$$\left\langle E\right\rangle=\left\langle \varepsilon \right\rangle \equiv {\int }_{0}^{{{{{{\rm{}}}}}}{{\infty }}}\varepsilon \left(r\right)P(r){dr},$$
(5)
where \(\varepsilon \left(r\right)={R}_{0}^{6}/({R}_{0}^{6}+{r}^{6})\), and \({R}_{0}\) is the Förster radius (5.4 nm for Alexa 488/594)26,38,65,66. See Supplementary Table 3 for the values of the parameters used (\({l}_{c}\), \(b\), \(n\)) and inferred (\(R\), \({l}_{p}\), \(\nu\)) by solving Eq. 5 numerically for the corresponding variable. Time-resolved fluorescence anisotropy measurements (Supplementary Fig. 4) indicate high mobility of the fluorophores (Supplementary Table 2), suggesting \({\kappa }^{2}\, \approx \,2/3\).Effect of glycerol on conformational free energyViscogens can affect intramolecular interactions and thus lead to changes in conformational free energy. The changes in transfer efficiency upon addition of glycerol were small under the conditions used here; we estimated the order of magnitude of the effect based on a simple approximation. Distance distributions, \(P(r)\), can be converted into potentials of mean force, \(F\left(r\right)\), through Boltzmann inversion (Fig. 1f, Supplementary Fig. 5):$$F\left(r\right)=-{k}_{B}T\, {{{{\mathrm{ln}}}}}\,P(r),$$
(6)
where \({k}_{B}\) is the Boltzmann constant and \(T\) the temperature. To estimate the change in conformational free energy upon addition of glycerol, we utilized Eq. 6 for the ssNA that exhibited the largest influence of glycerol on transfer efficiency, \({{{{{\rm{d}}}}}}{{{{{{\rm{T}}}}}}}_{19}(\Delta E=0.07)\), assuming a Gaussian chain distance distribution, with \({R}_{0}\) corrected for the refractive index change due to glycerol. The free energy change was estimated from$$\frac{\Delta F}{{k}_{B}T}={\int }_{o}^{{{\infty }}}{P}_{{{{{{\rm{GC}}}}}}}^{(35\%)}\left(r\right)\, {{{{\mathrm{ln}}}}}{P}_{{{{{{\rm{GC}}}}}}}^{(35\%)}\left(r\right)\,{dr}-{\int }_{o}^{{{\infty }}}{P}_{{{{{{\rm{GC}}}}}}}^{\left(0\%\right)}\left(r\right)\, {{{{\mathrm{ln}}}}}{P}_{{{{{{\rm{GC}}}}}}}^{\left(0\%\right)}\left(r\right)\,{dr}={{{{\mathrm{ln}}}}}\sqrt{\frac{\left\langle {r}_{35\%}^{2}\right\rangle }{\left\langle {r}_{0\%}^{2}\right\rangle }},$$
(7)
where \({P}_{{{{{{\rm{GC}}}}}}}^{(0\%)}(r)\) and \({P}_{{{{{{\rm{GC}}}}}}}^{(35\%)}(r)\) are the probability density functions of the chain at 0% and 35% glycerol, respectively, and \(\left\langle {r}_{0\%}^{2}\right\rangle\) and \(\left\langle {r}_{35\%}^{2}\right\rangle\) are the corresponding mean squared end-to-end distances. The resulting conformational free energy change is \(\Delta F\approx 0.1{k}_{B}T\), corresponding to a change in \(R={\left\langle {r}^{2}\right\rangle }^{1/2}\) by ~\(0.7\) nm. We thus conclude that the energetic changes within the chain upon glycerol addition are unlikely to affect internal friction.Single-molecule fluorescence lifetime analysisFrom PIE experiments, the donor and acceptor fluorescence lifetimes, \({\tau }_{{{{{{\rm{D}}}}}}}\) and \({\tau }_{{{{{{\rm{A}}}}}}}\), for each burst were determined from the mean detection times, τ’D and τ′A, of all photons of a burst detected in the donor and acceptor channels. These times are measured relative to the preceding pulses of the laser triggering electronics. Photons of orthogonal polarization