Resolving Mixtures in Solution by Single-Molecule Rotational

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Resolving Mixtures in Solution by Single-Molecule Rotational Diffusivity Hsiang-Yu Yang, and William Esco Moerner Nano Lett., Just Accepted Manuscript • DOI: 10.1021/acs.nanolett.8b02280 • Publication Date (Web): 12 Jul 2018 Downloaded from http://pubs.acs.org on July 13, 2018

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Resolving Mixtures in Solution by Single-Molecule Rotational Diffusivity Hsiang-Yu Yang and W. E. Moerner* Department of Chemistry, Stanford University, Mail Code 5080, Stanford, California 943055080 CORRESPONDING AUTHOR FOOTNOTE Phone: 650-723-1727 Fax: 650-725-0259 Email: [email protected]

Abstract: Sensing the size of individual molecules in an ensemble has proven to be a powerful tool to investigate biomolecular interactions and association-dissociation processes. In biologically-relevant solution environments, molecular size is often sensed by translational or rotational diffusivity. The rotational diffusivity is more sensitive to the size and conformation of the molecules as it is inversely proportional to the cube of the hydrodynamic radius, as opposed to the inverse linear dependence of the translational diffusion coefficient. Single-molecule rotational diffusivity has been measured with time-resolved fluorescence anisotropy decay, but the ability to sense different sizes has been restricted by the limited number of photons available or has required surface attachment to observe each molecule longer, and the attachment may be perturbative. To address these limitations, we show how to measure and monitor single-molecule rotational diffusivity by combining the solution-phase Anti-Brownian ELectrokinetic (ABEL) trap and maximum likelihood analysis of time-resolved fluorescence anisotropy based on the information inherent in each detected photon. We demonstrate this approach by resolving a mixture of single- and double-stranded fluorescently-labeled DNA molecules at equilibrium, freely rotating in a native solution environment. The rotational diffusivity, fluorescence brightness and lifetime, and initial and steady-state anisotropy, are simultaneously determined for each trapped single DNA molecule. The time resolution and precision of this method were analyzed using statistical signal analysis and simulations. We present key parameters that define the usefulness of a particular fluorescent label for extracting molecular size information from single-molecule rotational diffusivity.

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Keywords: single molecule, ABEL trap, molecular size, rotational diffusion, fluorescence anisotropy decay Manuscript: Biomolecular interactions are involved in virtually all biological processes, and quantitative methods to study the association and dissociation of biomolecules are important to unravel the fundamental mechanisms of biological phenomena. Methods such as single-molecule FRET1, 2 and stepwise photobleaching of multiply-labeled oligomers3 encode conformation or size information into spectral changes and/or fluorescence brightness. More standard biophysical methods monitor the hydrodynamic radius4, for example by translational diffusivity measurements using fluorescence correlation spectroscopy (FCS)5 or dynamic light scattering (DLS)6. Bulk techniques often rely on fitting specific models to ensemble measurements of the correlation function, which can be hard to apply to polydisperse mixtures. Thus, there is a continuing need to develop new and complementary direct methods to sense molecular size changes driven by biomolecular interactions. To explore the detailed behavior on the nanoscale for heterogeneous mixtures, the advent of single-molecule optical methods has removed the need for ensemble averaging. Recently, the development of single-molecule translational diffusometry in the Anti-Brownian Electrokinetic (ABEL) trap7, 8 has provided a powerful tool for the study of mixtures or for monitoring timedependent association dynamics. In the ABEL trap9, 10, high-speed feedback electronics sense position and compensate Brownian motion of a single molecule by applying electrokinetic (electrophoretic and/or electro-osmotic) forces, which keep the molecule in the trapping region for an extended period of time. By monitoring the residual motion of the particle and the pattern of feedback forces required for trapping, the translational diffusivity, , of the particle can be determined, and through its inverse dependence on , the hydrodynamic radius can be monitored in real time. This inverse linear dependence on radius ∝ has also enabled trapping of very small objects down to a single nm-sized fluorophore11, as the ABEL trap, without the use of optical forces relying upon volume-dependent molecular polarizability, only needs to overcome the loss of a particle due to its translational diffusion. If the object is 1000 times smaller in volume, the trap only needs to update 10 times faster. However, the use of may not be the most favorable for molecular size measurement. According to the Stokes– Einstein–Debye relation, = ⁄8 , the rotational diffusivity, , is more sensitive to the size and conformation of a molecule than the translational diffusivity, as is inversely proportional to the cube of , where is the viscosity. Sensing the volume, rather than the radius, of individual molecules is a primary goal of this work. As is well-known, rotational diffusivity has been measured at the bulk level using several approaches: fluorescence anisotropy12, fluorescence intensity fluctuation13, 14, or NMR lineshape analysis15. Single-molecule orientation measurements by polarized excitation and/or detection have a long history16, but here we are interested in the time-dependent changes arising from rotation. Rotational random walks have been measured for single metallic nanoparticles using scattering angle-dependent spectra17. For single molecules, in a “single-burst” approach, only a limited number of photons are detected as a single molecule diffused through a laser focus, so it 2 ACS Paragon Plus Environment

