Quantitative Single-Molecule Three-Color FÃ¶rster Resonance Energy

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Quantitative Single-Molecule Three-Color Förster Resonance Energy Transfer by Photon Distribution Analysis Anders Barth, Lena Voith von Voithenberg, and Don C Lamb J. Phys. Chem. B, Just Accepted Manuscript • DOI: 10.1021/acs.jpcb.9b02967 • Publication Date (Web): 22 May 2019 Downloaded from http://pubs.acs.org on June 3, 2019

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The Journal of Physical Chemistry

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Quantitative Single-Molecule Three-Color Förster Resonance

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Energy Transfer by Photon Distribution Analysis

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Anders Barth1,a, Lena Voith von Voithenberg1,b and Don C. Lamb1,*

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1 Department

of Chemistry, Center for Integrated Protein Science Munich, Nanosystems Initiative

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Munich and Center for Nanoscience, Ludwig-Maximilians-Universität München, Butenandtstr. 5-13,

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81377 Munich, Germany

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a Current

address: Lehrstuhl für Molekulare Physikalische Chemie, Heinrich-Heine-Universität, Universitätsstraße1,40225 Düsseldorf, Germany

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b

Current address: IBM Research - Zurich, Säumerstr. 4, CH-8803 Rüschlikon, Switzerland

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* Corresponding author: [email protected]

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Abstract

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Single-molecule Förster resonance energy transfer (FRET) is a powerful tool to study conformational

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dynamics of biomolecules. Using solution-based single-pair FRET by burst analysis, conformational

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heterogeneities and fluctuations of fluorescently labeled proteins or nucleic acids can be studied by

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monitoring a single distance at a time. Three-color FRET is sensitive to three distances simultaneously

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and can thus elucidate complex coordinated motions within single molecules. While three-color FRET

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has been applied on the single-molecule level before, a detailed quantitative description of the obtained

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FRET efficiency distributions is still missing. Direct interpretation of three-color FRET data is additionally

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complicated by an increased shot noise contribution when converting photon counts to FRET

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efficiencies. However, to address the question of coordinated motion, it is of special interest to extract

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information about the underlying distance heterogeneity, which is not easily extracted from the FRET

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efficiency histograms directly. Here, we present three-color photon distribution analysis (3C-PDA), a

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method to extract distributions of inter-dye distances from three-color FRET measurements. We present

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a model for diffusion-based three-color FRET experiments and apply Bayesian inference to extract

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information about the physically relevant distance heterogeneity in the sample. The approach is verified

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using simulated data sets and experimentally applied to triple-labeled DNA duplexes. Finally, three-

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color FRET experiments on the Hsp70 chaperone BiP reveal conformational coordinated changes

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between individual domains. The possibility to address the co-occurrence of intramolecular distances

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makes 3C-PDA a powerful method to study the coordination of domain motions within biomolecules

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undergoing conformational dynamics.

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Introduction:

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Förster resonance energy transfer (FRET) is a powerful tool to measure intra- or intermolecular

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distances in the range of 2-10 nm on the single-molecule level. Single molecule FRET has been widely

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applied to study the conformational landscape of proteins and nucleic acids using surface-

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immobilization1-4 or in solution5-9 . In the latter case, molecules are studied at picomolar concentrations

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as they diffuse through the observation volume of a confocal microscope, resulting in bursts of 1 ACS Paragon Plus Environment

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fluorescence signal from single molecules (Figure 1A-B)10-13 . The photons from a single molecule burst

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are analyzed and the molecule-wise distribution of FRET efficiencies provides information regarding

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the conformational states and dynamics of the biomolecules. Extensions of this method have included

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lifetime and polarization information9 as well as direct probing of the acceptor fluorophore using

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alternating excitation schemes14,15 to increase sorting capabilities and informational content of the

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measurements. Accurate FRET measurements require carefully determined correction factors for

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spectral crosstalk, direct excitation of the acceptor fluorophore, and differing photon detection

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efficiencies and quantum yields16. Using the methods of alternating laser excitation (ALEX) or pulsed

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interleaved excitation (PIE), these correction factors can be determined from the experiment

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directly14,17. Diffusion-based FRET measurements avoid possible artifacts related to surface

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immobilization. This advantage comes at the cost of a limited detection time of a few milliseconds per

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molecule, restricted by the diffusion time through the observation volume as well as fewer photons and

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thus increased shot noise. The shot-noise contribution can be accounted for in the analysis. Thus, it is

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possible to elucidate the contributions of physically relevant broadening of the FRET efficiency

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histogram18,19 to study static conformational heterogeneity as well as dynamical interconversion

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between distinct states20,21. This method, named the photon distribution analysis (PDA), has been

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widely applied in the field22-29.

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Two-color FRET is limited to the observation of a single distance at a time, while multiple distance

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readouts are available only in separate experiments. Three-color FRET enables three distances to be

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monitored in one experiment30-34. While the same distance distributions are obtainable from three two-

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color FRET experiments, information about the correlation of distance changes and thus the

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coordination of molecular movements is only obtained using three-color FRET. This is possible since

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three-color FRET experiments contain information about the co-occurrence of distances for the

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individual FRET pairs. During the last decade, a number of single-molecule multicolor FRET studies

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have been performed both in solution and using surface immobilization35-41. Proof-of-principle

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experiments showing the possibility to extract accurate distances from solution-based experiments,

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however, have been limited to the extraction of average values35,36, while a complete description of the

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shape of the FRET efficiency histogram is still missing. Three-color FRET using surface immobilization

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has been applied to study coordinated motion in different nucleic acid systems36-39,42 and the folding

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and oligomerization state of proteins43,44. With the development of new strategies for protein labeling45-

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48,

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opening up the possibility to study coordinated movements within single proteins during their

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conformational cycle.