with respect to the excitation polarization were weighted by 2 G to correct for fluorescence anisotropy effects; G corrects for the polarization-dependence of the detection efficiencies. For obtaining the mean fluorescence lifetimes, we further corrected for the effect of background photons and for a time shift due to the instrument response function (IRF) with \({\tau }_{x={{{{{\rm{D}}}}}},{{{{{\rm{A}}}}}}}=\frac{{\tau }_{x}^{{\prime} }-\alpha {\left\langle t\right\rangle }_{{bg},x}}{1-\alpha }-{\left\langle t\right\rangle }_{{{{{{\rm{IRF}}}}}}}\), with \(\alpha={n}_{{bg},x}\Delta /{N}_{x}\). Here, \({\left\langle t\right\rangle }_{{bg},x}\) is the mean arrival time of the background photons, \({\left\langle t\right\rangle }_{{{{{{\rm{IRF}}}}}}}\) is the mean time of the IRF, \({n}_{{bg},x}\) is the background photon detection rate, \(\Delta\) the burst duration, and \({N}_{x}\) the uncorrected number of photons in the donor (x = D) or acceptor (x = A) channels67. The distributions of relative lifetimes, \({\tau }_{{{{{{\rm{D}}}}}}}\)/\({\tau }_{{{{{{\rm{D}}}}}}0}\) and (\({\tau }_{{{{{{\rm{A}}}}}}}\) − \({\tau }_{{{{{{\rm{A}}}}}}0}\))/\({\tau }_{{{{{{\rm{D}}}}}}0}\), versus transfer efficiency for the FRET-active population are shown in Figs. 1e, 2a and Supplementary Fig. 3. \({\tau }_{{{{{{\rm{D}}}}}}0}\) and \({\tau }_{{{{{{\rm{A}}}}}}0}\) are the fluorescence lifetimes of donor and acceptor in the absence of FRET, respectively (see Supplementary Table 2). Supplementary Fig. 3 shows the distributions of relative donor lifetime versus transfer efficiency including the donor-only population. \({\tau }_{{{{{{\rm{A}}}}}}0}\) and \({\tau }_{{{{{{\rm{D}}}}}}0}\) were obtained from independent ensemble lifetime measurements as described below. The figures show dynamic FRET lines42 that were calculated assuming end-to-end distance distributions, \(P\left(r\right)\), for a Gaussian chain8 (GC), a worm-like chain8 (WLC), and for the SAW-\(\nu\) polymer63 (SAW-\(\nu\)) models. For the case that \(P\left(r\right)\) is sampled faster than the interphoton time ( ~ 10 μs) but slowly compared to \({\tau }_{{{{{{\rm{D}}}}}}}\) (3.5–4 ns; Supplementary Table 2), it has been shown that36$$\frac{{\tau }_{{{{{{\rm{D}}}}}}}}{{\tau }_{{{{{{\rm{D}}}}}}0}}=1-\left\langle \varepsilon \right\rangle+\frac{{\sigma }_{\varepsilon }^{2}}{1-\left\langle \varepsilon \right\rangle },$$and$$\frac{{\tau }_{{{{{{\rm{A}}}}}}}-{\tau }_{{{{{{\rm{A}}}}}}0}}{{\tau }_{{{{{{\rm{D}}}}}}0}}=1-\left\langle \varepsilon \right\rangle -\frac{{\sigma }_{\varepsilon }^{2}}{\left\langle \varepsilon \right\rangle },$$
(8)
where \({\sigma }_{\varepsilon }^{2}={\int } ^{\infty }_{0}{(\varepsilon (r)-\langle \varepsilon \rangle )}^{2}P(r){dr}\) is the variance of the transfer efficiency distribution corresponding to \(P(r)\). The dynamic FRET lines were obtained by varying the model parameters of the corresponding distributions, \(\left\langle {r}^{2}\right\rangle\) for the GC; the persistence length, lp, for the WLC; and the scaling exponent, \(\nu\), for the SAW-\(\nu\) model, respectively. The static FRET line, \({\tau }_{{{{{{\rm{D}}}}}}}/{\tau }_{{{{{{\rm{D}}}}}}0}=({\tau }_{{{{{{\rm{A}}}}}}}-{\tau }_{{{{{{\rm{A}}}}}}0})/{\tau }_{{{{{{\rm{D}}}}}}0}=1-\left\langle \varepsilon \right\rangle\), corresponds to fixed interdye distances. Note that this type of fluorescence lifetime analysis is only valid for the regime where \({\tau }_{{{{{{\rm{rot}}}}}}}\) is short relative to \({\tau }_{{{{{{\rm{D}}}}}}}\), and \({\tau }_{{{{{{\rm{D}}}}}}}\) is short relative to \({\tau }_{{{{{{\rm{r}}}}}}}\), i.e., \({\tau }_{{{{{{\rm{rot}}}}}}} \, < \, {\tau }_{{{{{{\rm{D}}}}}}} \, < \, {\tau }_{{{{{{\rm{r}}}}}}}\) (Supplementary Table 2, Supplementary Fig. 4).Hierarchical chain growth and fluorophore modelingTo carry out hierarchical chain growth (HCG), we created a molecular dynamics (MD) fragment library. Subsequently, we built heterotetramers with sequence d/rGXYZ. G served as a fixed head group at the 5ʹ end. For the other nucleotides “XYZ”, we used all \({4}^{3}\) combinations of thymine, uracil, cytosine and adenine. The heteromeric fragment library was extensively sampled via temperature replica exchange MD simulations, utilizing the parmBSC1 force-field68 for DNA and the DESRES force-field30 for RNA. For both DNA and RNA, the TIP4P-D water model69 was used. Fragments were placed in a dodecahedral box, solvated with 150 mM NaCl and neutralized, resulting in a system comprising ~6600 atoms. Depending on the fragment sequence, the total number varied by about 50 atoms. The fragment with the abasic site was parameterized as described by Heinz et al.70. MD simulations were performed using GROMACS/2019.6.71 For each system, we ran 24 replicas over a temperature range of 300–420 K for 100 ns as described before32. Afterwards, we randomly selected fragment conformations from the MD fragment library at 300 K to assemble disordered ssNAs with HCG in a hierarchical manner32.We also used HCG to build libraries of dye-labeled DNA and RNA 4-mer fragments. As inputs, we used the 4-mer libraries built here and the libraries built by Grotz et al.29 for the dyes Alexa Fluor 594 and Alexa Fluor 488 attached to dideoxyadenosine monophosphate (dA2) and dideoxythymidine monophosphate (dT2) at the 5ʹ and 3ʹ ends, respectively. The use of dA2- and dT2-dye fragments to model fluorophores attached to DNA and RNA chains has been validated by Grotz et al.29 We used the dA2 library for purines (A, G) at the respective end and the dT2 library for pyrimidines (U, C). Pairs of random structures were repeatedly drawn from the library of DNA or RNA 4-mer fragments and from library of dA2 or dT2 labeled with Alexa 594 or Alexa 488. For each pair, we performed a rigid body superposition of the heavy atoms of the terminal sugar moiety and nucleobase, leaving out non-matching atoms of the base. If the RMSD of the superposition was below 0.8 Å, we searched for clashing heavy atoms within a pair distance of 2.0 Å. If no clashing atoms were detected, the dye was attached to the DNA or RNA 4-mer fragment according to the superposition, excluding the terminal oxygen atoms of the 4-mer. The resulting libraries of DNA and RNA 4-mer structures with the FRET dyes Alexa Fluor 594 and Alexa Fluor 488 attached at their 5ʹ or 3ʹ ends were subsequently used to build dye-labeled DNA and RNA chains by HCG.Bayesian ensemble refinementTo optimize the agreement of the HCG ensembles with the experimental data, we reweighted the ensembles of configurations based on two experimental