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was only possible in one study to differentiate objects (single dye vs. fluorescent protein) with two orders of magnitude difference in rotational diffusivity18. Unfortunately, the immobilization of single biomolecules on surfaces that usually enables long observation time is often undesirably perturbative to rotational dynamics. For example, in a surface-attached experiment19, observations were limited to the study of restricted molecular rotation in the highly confined environment near a membrane. Although many molecules were found to sample the solution, inhomogeneities in fluorescence anisotropy and fluctuations in rotational rate were observed due to the use of a linker and interactions with the nearby surface. Previous studies have also noted that, in some cases, surface attachment can alter the structure or function of biomolecules20, 21. Here we demonstrate that the solution-based ABEL trap may also be used to measure the rotational diffusion of single trapped particles by monitoring their time-resolved fluorescence anisotropy decay. Perturbations from surface attachment are avoided, and the free, unrestricted molecular rotation allows easier modeling of rotational dynamics and anisotropy decay.

Figure 1. a) Apparatus for single-molecule rotational diffusivity measurements in the ABEL trap: A linearly polarized, pulsed excitation beam (620 nm, 76 MHz repetition rate, pulse width ~10ps) scans the trapping region in a microfluidic chamber. Emitted fluorescence photons passing through the dichroic mirror (DM) are split into two polarizations by a polarizing beam splitter (PBS). The absolute photon arrival time, pulse-photon delay and polarization of each detected photon are recorded by a time-tagged time-resolved (TTTR) module. The position of the moving laser beam at the moment of photon detection is used to track the molecular position. A high speed active feedback mechanism compensates Brownian motion by electrokinetic forces, keeping the molecule in the trapping region for an extended period of time. b) Schematic


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illustration of the custom ss- and dsDNA oligomers, internally labeled with Cy5 (IDT). Inset: Schematic model of the internally labeled dsDNA oligomer.

In our approach, the standard implementation of an ABEL trap is modified by using pulsed, polarized excitation, and recording both absolute photon arrival times () and pulsephoton delays () in both s- and p-polarized detection channels (Fig. 1a). Because the trapping feedback forces are translational (if the reorientation energy caused by the applied electric field is small compared to thermal energy, see Supporting Information), a trapped particle still freely rotates. As a result, photons emitted at short relative to the rotational correlation time () have high anisotropy, which decays from its initial value, , to zero at ≫ . The rotational correlation time can be directly determined (along with the two additional parameters, fluorescence lifetime, , and initial anisotropy, ), by maximum likelihood (MLE) analysis of the stream of photons for each trapped molecule, and therefore the volume, , and rotational diffusivity of the molecule can be monitored through their direct relation to the rotational correlation time, = 1⁄6 = ⁄ . The probability function for this MLE problem considers the underlying dependence of the parallel and perpendicular emitted photons on the (high NA) collection geometry and the rotational dynamics (anisotropy decay). Independently, fluorescence brightness and the usual steady-state anisotropy (̅ ) (integrated over the timedependent fluorescence intensity decay) are determined using the time-averaged fluorescence brightness in both polarization channels for comparison and validation. Here, we demonstrate the power of this method in a single-biomolecule identification task. We resolve a mixture of single- (ssDNA) and double-stranded fluorescently-labeled DNA (dsDNA) molecules (Fig. 1b), freely rotating in solution at equilibrium, where one strand is internally labeled with a doubly-attached fluorophore. The slowing down of rotational diffusivity upon hybridization yields changes in (and other molecular parameters) that allow each object in the trap to be identified as a double-stranded or single-stranded oligonucleotide. We analyze the precision of single-molecule rotational diffusivity measurements in the ABEL trap using the theory of Fisher information and simulations. The analysis quantifies the information carried by each detected photon and provides guidance toward the optimal labeling strategy for other situations in future applications of this method. Results and Discussion We applied our approach to resolve a mixture of ssDNA and dsDNA molecules in solution. One centrally located nucleotide of a 12-nt ssDNA oligomer is replaced with a Cy5 molecule, which has absorption and emission dipoles along its long axis (Fig. 1b). This internal Cy5 label is doubly attached to the DNA backbone, thus its orientation relative to the DNA backbone remains fixed22. As a result, the observed is the rotational correlation time of the host DNA molecule, which is ~ tenfold larger compared to of free Cy5 (see Supporting Information). Experiments were conducted at different concentrations of an unlabeled strand, which binds to the ssDNA to form dsDNA, and naturally also dissociates at equilibrium. It is the goal of our study to recognize the two species, ssDNA and dsDNA, molecule by molecule, in the 4 ACS Paragon Plus Environment