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However, the extension of two-color FRET to three-color FRET is not as straightforward as it appears

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and there are many challenges that limit the quantitative analysis of three-color FRET data. First of all,

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the signal fractions in a three-color FRET experiment are not easily interpreted in terms of distances,

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as is the case in a two-color FRET experiment. Moreover, any heterogeneity observed for these

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quantities is difficult to relate to physically relevant distance heterogeneity of the studied system. To

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complicate the issue, shot noise is a larger problem in three-color FRET measurements because signal

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after excitation of the donor dye is distributed among three channels, lowering the achievable signal-to-

it is becoming more feasible to site-specifically label proteins with three or more fluorophores,

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noise ratio. Fluorophores, detectors and optics have been optimized for the visible region of the

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spectrum, which minimizes the obtainable spectral separation of the three channels used for three-color

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FRET. Hence, larger corrections need to be applied to account for spectral crosstalk and direct

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excitation of the fluorophores. Thus, extracting accurate distances from three-color FRET experiments

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remains challenging. To this end, a detailed statistical analysis of the experiment with respect to the

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underlying three-dimensional distance distribution is needed, which opens up the possibility to study

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coordinated movements within single biomolecules by analyzing correlations between measured

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distances.

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Here, we present three-color photon distribution analysis (3C-PDA), an extension of the framework of

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photon distribution analysis to three-color FRET systems. The method is based on a likelihood function

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for the three-color FRET process in diffusion-based experiments. We apply model-based Bayesian

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inference using the Metropolis-Hastings algorithm49,50 to infer the model parameters. The method is

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tested using synthetic data and applied experimentally to triple-labeled double-stranded DNA as a

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model system and used to investigate the coordinated motions with the Hsp70 chaperone BiP.

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Theory:

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Three-color FRET

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FRET is the non-radiative energy transfer from a donor fluorophore 𝐷 to an acceptor fluorophore 𝐴

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mediated by dipole-dipole interactions with a strong dependence on distance51: 𝐸=

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1

#(1)

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()

𝑅 1+ 𝑅0

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where 𝑅 is the inter-dye distance and 𝑅0 is the Förster distance. In the experiment, the amount of energy

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transfer can be measured by separating the fluorescence signal according to the emission spectra of

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the donor and acceptor fluorophore. A qualitative measure of the FRET efficiency can be obtained from

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the raw signal fractions in the donor and acceptor channels after donor excitation (𝐼𝐷ex, 𝐼𝐴 ex), which we

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𝐷

𝐷

call the proximity ratio 𝑃𝑅:

𝑃𝑅 =

𝐼𝐷𝐴 ex 𝐼𝐷𝐷ex + 𝐼𝐷𝐴 ex

#(2)

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When alternating laser excitation is employed using PIE or ALEX17,52, accurate intensity-based FRET

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efficiencies can be determined when the appropriate corrections are applied to the photon counts.

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In three-color FRET, energy can be transferred from a donor fluorophore 𝐷 to two spectrally different

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acceptor fluorophores 𝐴1 and 𝐴2 (see Figure 1C-D). One of the acceptors, which we define as 𝐴1, may

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also transfer energy to the lower energy acceptor 𝐴2. Thus, the signal observed for 𝐴2 contains

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contributions from two possible pathways, either from direct energy transfer from 𝐷, or from two-step

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energy transfer mediated by 𝐴1. The signal fractions after excitation of the donor dye are defined by:



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𝑃𝑅𝐷𝐴1 =

𝐼𝐷𝐴1ex 𝐼𝐷𝐷ex + 𝐼𝐷𝐴1ex + 𝐼𝐷𝐴2ex

;

𝑃𝑅𝐷𝐴2 =

𝐼𝐷𝐴2ex 𝐼𝐷𝐷ex + 𝐼𝐷𝐴1ex + 𝐼𝐷𝐴2ex

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#(3)

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Contrary to two-color FRET, these quantities are not directly related to distances. Since in three-color

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FRET systems, energy transfer may occur from the donor dye to either acceptor, the efficiency of

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energy transfer to one acceptor dye is thus modified by the quenching effect of the second acceptor

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dye, effectively reducing the Förster radius of the dye pair. The apparent FRET efficiencies from the

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donor dye to either acceptor 𝐸′𝐷𝐴𝑖, which describe the transition probabilities in the three-color FRET

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system as shown in Figure 1C, are related to the single-pair FRET efficiencies 𝐸𝐷𝐴𝑖 by (see

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Supplementary Note 1):

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𝐸′𝐷𝐴1 =

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𝐸′𝐷𝐴2 =

𝐸𝐷𝐴1(1 ― 𝐸𝐷𝐴2)

#(4)

1 ― 𝐸𝐷𝐴1𝐸𝐷𝐴2 𝐸𝐷𝐴2(1 ― 𝐸𝐷𝐴1) 1 ― 𝐸𝐷𝐴1𝐸𝐷𝐴2

#(5)

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′ ′ Similar to the two-color FRET efficiency 𝐸 (eq. 1), the apparent FRET efficiencies 𝐸𝐷𝐴 and 𝐸𝐷𝐴 describe 1 2

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the probability of energy transfer from the donor dye 𝐷 to acceptor 𝐴1 or 𝐴2, but have a complex relation

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to the distances between the donor and both acceptors. Thus, when the distances are known, the

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transition probabilities in the three-color FRET system can be calculated by means of equations 1, 4

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and 5.