observables from the single-molecule measurements: the mean transfer efficiency, \(\left\langle E\right\rangle\), and the variance of the underlying transfer efficiency distribution, \({\sigma }_{\varepsilon }^{2}\), as described in Single-molecule fluorescence lifetime analysis (for experimental \(\left\langle E\right\rangle\) and \({\sigma }_{\varepsilon }^{2}\) of the individual constructs, see Supplementary Table 6). The transfer efficiency was calculated for each of the \(N\) ensemble members, and uniform weights \({w}_{\alpha }^{0}=1/N\) were initially assigned to all of them. Optimal weights were found using Bayesian inference of ensembles43 (BioEn) by minimizing$$\begin{array}{c}\Delta G({w}_{1},\ldots,{w}_{N})=\frac{1}{2}{\chi }^{2}-\theta \Delta S\\ {{{{{\rm{with}}}}}}\, {\chi }^{2}=\frac{{\left({\left\langle \varepsilon \right\rangle }_{{{{{{\rm{BioEn}}}}}}}-\left\langle E\right\rangle \right)}^{2}}{{Var}(\left\langle E\right\rangle )}+\frac{{({{\sigma }_{\varepsilon }^{2}}_{{{{{{\rm{BioEn}}}}}}}-{\sigma }_{\varepsilon }^{2})}^{2}}{{Var}({\sigma }_{\varepsilon }^{2})}\\ {{{{{\rm{and}}}}}}\, \Delta S=-{\sum }_{\alpha=1}^{N}{w}_{\alpha }\, {{{{\mathrm{ln}}}}}\frac{{w}_{\alpha }}{{w}_{\alpha }^{0}}\end{array}$$
(9)
The optimal weights \({w}_{\alpha }\) of ensemble members \(\alpha\) are written in terms of two generalized forces43 \(f\) and \(g\) for the first and second power of the respective transfer efficiency \({\varepsilon }_{\alpha }\), i.e., \({w}_{\alpha }\propto {w}_{\alpha }^{0}\exp \left(f{\epsilon }_{\alpha }+g{\epsilon }_{\alpha }^{2}\right)\) with \({\sum }_{\alpha=1}^{N}{w}_{\alpha }=1\). The minimum of \(\Delta G\) as function of \(f\) and \(g\) was found with a 2D Newton–Raphson solver, staying in the convex region by first increasing and then step-wise decreasing \(\theta\) to the target value. Reweighted values are given by \({\left\langle \epsilon \right\rangle }_{{{{{{\rm{BioEn}}}}}}}={\sum }_{\alpha=1}^{N}{w}_{\alpha }{\varepsilon }_{\alpha }\) and \({{\sigma }_{\varepsilon }^{2}}_{{{{{{\rm{BioEn}}}}}}}={\sum }_{\alpha=1}^{N}{w}_{\alpha }{\varepsilon }_{\alpha }^{2}-{\left\langle \varepsilon \right\rangle }_{{{{{{\rm{BioEn}}}}}}}^{2}\), where \({w}_{\alpha }\) and \({\varepsilon }_{\alpha }\) are the weight and the transfer efficiency of the \(\alpha\) th ensemble member, respectively, with \({\sum }_{\alpha=1}^{N}{w}_{\alpha }=1\). For each ensemble, we chose the largest value of \(\theta\) for which \({\left\langle \varepsilon \right\rangle }_{{{{{{\rm{BioEn}}}}}}}\) and \({{\sigma }_{\varepsilon }^{2}}_{{{{{{\rm{BioEn}}}}}}}\) agreed with the measured values within the experimental uncertainties of \({Var}{\left(\left\langle E\right\rangle \right)}^{1/2}=0.03\) and \({Var}{\left({\sigma }_{\varepsilon }^{2}\right)}^{1/2}=0.003\), respectively. The dye-to-dye distance distributions of initial and reweighted ensembles for all ssRNAs and \({{{{{{\rm{dT}}}}}}}_{19}^{{{{{{\rm{ab}}}}}}}\) are depicted in Supplementary Fig. 5. A useful quantity to estimate the quality of the prior distribution is the effective fraction of configurations used from the initial ensemble, \({\phi }_{{{{{{\rm{eff}}}}}}}={e}^{\Delta S}\). For the 19 mer ssRNA ensembles and \({{{{{{\rm{dT}}}}}}}_{19}^{{{{{{\rm{ab}}}}}}}\), \({\phi }_{{{{{{\rm{eff}}}}}}}\) was between 75% and 90%, for the 19 mer ssDNA ensembles, \({\phi }_{{{{{{\rm{eff}}}}}}}\) was between 65% and 71%, indicating that the prior distributions for ssRNA were in better agreement with the experimental data than for ssDNA. In view of the strong HCG ensemble reweighting required for ssDNA, which in the case of dT38 with its very low transfer efficiency resulted in a bimodal end-to-end distance distribution, we instead reweighted the transfer efficiency distributions obtained from the analytical polymer models for ssDNA. To achieve this, we discretize the transfer efficiency range uniformly between 0 and 1, \({\varepsilon }_{\alpha }=(\alpha -1)\Delta \varepsilon\), with \(\Delta \varepsilon=0.03\), and \(\alpha\) ranging from 1 to \(N=33\). We used as priors the distance distributions, \(P(r)\), from the analytical polymer models (Eqs. 2–4) to obtain the initial weights \({w}_{\alpha }^{0}={a\,P}\left(r({\varepsilon }_{\alpha })\right)|\frac{{dr}}{d{\varepsilon }_{\alpha }}|\), where \(r\left({\varepsilon }_{\alpha }\right)={R}_{0}{(1/{\varepsilon }_{\alpha }-1)}^{1/6}\) and \(a\) is a normalization constant ensuring \({\sum }_{\alpha=1}^{N}{w}_{\alpha }^{0}=1\). We then minimized \(\triangle G\) with respect to \({w}_{1},\ldots,\,{w}_{N}\) as described above. For all three different prior distributions, we found very similar reweighted distributions (i.e., values of \({w}_{\alpha }\)). The end-to-end distance distributions of prior and reweighted polymer models for all ssDNAs are depicted in Supplementary Fig. 5. We used the reweighted HCG distributions (HCGBioEn) for ssRNA and the reweighted polymer model distributions (PMBioEn) for ssDNA to convert \({\tau }_{{cd}}\) to \({\tau }_{r}\) (see below, Eqs. 14, 15, Fig. 2, Supplementary Table 4). We note, however, that even with the reweighted distance distributions from HCG for ssDNA, the resulting values of \({\tau }_{r}\) are very similar to those from the alternative analyses (Supplementary Table 4).Fluorescence correlation spectroscopy (FCS)FCS measurements were performed on freely diffusing Alexa 488- and Alexa 595-labeled oligonucleotides at concentrations and buffer conditions as described in “Free diffusion single-molecule spectroscopy”. Additionally, we included appropriate concentrations of glycerol to increase the solvent viscosity (measurements performed without ZMWs). The correlation between two time-dependent signal intensities, \({I}_{i}\left(t\right)\) and \({I}_{j}\left(t\right)\), measured on two detectors \(i\) and \(j\), is defined as:$${G}_{{ij}}\left(\tau \right)=\frac{\left\langle {I}_{i}\left(t\right){I}_{j}\left(t+\tau \right)\right\rangle }{\left\langle {I}_{i}\left(t\right)\right\rangle \left\langle {I}_{j}\left(t\right)\right\rangle }-1,$$
(10)
where the pointed brackets indicate averaging over \(t\). In our experiments, we use two acceptor and two donor detection channels, resulting in the autocorrelations \({G}_{{AA}}\left(\tau \right)\) and \({G}_{{DD}}\left(\tau \right)\), and cross correlations \({G}_{{AD}}\left(\tau \right)\) and \({G}_{{DA}}\left(\tau \right)\). By correlating detector pairs, and not the signal from a detector with itself, contributions to the correlations from deadtimes and afterpulsing of the detectors are