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resulting mixture. Due to the size difference, they have distinct rotational correlation times which can be utilized to identify each molecule. The association/dissociation times were designed to be two orders of magnitude longer than the trapping time (see Supporting Information) so that switching events between ssDNA and dsDNA are unlikely to be detected.

Figure 2. a) Representative fluorescence brightness traces of single trapped dsDNA oligomers for the parallel (light blue) and perpendicular (light red) polarization channels. The brightness levels (dark blue/red) are identified by a change-point finding algorithm. b) Normalized instrument response functions (light blue/red) and normalized background decay curves (dark blue/red) for the parallel and perpendicular polarization channels. The total integrated count in each curve was normalized to one. c) Top: Representative time-correlated single photon counting (TCSPC) histograms (dots) of a single trapped dsDNA oligomer (shaded area in a), ~20000 photons) for the parallel (blue) and perpendicular (red) polarization channels. MLE fitting (solid lines) to the data gives = 1.64 ± 0.08ns, = 4.2 ± 0.4ns, and = 0.3 ± 0.1. Bottom: Weighted (divided by square root of signal) residuals for each polarization channel.

The time-averaged intensity of the laser scanning pattern is uniform across the trapping region, enabling well-defined fluorescence brightness (photons/time) to be monitored for trapped single molecules23. An example of a 10ms binned fluorescence brightness trace from a dsDNA sample, in the presence of 5µM unlabeled strand, is shown in Fig. 2a, which contains 7 singlemolecule events separated by periods of background only. One consistent brightness level was identified from individual trapped dsDNA molecules with a steady-state anisotropy ̅ = 0.23, using a change-point finding algorithm24. This value is extracted from the signals in the two polarization channels using a correction for high NA (Eqn. S7 in Supporting Information). The signal-to-background ratio observed was typically 9-11. For each level (which corresponds to a 5 ACS Paragon Plus Environment

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single DNA molecule), photon delays were fed to the maximum likelihood procedure to extract molecular parameters, given the measured instrument response functions (IRFs) and expected background fluorescence intensity decays (∥ !" and # !") shown in Fig. 2b. The IRF was measured using the very short-lived fluorescence of malachite green to minimize systematic error (see Supporting Information). ∥ !" and # !" were generated using all photons in the periods of background signals. In the maximum likelihood analysis, the raw photon delays and polarizations were directly used without calculating the time-dependent anisotropy !" = $%∥ !" & %# !"' /% !" because !" has a higher variance due to the small intensity difference

in the numerator. One example pair of fluorescence intensity decays for dsDNA is shown in Fig. 2c, with MLE estimates = 1.64 ± 0.08ns, = 4.2 ± 0.4ns, and = 0.3 ± 0.1. The rotational correlation time of this 12 bp dsDNA molecule is similar to the literature value of 3.5-4ns measured using 11 bp DNA oligomers with an intercalating dye25.

Figure 3. 1D densities of rotational correlation time and steady-state anisotropy ̅ , and 2D density map of their joint distribution, for a) ssDNA alone, b) dsDNA/ssDNA mixture (10nM unlabeled strand), and c) dsDNA (5µM unlabeled strand). For the mixture in b), the 1D density of and the 2D density map both show two populations, corresponding to dsDNA and ssDNA, respectively. Each density map was estimated using ~300 events and was peak-normalized. d) Distribution of fluorescence lifetime for ssDNA and dsDNA. e) Distribution of initial anisotropy for ssDNA and dsDNA. f) 2D density map of fluorescence brightness level,


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identified using all photons emitted during the level, for dsDNA/ssDNA mixture. g) 2D density map of fluorescence brightness for dsDNA/ssDNA mixture using a 50ms-sliding window.