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Excitation of the donor dye yields three signal streams and thus two independent intensity ratios, which

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is insufficient to solve the three-color FRET system for the three inter-dye distances. Thus, to calculate

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distance-related three-color FRET efficiencies, the FRET efficiency between the two acceptor dyes

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𝐸𝐴1𝐴2 needs to be determined independently. In the experiment, this information is obtained by

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alternating the excitation of the blue and green dye (Figure 1C-D and Supplementary Figures S8-9).

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The FRET efficiencies in the three-color FRET system can then be calculated from the measured

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photon counts as described in Supplementary Note 1. Extensions of the three-color FRET efficiency

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calculations including correction factors for real experimental conditions are given in Supplementary

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Note 2.

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In our experiments, a blue, green and red dye are used. Hence, we change the notation from the general

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case of one donor dye with two acceptor dyes (𝐷, 𝐴1, 𝐴2) to the specific notation indicating the dye

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colors (𝐵, 𝐺, 𝑅). The emission probabilities 𝜀 in the three-color FRET system can be calculated

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according to:

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𝜀𝐵𝐵 = 1 ― 𝐸′𝐵𝐺 ― 𝐸′𝐵𝑅 #(6)

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𝜀𝐵𝐺 = 𝐸′𝐵𝐺(1 ― 𝐸𝐺𝑅)#(7)

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𝜀𝐵𝑅 = 𝐸′𝐵𝐺𝐸𝐺𝑅 + 𝐸′𝐵𝑅#(8)


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where 𝜀𝑖𝑗 describes the probability to detect a photon in channel 𝑗 after excitation of dye 𝑖. The emission

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probabilities are not to be confused with the proximity ratios defined in eq. 3. Proximity ratios are

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calculated from the detected photon counts for each observation and are experimental observables that

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are subject to shot-noise. The emission probabilities, on the other hand, are the parameters of the

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probability distribution that determines the distribution of the photon counts on the different color

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channels (and thus the proximity ratios).

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These expressions hold true only for an ideal system. As also is the case for two-color FRET, a number

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of correction factors have to be considered. Additional signal in the FRET channels occurs due to

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spectral crosstalk of the shorter wavelength dyes into the detection channels of the longer wavelength

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dyes and direct excitation of dyes by shorter wavelength lasers. Additionally, the different detection

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efficiencies and quantum yields of the dyes need to be considered. Modified emission probabilities in

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the non-ideal case are calculated as described in Supplementary Notes 3 and 4.

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A likelihood expression for three-color FRET

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To extract detailed information about the system in terms of distance heterogeneity, it is necessary to

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sufficiently sample the FRET efficiency histogram. For three-color FRET, the histogram spans three

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independent dimensions, requiring the cube of the number of data points as compared to two-color

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FRET (see Figure S11). In practice, this amount of sampling is usually not accessible. When using the

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reduced chi-squared (𝜒2red) as a determinate for the goodness-of-fit, large counting errors can result in

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misleadingly small values of 𝜒2red and lead to an over-interpretation of insufficient or bad quality data

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(see Supplementary Note 5). In such cases, it is advisable to use a maximum likelihood estimator (MLE)

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to subject the analysis to the likelihood that the observed data was generated by a given model. The

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MLE approach generally performs better at low statistics and does not require binning of the data53.

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In a three-color FRET experiment, the registered data is a time series of photons in the different

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detection channels, which are processed into a time-binned set of photon counts {𝐼}𝑖 =

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{𝐼𝐵𝐵, 𝐼𝐵𝐺,𝐼𝐵𝑅, 𝐼𝐺𝐺,𝐼𝐺𝑅}𝑖 with a typical time interval of 1 millisecond. The total likelihood ℒ is then the product

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over the probabilities for all observations that yield the given combination of photon counts {𝐼}𝑖 for a

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given model 𝑀: ℒ=

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∏ 𝑃({𝐼} |𝑀)#(9) 𝑖

all bins 𝑖

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Since excitation of the blue and green dyes is performed alternatingly, both photon emission processes

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are statistically independent. However, the underlying parameters of the two probability distributions

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are linked (see equations 6-8). The probability of observing a combination of photon counts after green

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excitation and after blue excitation is then the product of the individual probabilities:

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𝑃(𝐼𝐵𝐵, 𝐼𝐵𝐺,𝐼𝐵𝑅, 𝐼𝐺𝐺,𝐼𝐺𝑅|𝑀) = 𝑃(𝐼𝐺𝐺,𝐼𝐺𝑅|𝑀)𝑃(𝐼𝐵𝐵,𝐼𝐵𝐺,𝐼𝐵𝑅|𝑀)#(10)

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The joint probabilities of the photon counts on the right-hand side can further be expressed using the

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conditional probabilities:

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𝑃(𝐼𝐺𝐺,𝐼𝐺𝑅|𝑀) = 𝑃(𝐼𝐺𝐺 + 𝐼𝐺𝑅)𝑃(𝐼𝐺𝑅|𝐼𝐺𝐺 + 𝐼𝐺𝑅,𝑀)#(11) 5 ACS Paragon Plus Environment


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𝑃(𝐼𝐵𝐵,𝐼𝐵𝐺,𝐼𝐵𝑅|𝑀) = 𝑃(𝐼𝐵𝐵 + 𝐼𝐵𝐺 + 𝐼𝐵𝑅)𝑃(𝐼𝐵𝐺,𝐼𝐵𝑅|𝐼𝐵𝐵 + 𝐼𝐵𝐺 + 𝐼𝐵𝑅,𝑀)#(12)

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Here, the probabilities 𝑃(𝐼𝐺𝐺 + 𝐼𝐺𝑅) and 𝑃(𝐼𝐵𝐵 + 𝐼𝐵𝐺 + 𝐼𝐵𝑅) describe the probability to see a total number

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of photons after blue or green excitation. An analytic description of these photon count distribution would

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require a precise knowledge of shape of the confocal volume, the diffusion properties of the molecule

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and the brightness of the fluorophores54,55. However, the photon count distributions can be obtained

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from the experimental data directly. In the original implementation of photon distribution analysis18, the

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distribution of photon counts was required to model the contribution of shot-noise to the experimental

183

histogram. In the likelihood formalism presented here, the photon count distributions represent a

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constant factor. They have no influence on the likelihood function and are not considered in the

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following.