eliminated5,72. Full FCS curves with logarithmically spaced lag times ranging from nanoseconds to seconds (Fig. 1c) were fitted with73,74$${G}_{{ij}}\left(\tau \right)={a}_{{ij}}\frac{\left(1-{c}_{{ab}}^{{ij}}{e}^{-\frac{\left|\tau \right|}{{\tau }_{{ab}}^{{ij}}}}\right)\left(1+{c}_{{cd}}^{{ij}}{e}^{-\frac{\left|\tau \right|}{{\tau }_{{cd}}}}\right)\left(1+{c}_{T}^{{ij}}{e}^{-\frac{\left|\tau \right|}{{\tau }_{T}^{{ij}}}}\right)}{\left(1+\frac{\left|\tau \right|}{{\tau }_{D}^{{ij}}}\right){\left(1+\frac{\left|\tau \right|}{{{s}^{2}\tau }_{D}^{{ij}}}\right)}^{1/2}}$$
(11)
The three terms in the numerator with amplitudes \({c}_{{ab}}\), \({c}_{{cd}}\), \({c}_{T}\) and timescales \({\tau }_{{ab}},{\tau }_{{cd}},\,{\tau }_{T}\) describe photon antibunching, chain dynamics, and triplet blinking, respectively. \({\tau }_{D}\) is the translational diffusion time of the labeled molecules through the confocal volume; a point spread function (PSF) of 3-dimensional Gaussian shape is assumed, with a ratio of axial over lateral radii of s = ωz/ωxy (\(s\) = 5.3 without and \(s\) = 1.0 with ZMW; note that this PSF is not expected to be a good approximation for the confocal volume in the ZMWs but has been commonly used owing to a lack of suitable alternatives54,75), and \({a}_{{ij}}\) are the amplitudes of the correlation functions. Parameters without indices ij are treated as shared parameters in the global fits of the auto- and crosscorrelation functions. To study the dynamics in more detail, donor and acceptor fluorescence auto- and crosscorrelation curves were computed and analyzed over a linearly spaced range of lag times, \(\tau\), up to a maximum, \({\tau }_{\max }\), that exceeds \({\tau }_{cd}\) by an order of magnitude (Supplementary Fig. 2). For the subpopulation-specific analysis, we used only photons of bursts with E in the range of ±0.2 of the mean transfer efficiency of the FRET-active population, which reduces the contribution of donor-only and acceptor-only signal to the correlation. For direct comparison, correlation curves were normalized to unity at \({\tau }_{\max }\). After normalization and in the limit of \(\left|\tau \right|\, \ll \, {\tau }_{T}^{{ij}}\) and \(\left|\tau \right|\, \ll \, {\tau }_{D}^{{ij}}\), Eq. 11 reduces to:$${g}_{{ij}}\left(\tau \right)={b}_{{ij}}\left(1-{c}_{{ab}}^{{ij}}{e}^{-\frac{\left|\tau \right|}{{\tau }_{{ab}}^{{ij}}}}\right)\left(1+{c}_{{cd}}^{{ij}}{e}^{-\frac{\left|\tau \right|}{{\tau }_{{cd}}}}\right),$$
(12)
where \({{b}_{{ij}}=1/G}_{{ij}}\left({\tau }_{\max }\right)\) are the normalization constants.For quantifying the solvent viscosity directly in the samples as a function of glycerol concentration, we used the information available from the FCS measurements. The average diffusion time of the labeled oligonucleotides through the confocal volume is directly proportional to the solvent viscosity, η, so η can be estimated from an FCS-based calibration curve. Calibration curves were obtained by measuring the diffusion time by means of acceptor and donor autocorrelations and acceptor-donor cross correlations of double-labeled \({{{{{{\rm{dT}}}}}}}_{19}^{{{{{{\rm{ab}}}}}}}\) at five different known solvent viscosities adjusted with glycerol. The viscosity of each solution was determined using a