Extracted parameters from each level can be represented as a point in the parameter space of all measured variables (multi-dimensional scatter plot), but we focus on θ and ̅ due to their direct relation to the rotational diffusivity (for ̅ , see Perrin equation12). The data was filtered by photon number to exclude short trapping events (
(2)

where ̂ , ̂ , ? are the estimated values of the key parameters. @̂∥ and @̂# are estimates of the IRF shift (see Supporting Information) for each avalanche photodiode detector needed to compensate for the differences between the emission spectra for MG and the internal DNA label (color shift33), as well as other sources of IRF shift during experiments. N and OPQ are ,-dimensional vectors of time delay and polarization, where , is the total number of collected photons for the specific single-molecule level. Each photon delay relative to the pump pulse was recorded with a time resolution of 4ps but further combined into 40ps time bins. To model TSCPC data, two maximum likelihood approaches are often used34: a) If the total number of photon , is unknown, a Poisson counting noise for each time bin in the TCSPC histogram is assumed. The overall log-likelihood is a sum of log-likelihoods, where is the total number of time bins. b) If the photon number , is known, this additional information can be utilized, and the counts in TCSPC histogram will strictly follow a single multinomial distribution, rather than independent Poisson distributions. However, in the present problem, the collected photons are time-tagged, and therefore distinguishable from each other even if they fall into the same delay time bin. As a result, the log-likelihood can be represented as a sum of , single-photon log-likelihoods: 2

log M!N, OPQ" = 3 log )!4 , )RS4 " 45

(3) Note that the only different between this log-likelihood model and the one from a multinomial distribution is the disappearance of the factorial factors coming from indistinguishability, which are independent of all fitting parameters and therefore have no effect during maximization. Our 12 ACS Paragon Plus Environment

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model and the multinomial distribution approach yield exactly the same estimates with the same precision.

To construct the probability model )!, )RS" for the photon stream for each polarization channel, we utilized a fluorescence intensity decay function corrected for high NA. Proper high NA corrections for time-resolved fluorescence anisotropy decay measurements have been discussed extensively in the literature35-37. Only the correction for detection optics was considered here as we did not back fill the objective for laser excitation. The effect of high NA detection is mixing of different polarizations, and, as a result, a reduced anisotropy is observed; the magic angle TU , at which the detected fluorescence intensity decay only depends on fluorescence lifetime, shifts to a smaller angle. With collection half angle 8 (for our choice of objective and immersion oil, 8 = 68° ), the fluorescence intensity decay functions can be expressed compactly using the total fluorescence decay and TU (see SI for the expression of TU !8") as 1 %∥ !; 8" = % !" X + [sin_ TU !8"`!; 8"a 3

1 %# !; 8" = b% !" X & [cos _ TU !8"`!; 8"a 3 where % !" is the total fluorescence intensity decay for unpolarized excitation

(4)

(5)

% !"~ exp!&/"

(6)

and b is the usual G-factor quantifying the ratio of detection efficiencies in the two channels. The measured G-factor of 0.99 of our optical setup is close to 1, and was omitted in the analysis. The fluorescence anisotropy !; 8" was modeled as a single exponential decay !; 8" = @ !8" exp!&/"

(7)

where @ !8" is a correction factor (see Supporting Information) to model the reduced observed anisotropy due to high NA. To obtain our complete model for the measured quantities, %∥ !; 8" and %# !; 8" were then convolved with shifted instrument response functions IRF∥ !; @∥ " and IRF# !; @# " (see Supporting Information), and background contribution was added: %∥j !; 8" = %∥ !; 8" ∗ IRF∥ !; @∥ " + 3 %∥ ! j ; 8" l

m∥ !" 1 & m∥ ∥

(8)


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%#j !; 8" = %# !; 8" ∗ IRF# !; @# " + 3 %# ! j ; 8" l

m# !" 1 & m# # (9)

where m∥ and m# are the fraction of background photons and ∥ !" and # !" the normalized background decay extracted from all photons in periods of background. For each channel, the fraction of background photons was computed in the trapping experiment from the singlemolecule brightness level and the nearby background level (Fig. 2a). This procedure ensures that the ratio of signal photons in the two polarization channels, as well as the amount of background photons in each channel, are both correct. The two model functions %∥j !; 8" and %#j !; 8" can be recognized as unnormalized marginal probabilities, and are further normalized collectively to yield the probability model )!, )RS": )!, ∥" = %∥j !; 8"pn3 %∥j ! j ; 8" + 3 %#j ! j ; 8"o l

l

(10) )!, ⊥" = %#j !; 8"pn3 %∥j ! j ; 8" + 3 %#j ! j ; 8"o l

l

(11) For each period of constant brightness (i.e. level), delays of all photons within this period were used for the maximum likelihood analysis to generate , , and estimates for that period. Fisher information. We are also interested in the amount of information about the molecular orientation contained in each detected photon. The observed Fisher information matrix 1!2" can be decomposed into contributions from individual collected photons: 1