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The model, M, in its simplest form, is described by a triple of distances 𝑹 = (𝑅𝐵𝐺,𝑅𝐵𝑅,𝑅𝐺𝑅), which are

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converted into a set of emission probabilities after blue and green excitation 𝜺 = (𝜀𝐵𝐺,𝜀𝐵𝑅,𝜀𝐺𝑅) according

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to equations 1, 7 and 8. To model the FRET processes, a binomial distribution (𝐵) is applied for two-

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color FRET after green excitation. Analogously, a trinomial distribution (𝑇) is used to describe the three-

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color FRET process after blue excitation:

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𝑃(𝐼𝐺𝑅|𝜀𝐺𝑅,𝐼𝐺𝐺 + 𝐼𝐺𝑅) = 𝐵(𝐼𝐺𝑅|𝜀𝐺𝑅,𝐼𝐺𝐺 + 𝐼𝐺𝑅) =

(𝐼𝐺𝐺 + 𝐼𝐺𝑅)! 𝐼𝐺𝑅!𝐼𝐺𝐺!

𝐺𝑅 (1 ― 𝜀𝐺𝑅)𝐼𝐺𝐺#(13) 𝜀𝐼𝐺𝑅

192

𝑃(𝐼𝐵𝐺, 𝐼𝐵𝑅|𝜀𝐵𝐺,𝜀𝐵𝑅,𝐼𝐵𝐵 + 𝐼𝐵𝐺 + 𝐼𝐵𝑅) = 𝑇(𝐼𝐵𝐺,𝐼𝐵𝑅|𝜀𝐵𝐺,𝜀𝐵𝑅,𝐼𝐵𝐵 + 𝐼𝐵𝐺 + 𝐼𝐵𝑅) (𝐼𝐵𝐵 + 𝐼𝐵𝐺 + 𝐼𝐵𝑅)! 𝐵𝐺 𝐼𝐵𝑅 (1 ― 𝜀𝐵𝐺 ― 𝜀𝐵𝑅)𝐼𝐵𝐵𝜀𝐼𝐵𝐺 = 𝜀𝐵𝑅#(14) 𝐼𝐵𝐵!𝐼𝐵𝐺!𝐼𝐵𝑅!

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Experimental correction factors are accounted for in the calculation of emission probabilities from

194

distances as described in Supplementary Note 3.

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In addition, constant background signal in the different detection channels, originating from scattered

196

laser light or detector dark counts, can have a substantial influence on the determined FRET

197

efficiencies, especially in the case of high or low FRET efficiencies where the fluorescence signal in the

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donor or acceptor channel is low. Since the data is processed into time windows of equal length,

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background count distributions are assumed to be Poissonian with a mean value of 𝜆:

200

𝜆𝑏 ―𝜆 𝑃(𝑏|𝜆) = 𝑒 #(15) 𝑏!

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Background contributions are included into the model by summing over all combinations of photon and

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background counts, whereby the fluorescence signal used to evaluate the FRET processes is reduced

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by the respective number of background counts. 𝐼𝐺𝐺

204

𝑃(𝐼𝐺𝑅|𝜀𝐺𝑅,𝐼𝐺𝐺 + 𝐼𝐺𝑅) =

𝐼𝐺𝑅

∑ ∑ 𝑃(𝑏

𝐺𝐺|𝜆𝐺𝐺)𝑃(𝑏𝐺𝑅|𝜆𝐺𝑅)𝐵(𝐼𝐺𝑅

― 𝑏𝐺𝑅|𝜀𝐺𝑅,𝐼𝐺𝐺 + 𝐼𝐺𝑅 ― 𝑏𝐺𝐺 ― 𝑏𝐺𝑅)#(16)

𝑏𝐺𝐺 = 0𝑏𝐺𝑅 = 0 𝐼𝐵𝐵

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𝑃(𝐼𝐵𝐺,𝐼𝐵𝑅│𝜀𝐵𝐺,𝜀𝐵𝑅,𝐼𝐵𝐵 + 𝐼𝐵𝐺 + 𝐼𝐵𝑅) =

𝐼𝐵𝐺

𝐼𝐵𝑅

∑ ∑ ∑ 𝑃(𝑏

𝐵𝐵│𝜆𝐵𝐵)𝑃(𝑏𝐵𝐺│𝜆𝐵𝐺)𝑃(𝑏𝐵𝑅│𝜆𝐵𝑅)

∗…

𝑏𝐵𝐵 = 0𝑏𝐵𝐺 = 0𝑏𝐵𝑅 = 0

𝑇(𝐼𝐵𝐺 ― 𝑏𝐵𝐺,𝐼𝐵𝑅 ― 𝑏𝐵𝑅|𝜀𝐵𝐺,𝜀𝐵𝑅,𝐼𝐵𝐵 + 𝐼𝐵𝐺 + 𝐼𝐵𝑅 ― 𝑏𝐵𝐵 ― 𝑏𝐵𝐺 ― 𝑏𝐵𝑅)#(17) 6 ACS Paragon Plus Environment

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206

In practice, additional effects have to be accounted for in the likelihood expression. The observed FRET

207

efficiency or proximity ratio histograms also usually show broadening beyond the shot noise limit.