cone/plate viscometer (DV-I + , Brookfield Engineering Laboratories, Middleboro, MA, USA). Diffusion times were normalized to the diffusion time in buffer and their dependence on viscosity fitted linearly. The solvent viscosity of all other solutions was obtained based on this calibration from the diffusion times of the samples. The values and uncertainties plotted in Fig. 2d represent the resulting means and standard deviations.Fluorescence lifetime measurementsTo determine the relevant timescales for fluorescence lifetime analysis, we performed polarization-resolved ensemble lifetime measurements of all ssNAs on a custom-built fluorescence lifetime spectrometer74, which allowed us to determine the fluorescence lifetimes of Alexa Fluor 488 and 594 as well as the fluorescence anisotropy decays of the dyes conjugated to the different ssNAs. Fluorescence decays of the donor fluorophore were measured on constructs labeled only with Alexa 488. The acceptor fluorescence lifetime decays and corresponding anisotropy decays were measured upon acceptor excitation of double-labeled constructs. All measurements were performed at 150 mM NaCl, 0.01% Tween 20, 0.143 mM BME in 10 mM HEPES buffer with sample concentrations of 50–200 nM. Alexa 488 was excited by a picosecond diode laser (LDH DC 485) at 488 nm with a pulse repetition rate of 40 MHz. Alexa 594 was excited by a supercontinuum light source (SC450-4, Fianium, Southampton, UK), with the wavelength selected using a z582/15 band pass filter and a pulse frequency of 40 MHz. The emitted donor fluorescence was filtered with an ET 525/50 filter (Chroma Technology), and the acceptor fluorescence with an HQ 650/100 filter (Chroma Technology). The emitted photons were detected with a microchannel plate photomultiplier tube (R3809U-50; Hamamatsu City, Japan), and the arrival times were recorded with a PicoHarp 300 photon-counting module (PicoQuant). Intensity decays, \({I}_{{VH}}\left(t\right)\) and \({I}_{{VV}}\left(t\right)\), with horizontal and vertical polarizer orientation, respectively, were measured with vertically polarized excitation (Supplementary Fig. 4). The decays were fitted globally with$$\begin{array}{c}{I}_{{VH}}\left(t\right)=\beta \left[1-{r}_{0}\left[(\alpha {e}^{-t/{\tau }_{{rot}}}+(1-\alpha )){e}^{-t/{\tau }_{M}}\right]\right]{e}^{-t/{\tau }_{{fl}}}+{c}_{{VH}}\\ {I}_{{VV}}\left(t\right)=G\beta \left[1+2{r}_{0}\left[(\alpha {e}^{-t/{\tau }_{{rot}}}+(1-\alpha )){e}^{-t/{\tau }_{M}}\right]\right]{e}^{-t/{\tau }_{{fl}}}+{c}_{{VV}},\end{array}$$
(13)
convolved with the instrument response function (IRF, measured with scattered light). r0 = 0.38 is the limiting anisotropy of the dyes76; G accounts for the different detection efficiencies of vertically and horizontally polarized light and was obtained for the donor and acceptor intensities from the ratio of the vertical and horizontal emission after horizontal excitation, \(G={I}_{{HV}}/{I}_{{HH}}\). The offsets \({c}_{{VV}}\) and \({c}_{{VH}}\) account for background signal. The two rotational correlation times, \({\tau }_{{rot}}\) and \({\tau }_{M}\), account for fast fluorophore rotation and slower tumbling of the entire labeled molecule, respectively. \(\alpha\) represents the fractional amplitude of the fast component; \(\beta\) and \({\tau }_{{fl}}\) represent the amplitude and relaxation time of the total fluorescence intensity decay, respectively (Supplementary Table 2).Chain reconfiguration time \({\tau }_{r}\)