!2"

2

2

= &∇∇ log M!N, OPQ"|Fu,̂G,Hv,Î∥ ,ÎJ = 3&∇∇ log )!4 , )RS4 "|Fu,̂G,Hv,Î∥ ,ÎJ = 3 14 s

45

s

45

! "

(12)

Note that the MLE-determined estimates of ̂ , ̂ , ?, @̂∥ , @̂# using , photons are used to evaluate ! "

the single-photon 14 ’s. The IRF shift estimates from our measurements do not deviate much from zero and are ignored in the Fisher information analysis, and x ∇= w x

x x

x s y x

! "

(13)

Therefore, the resulting 1!2" and 14 ’s are 3-by-3 matrices of second derivatives with respect to the parameters. The Fisher information matrix % !2" , which is the expected value of observed 14 ACS Paragon Plus Environment

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Fisher information 1!2" over all photon delay times and polarizations, is also a 3-by-3 matrix, and can similarly be decomposed: % !2" = z{1!2" | = ,z{1! " | = ,% ! " (14) where % ! " = z{1! " | = 3 )!, )RS"|Fu,̂G ,Hv,Î∥ ,ÎJ $&∇∇s log )!, )RS"|Fu,̂G ,Hv,Î∥ ,ÎJ ' ,67

(15) Equivalently, the Fisher information can be expressed as the covariance of the score vector, or first derivative, of the probability function: % ! " = 3 )!, )RS"|Fu,̂G ,Hv,Î∥ ,ÎJ $∇ log )!, )RS" ∇s log )!, )RS"|Fu,̂G,Hv,Î∥ ,ÎJ ' ,67

(16) To evaluate the importance of photons at different delay, we calculate the partial Fisher ! " information %647 !" without taking the sum over photon delays %647 !" = 3)!, )RS"|Fu,̂G,Hv,Î∥ ,ÎJ $∇ log )!, )RS" ∇s log )!, )RS"|Fu,̂G ,Hv,Î∥ ,ÎJ ' ! "

67

The component }%647 !"~ ! "

HH

and component }%647 !"~ ! "

(17) G G

of the matrix are shown in

Fig. 4b as functions of /. For the partial Fisher information illustration, the probability model was constructed using ̂ = 100ns, ̂ = 0.4, ? = 4ns, @̂∥ = @̂# = 0 with no background photons. A long lifetime was used for the illustration to generate a flat photon delay distribution over the timescale of the rotational dynamics. For completeness, the single-photon Fisher information can also be formulated in the following way to emphasize the background contribution: % ! " = 3

,67

)4U7 !, )RS" u

1

v ,Î∥ ,ÎJ F,̂G ,H

+ )U !, )RS"

$&∇∇s )4U7 !, )RS" u

where )4U7 !, )RS" + )U !, )RS" = )!, )RS".

v ,Î∥ ,ÎJ F,̂G ,H

' (18)


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Cramér–Rao lower bound (CRLB). To estimate the best-case variance of the rotational correlation time estimate, the CRLB (for one photon) was computed from the corresponding

element of the inverted Fisher information matrix: CRLB() = }[% ! " ` ~ . The calculation was repeated for various values of .

HH

Simulation. (Fig. 4ad) For each photon in , simulated photons for a chosen and , the polarization and delay are randomly determined using the described probability model )!, )RS", where the delay time is discretized into 40ps bins over a 13.2ns window. The simulated polarizations and delays were analyzed using the same maximum likelihood procedure to extract molecular parameters.

Supporting Information. Corrections for high NA depolarization effects. Control experiments. DNA sequence design and dissociation rate estimate. IRF measurement. dsDNA/ssDNA classification. Reorientation energy estimate. Absorption spectra. Time duration of constant brightness distribution. This material is available free of charge via the Internet at http://pubs.acs.org. Acknowledgements. This material is based on work supported in part by the U.S. Department of Energy, Office of Science, Office of Basic Energy Sciences, Division of Chemical Sciences, Geosciences, and Biosciences under Award Number DE-FG02-07ER15892 (to W.E.M). The authors warmly thank Allison Squires for a close reading of the manuscript and assistance during alignment, and Peter Dahlberg, Abhijit Lavania, Allison Squires, and Quan Wang for helpful discussions.

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