208

Photophysical artifacts of the acceptor dye can cause additional broadening, which can be described

209

by a Gaussian distribution of distances instead of a single distance with widths of approximately 4-8%

210

of the center distance18,56. Broadening beyond this width is typically attributed to conformational

211

heterogeneity of the sample, e.g. due to the existence of conformational substates. To describe

212

additional broadening, a trivariate normal distribution is used to describe a distribution of distances.

213

𝑃(𝑹) = (2𝜋)

―

3 2

|Σ|

―

1

(

)

1 𝑇 exp ― (𝑹 ― 𝑹) Σ ―1(𝑹 ― 𝑹) #(18) 2

2

214

Here, 𝑹 = (𝑅𝐵𝐺,𝑅𝐵𝑅,𝑅𝐺𝑅) is the column vector of distances, 𝑹 is the column vector of center distances

215

and Σ is the covariance matrix given by:

216

Σ𝑖𝑗 = COV(𝑅𝑖,𝑅𝑗) = E[(𝑅𝑖 ― 𝑅𝑖)(𝑅𝑗 ― 𝑅𝑗)] =

∫ 𝑃(𝑹)(𝑅 ― 𝑅 )(𝑅 ― 𝑅 )d𝑹#(19) 𝑖

𝑖

𝑗

𝑗

𝑹

217

where E[𝑋] is the expected value of 𝑋.

218

By normalizing the covariance with respect to the individual variances 𝜎2𝑖 , one obtains the Pearson

219

correlation coefficient that quantifies the observed correlation in the interval [-1,1]: COR(𝑅𝑖,𝑅𝑗) =

220

COV(𝑅𝑖,𝑅𝑗) 𝜎𝑖𝜎𝑗

#(20)

221

This quantity is a more direct measure of the correlation between the observed distances. The

222

implementation of a Gaussian distribution of distances is achieved by constructing a linearly spaced

223

grid of distances with a given number of bins 𝐺 and width of 2𝜎, thus covering more than 95 % of the

224

probability density, and numerically integrating over the three dimensions: 𝑃({𝐼}𝑖|𝑀) = 𝐺

225

=

𝐺

∫𝑃(𝑹)𝑃({𝐼} |𝑹)d𝑹 𝑖

𝐺

∑ ∑ ∑𝑃(𝑹

𝑗𝑘𝑙)𝑃({𝐼}𝑖|𝑹𝑗𝑘𝑙)#(21)

𝑗 = 1𝑘 = 1𝑙 = 1

226

Generally, a grid with 𝐺 = 5 sample points at 𝑅 + [ ―2𝜎, ― 1𝜎,0, 1𝜎, 2𝜎] is employed to implement the

227

distance distribution.

228

Proteins and nucleic acids also often exist in multiple conformational states, each of which may show

229

different degrees of static heterogeneity, each described by a distribution of distances through model 𝑀𝑗

230

. When 𝑁𝑠 species are present, the individual likelihoods are summed up and weighted by the

231

normalized contributions 𝐴𝑗 of the respective species: 𝑁𝑠

232

𝑃({𝐼}𝑖|𝑀) =

∑𝐴 𝑃({𝐼} |𝑀 ), 𝑗

𝑖

𝑗

with

∑𝐴 = 1#(22) 𝑗

𝑗=1

233

Multiple species with different inter-dye distances exhibit different distributions of the signal on the

234

detection channels. Due to different quantum yields and detection efficiencies, this results in differences 7 ACS Paragon Plus Environment


Page 8 of 39

235

of the photon count distribution57, which we have previously not considered in the description of a single

236

species (see equations 11 and 12). As bright species have a higher likelihood to show a higher number

237

of photons than dim species, it is more likely that a burst with a high number of photons shows the

238

conformational state of the bright species. This brightness effect is accounted for in the analysis by a

239

signal-dependent weighting factor as described in Supplementary Note 6.

240

Since the overall likelihood ℒ is generally very small, it is necessary to work with the logarithm of the

241

likelihood function to avoid underflow problems: logℒ =

242

∑ log[𝑃({𝐼} |𝑀)]#(23) 𝑖

all bins 𝑖

243

The algorithm for the calculation of the likelihood function is illustrated in Figure 2. The likelihood

244

function directly takes the center distances and covariance matrix as input parameters, which are varied

245

by the fitting algorithm to find the maximum of the likelihood. From the distances, the transition

246

probabilities in the three-color FRET system are evaluated and adjusted using the experimental

247

correction factors to obtain the emission probabilities. These probabilities are used as the parameters

248

of the binomial and trinomial distributions to evaluate the likelihood function under consideration of the

249

background signal. Finally, to account for the distance distribution, a numeric integration is performed

250

over the grid of distances, and the contributions of different species are summed up.

251

For comparison of the model to the experimental photon count distribution, we use a Monte Carlo

252

approach that simulates the photon emission process to generate expected photon count distributions

253

for the extracted model parameters (for details, see Supplementary Note 7)19. The simulated distribution

254

of the proximity ratios is visually compared to the experimental proximity ratio distribution to evaluate

255

the consistency of the maximum likelihood estimate with the experimental data. The Monte Carlo

256

approach is also initially used to find the region of high likelihood (see Supplementary Note 8). To infer

257

the model parameters and estimate their confidence intervals, we apply model-based Bayesian

258

inference using the Metropolis-Hastings algorithm (see Supplementary Note 9).