For any distance-dependent observable, \(f(r)\), the correlation time, \({\tau }_{f}\), is defined as$${\tau }_{f}\equiv {\int }_{0}^{{{\infty }}}\frac{{\left\langle \delta f\left(r\left(t\right)\right)\delta f\left(r\left(0\right)\right)\right\rangle }_{r}}{{\left\langle \delta f{\left(r\right)}^{2}\right\rangle }_{r}}{dt},$$
(14)
where \(\delta f\left(r\right)=f\left(r\right)-{\left\langle f(r)\right\rangle }_{r}\), and \({\left\langle \cdot \right\rangle }_{r}\) denotes \({\left\langle \cdot \right\rangle }_{r}=\int \cdot \,P\left(r\right){dr}\). The numerator is defined using the joint probability, \(P({r}_{0},{r}_{t})\), of populating at an arbitrary time zero the distance \({r}_{0}\) and at a later time \(t\) the distance \({r}_{t}.\) With these definitions, we have \({\left\langle \delta f\left(r\left(t\right)\right)\delta f\left(r\left(0\right)\right)\right\rangle }_{r}=\iint \delta f\left({r}_{t}\right)\delta f\left({r}_{0}\right)P\left({r}_{0},{r}_{t}\right)d{r}_{0}d{r}_{t}.\) If the dynamics of \(r(t)\) are well described as diffusive motion in a potential of mean force, \(F\left(r\right)=-{k}_{B}T\,{{{{\mathrm{ln}}}}}\,P(r)\), then \({\tau }_{f}\) can be calculated from39$${\tau }_{f}=\frac{{\int }_{0}^{{{\infty }}}P{\left(r\right)}^{-1}{\left[{\int }_{0}^{r}\delta f\left(\rho \right)P(\rho )d\rho \right]}^{2}{dr}}{D{\int }_{0}^{{{\infty }}}\delta f{\left(\rho \right)}^{2}P\left(r\right){dr}},$$
(15)
where \(D\) is the effective end-to-end diffusion coefficient. From fitting the nsFCS curves, we get the intensity correlation time, \({\tau }_{{cd}}={\tau }_{\epsilon }\), where \(f\left(r\right)=\epsilon (r)\) is the transfer efficiency. We can use Eq. 15 to convert \({\tau }_{{cd}}\) to the physically more interesting chain reconfiguration time, \({\tau }_{r}\), where \(f\left(r\right)=r.\) We calculated conversion ratios \(\theta={\tau }_{{cd}}\)/\({\tau }_{r}\) for all distance distributions used. \(\theta\) as a function of \(R/{R}_{0}\) was calculated for the GC, WLC, and the SAW-ν polymer models, as well as for the HCGBioEn ensembles by varying \({R}_{0}\) (Fig. 1g). Note that \(\theta\) is independent of \(D\). The resulting values of \({\tau }_{r}\) are given in Supplementary Table 4.End-to-end contact ratesFor comparing end-to-end distance dynamics measured here with published values of end-to-end contact formation rates23, we used \(D\) obtained using Eq. 15 for all polymer models and all ssNA variants to estimate end-to-end contact rates, \({k}_{{ee}}\) (see Supplementary Table 5), according to77$$\frac{1}{{k}_{{ee}}}=\frac{1}{{k}_{{{{{{\rm{R}}}}}}}}+\frac{1}{D}{\int }_{a}^{{{\infty }}}\frac{1}{P(r)}{\left[{\int }_{r}^{{{\infty }}}P(\rho )d\rho \right]}^{2}{dr},$$
(16)
where \({k}_{{{{{{\rm{R}}}}}}}={qP}(a)\) is the reaction-limited rate, with a quenching rate upon contact of \(q={10}^{12}{s}^{-1}\) and a quenching distance of \(a=\) 0.4 nm.3,26Reporting summaryFurther information on research design is available in the Nature Portfolio Reporting Summary linked to this article.
Nanosecond chain dynamics of single-stranded nucleic acids