259

Methods:

260

Simulations:

261

Simulations of freely diffusing single molecules through the observation volume, represented by a 3D

262

Gaussian function, were performed using a Monte Carlo approach. Photons for each excitation channel

263

were generated randomly at each time step with a probability proportional to the excitation intensity at

264

the current position. FRET and experimental artifacts were evaluated on photon emission as described

265

in Supplementary Note 3. Simulation parameters were chosen to approximately represent experimental

266

conditions. See Supplementary Methods for details.

267

Three-color FRET measurements:

268

Three-color FRET measurements with pulsed interleaved excitation (PIE)15,58 and multiparameter

269

fluorescence detection (MFD)8 were performed on a homebuilt confocal three-color dual-polarization

270

detection setup (SI Methods and Figure S8). Data processing was performed using the software


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271

package PAM written in MATLAB (The MathWorks, Natick, Massachusetts), which is freely available

272

(see Code Availability)59. Bursts were selected as described previously using a sliding time window

273

approach on the total signal, requiring at least 5 photons per time window of 500 µs and at least 100

274

photons in total per burst19. Triple-labeled bursts were selected based on stoichiometry thresholds (SBG,

275

SBR and SGR)35 using a lower boundary of 0.15 and an upper boundary of 0.9. To additionally remove

276

photobleaching and blinking events, the ALEX-2CDE filter60 was applied, which was calculated pairwise

277

for the three excitation channels. Fluorescence lifetimes and anisotropies for the blue, green and red

278

fluorophores after excitation by the respective laser were determined as described previously17. For the

279

photon distribution analysis, burst-wise data was processed into time bins of 1 ms length. Using three-

280

color MFD-PIE, we can determine all necessary correction factors directly from the experiment (see SI

281

Methods).

282

Determination of confidence intervals:

283

Inference uncertainties are given by 95% confidence intervals determined from long runs of the

284

Metropolis-Hastings algorithm (see Supplementary Notes 8 and 9) to estimate the posterior probability

285

density of the model parameters (> 10.000 steps). Statistically independent samples are drawn with a

286

spacing of 500 steps and parameter uncertainties are determined from the distribution of samples.

287

DNA sample preparation and measurement:

288

Single- and double-labeled ssDNA was purchased from IBA GmbH (Göttingen, Germany). DNA was

289

labeled either with Atto488, Atto565 and Atto647N (Atto-Tec, Siegen, Germany) or Alexa488, Alexa568

290

and Alexa647 (Thermofisher Scientific). DNA sequences and labeling positions are given in the SI.

291

Single-stranded DNA was annealed in TE buffer (10 mM Tris, 1 mM EDTA, 50 mM NaCl, pH 8.0) by

292

heating to 95 °C for 5 min and then gradually cooling down at a rate of 1 °C per minute to a final

293

temperature of 4 °C. The sample was diluted to a final concentration of 20 pM in PBS buffer

294

(Thermofisher

295

photobleaching and -blinking of the dyes61.

296

Protein expression:

297

For expression and purification of BiP, the BiP-PrK-pProEX and pEvol tRNAPyl PylRSWT plasmids62 were

298

co-expressed in E. coli BL21-AI as described previously23 with the following modifications: 1 mM N-

299

propargyl-lysine was added after an initial incubation at 37°C for approximately 30 min

300

(OD600nm(E.coli)=0.2). Induction was achieved by the combined addition of IPTG and 0.02% L-(+)-

301

arabinose. Fluorescent labeling was performed as described in the SI Methods.

302

Protein measurements:

303

Triple-labeled BiP was diluted to concentrations of 20-100 pM in 50 mM Hepes, pH7.5, 150 mM KCl,

304

and 10 mM MgCl2. Measurements were performed in the absence or in the presence of 1 mM ADP or

305

1 mM ATP.

Scientific)

containing

1

mM

Trolox

(Sigma

Aldrich)

to

reduce



Page 10 of 39

306

Results and Discussion:

307

Verification of 3C-PDA using simulations

308

To test the analysis method, we first applied it to synthetic datasets. Freely diffusing molecules were

309

simulated using a Monte Carlo approach (see SI Methods) assuming a molecular brightness of 200

310

kHz for all fluorophores, similar to what is experimentally observed for our setup. The resulting photon

311

stream was analyzed in the same manner as the experimental data. The simulation assumed fixed

312

distances between the fluorophores resulting in a FRET efficiency of 80% between the green and red

313

dye, 80% between the blue and red dye, and 25% between the blue and red dye. From the photon

314

counts, we calculated three-color FRET efficiencies according to equations S1.25-26 of Supplementary

315

Note 1 (Figure 3A-B, grey bars). The conversion of burst-wise photon counts to distance-related three-

316

color FRET efficiencies lead to very broad FRET efficiency distributions with nonsensical values outside

317

the interval [0,1] for the FRET efficiency between blue and red dye. Furthermore, the input value of

318

EBR = 0.25 did not coincide with the peak value in the respective distribution (Figure 3B) and inherent

319

correlations due to stochastic variations in the distribution of photons over the three detection channels

320

were apparent (Figure 3C and S6). Using 3C-PDA, we could determine the underlying inter-dye

321

distances (Table S1) and compare the measured distributions of the FRET efficiencies with expected

322

distributions given by the fitted values (black lines in Figure 3A-C), determined by Monte Carlo

323

simulations (see Supplementary Notes 7 and 8). 3C-PDA accounted for the observed shot-noise

324

broadening of the histograms and accurately captured the apparent correlations in the distributions. For

325

representation purposes, we found it more practical to plot the three-color FRET dataset using the

326

proximity ratios (or signal fractions) PRBG, PRBR and PRGR as defined in eq. 2-3 (Figure 3D-I) rather than

327

the FRET efficiency distributions, which are broad and contain the inherent correlations discussed

328

above. The data are then three dimensional, which we visualize using one- and two-dimensional

329

projections. In this parameter space, the data shows a single population that is well described by the

330

extracted distances. As this representation avoids artifacts introduced by the conversion of photon

331

counts into three-color FRET efficiencies, it poses a more convenient way of displaying the three-color

332

FRET data set. In fact, most single-molecule three-color FRET studies have been analyzed by

333

determining mean values of signal fractions or similar quantities, which are then used to calculate FRET

334

efficiencies and thus distances35,36,40. Using this approach, however, the extracted distances can be

335

very sensitive to errors in the determined mean values (Figure S10), and any information about sample

336

heterogeneity is discarded.

337

To accurately reflect the experimental system, we assumed a normal distribution of distances in the

338

simulations with a width of 2 Å around the center values. Using the distance distribution model, we were

339

able to recover the input distances as well as the additional broadening with high accuracy (Table S1).

340

To map the sensitivity of the method at various inter-dye distance combinations, we simulated different

341

combinations of distances with a constant distance distribution width of 2 Å and a Förster radius of 50

342

Å for all FRET pairs. To judge the performance at the sampled distances, we determined the precision

343

given by 95% confidence intervals and the accuracy given by the deviation of the inferred values from

344

the input. The precision and accuracy of the analysis depend on the gradient of the signal distribution


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345

with respect to the distances. When a change in distance invokes a large change in the distribution of

346

the photon counts over the three detection channels, the inferred values will be well defined. On the

347

other hand, when the photon count distribution is not sensitive to distance changes, the confidence

348

intervals will increase and inaccuracies may occur. To illustrate this effect and to map the sensitive

349

range of 3C-PDA, we plot the fraction of green and red photons after blue excitation as a function of

350

the distances RBG and RBR at a fixed distance RGR of 40 Å (Figure 4A-B). Generally, the sensitivity for the

351

distances RBG and RBR will be optimal when the FRET efficiency between the green and red dye is 0,

352

and worst when the FRET efficiency between green and red is high. In the latter case, the majority of

353

fluorescence signal after excitation of the blue dye is detected in the red channel while only a minor

354

fraction is seen in the green channel, resulting in reduced sensitivity. Mapping the uncertainty at

355

different distance combinations confirms the trend of the signal distributions (Figure 4A-B and Figure

356

S14). The highest uncertainty is observed for the distance RBR in the case of a small distance RBG and

357

a large distance RBR, where neither the red nor the green signal fraction shows high sensitivity to a

358

change in the distance RBR. However, for the case of RBG = 40 Å and RBR = 60 Å (Table S1), the input

359

values are still recovered with high accuracy and precision.

360

To address the minimum amount of data needed to accurately infer all three distances, we randomly

361

picked subsets of time bins from the simulation using input values of RGR = 40 Å, RBG = 60 Å and RBR =

362

60 Å, and performed 3C-PDA (Figure 4C) using an increasing number of time bins. Using 500 time

363

bins, the inferred distances already deviate by less than 2 % from the target values, however the

364

uncertainty is large, especially for the distance RBR. At 1000 time bins, the relative error is well below 1

365

% and the distance uncertainty is reduced to ≤ 2 %, further improving upon inclusion of more time bins.

366

For a simple 3C-PDA including a single population, 1000 time bins are thus sufficient to infer accurate

367

distances. Note that, depending on the duration of the single-molecule observations, one burst can

368

contribute multiple time bins to the 3C-PDA, so the number of sampled molecules can be lower.

369

To test the sensitivity of our approach with respect to correlated distance broadening, we simulated a

370

static distance distribution with a single non-zero covariance element, using a correlation coefficient of

371

0.5 between RBG and RGR. This corresponds to a system where the blue and red dye are fixed with

372

respect to each other, but the green dye position is flexible, leading to static heterogeneity in the

373

distances with respect to the green dye (Table S1 and Figure S15A-B). When all covariance elements

374

are set to zero, no satisfactory fit could be achieved (Figure S15C). By introducing the covariance matrix

375

as a fit parameter, there was indeed a correlation detected between RBG and RGR, while the other

376

distance combinations showed no correlation. The analysis yielded an inferred correlation coefficient of

377

0.53 ± 0.01, close to the input value of 0.5 (Table S1 and Figure S15D). Thus, in the ideal case, all input

378

values can be recovered with high confidence, showing that our method is capable of extracting

379

quantitative values for the separation of the three fluorophores as well as the correlation between them.

380

Experimental systems often coexist in different conformational states. To test to what limit small

381

subpopulations can be detected, we simulated a 9:1 mixture of molecules with a difference in the center

382

distances of 2 Å and 5 Å for all dye pairs and a distance distribution width of 2 Å (Figure S16-17). By

383

eye, both datasets could be fit reasonably well using a single population. However, in both cases 11 ACS Paragon Plus Environment


Page 12 of 39

384

addition of a second population increased the log-likelihood. To test whether this increase was justified

385

or just caused by the increased number of model parameters, we applied the Bayesian information

386

criterion (BIC)63 to decide between the two models. The BIC is defined as:

387

BIC = ―2lnℒ𝑀 + 𝑘ln𝑁#(24)

388

where ℒ𝑀 is the maximum value of the likelihood function, 𝑘 is the number of parameters of the model

389

and 𝑁 is the number of data points (here time bins). When comparing different models, the model with

390

the lowest value of the BIC is to be preferred. In this case, the number of model parameters increased

391

from 6 (three center distances and three distribution widths) to 13 (six center distances, six distribution

392

widths and one fractional amplitude). Indeed, the BIC was lower for the two-population model for a

393

separation of 5 Å (ΔBIC = -710, Table S2). However, for a distance difference of only 2 Å, the use of

394

the more complex model was not justified by the BIC (ΔBIC = 30).

395

In real experiments, a number of experimental artifacts have to be accounted for. Background noise

396

can lead to an error in the extracted distances, especially if the signal is low. In three-color FRET, this

397

situation can easily occur for the blue channel due to the presence of two FRET acceptors. To test the

398

robustness of our analysis with respect to background noise, we performed simulations using a

399

brightness of 200 kHz for all fluorophores at increasing Poissonian background signal (Figure 4D-F).

400

We define the signal-to-noise ratio (SNR) as the ratio of the average burst-wise count rate over the total

401

background count rate of all detection channels. Typical burst-wise count rates were above 100 kHz

402

and the background count rate per color channel was usually below 2 kHz, resulting in a SNR of 15-20

403

over all detection channels for our setup. The fitted distances deviated from the input value at low SNR.

404

However, at SNR  10, the relative error was below 1% (Figure 4D-E). The uncertainty of the extracted

405

distances, on the other hand, was independent of the SNR (Figure 4F), since it is primarily limited by

406

the amount of data available.

407

Additional experimental artifacts are given by crosstalk, direct excitation, and differences in detection

408

efficiency and quantum yield. To test whether we could still infer correct distances in the presence of

409

these artifacts, we included them into the simulations, choosing values close to those encountered in

410

the experiment (Table S3). We were able to still recover the correct center distances for the case of a

411

high FRET efficiency between the blue and green as well as the green and red dyes, and a low FRET

412

efficiency between the blue and red dyes (RGR = RBG = 40 Å, RBR = 60 Å, R0 = 50 Å, Table S1). There

413

was, however, some deviation for the inferred value of σBR and a larger uncertainty associated with both

414

RBR and σBR (RBR = 60 ± 1 Å and σBR = 1.3 ± 1.6 Å, input value 2 Å), likely due to the reduced sensitivity

415

of the red channel caused by the smaller simulated detection efficiency. We then simulated a system

416

showing coordinated broadening of the distance distribution in all dimensions in the presence of

417

experimental artifacts, using center distances of 50 Å for all dye pairs. Mean distances and distribution

418

widths were again accurately recovered (Table S1). The inferred covariance matrix elements showed

419

small deviations from the input values but were still found within the confidence intervals. Revisiting the

420

case of a small distance between blue and green dye and large distance between red and green dye,

421

we found that the existence of substantial coordination could cause deviation of the inferred distance

422

values exceeding the confidence intervals (Table S1). In this case, the distance RBR was inferred to be 12 ACS Paragon Plus Environment

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423

57.5 ± 1.9 Å while the input value was 60 Å. Likewise, the covariance elements for RBR also showed

424

deviations from the input values, resulting in an underestimation of the anti-correlation between RBR and

425

RBG (correlation coefficient 𝜌BG,BR = -0.1 ± 0.3, input value -0.32), and an overestimation for the

426

correlation between RBR and RGR (𝜌BR,GR = 0.8 ± 0.4, input value 0.50), while the correlation between RBG

427

and RGR was correctly recovered (𝜌BG,GR = 0.3 ± 0.1, input value 0.375). Even in this extreme case, the

428

method can thus still recover the correct trend of coordination, albeit the absolute values for the

429

elements of the covariance matrix may deviate from the input values to some extent.

430

The likelihood approach to photon distribution analysis

431

The 3C-PDA method is an extension of previous work on the statistics of photon emission in solution-

432

based two-color FRET experiments18,19. By using a likelihood approach, similar to a previous approach

433

for the analysis of conformational dynamics in single-molecule FRET before43,64,65, 3C-PDA can be

434

performed without the need of processing the data into histograms. As a result, the approach is more

435

stable at low statistics, which is of special importance for three-color FRET experiments. We note,

436

however, that our analysis is subjected to pre-processing of the data through burst detection. Thus,

437

determined population sizes will be biased when species differ in brightness because bright bursts are

438

preferentially selected by the burst search algorithm66. Previously, a likelihood approach had been

439

developed for the analysis of three-color FRET experiments of surface-immobilized molecules43, which

440

includes conformational dynamics in the model function. In our approach, we have not considered the

441

effect of dynamics, but provide a description for static conformational heterogeneity by means of the

442

distance distribution.

443

The likelihood analysis presented here may also be applied to two-color FRET experiments. Although

444

acquiring sufficient statistics to sample the proximity ratio histogram usually is not a problem in two-

445

color FRET experiments, we still find that using the likelihood estimator presented here, instead of a 𝜒2

446

goodness-of-fit estimator, yields a smoother optimization surface, leading to a faster convergence of

447

the optimization algorithm and generally more accurate results at low statistics. Furthermore, data from

448

two-color FRET measurements of the same distances can be incorporated into the three-color photon

449

distribution analysis to perform a global analysis over many data sets. The two-color FRET data may

450

be taken from incompletely labeled subpopulations, which are available in the three-color FRET

451

measurement, or from independent two-color FRET measurements. In such a global analysis,

452

distances are linked between two- and three-color experiments, however, the dye pairs or setup

453

parameters do not need to be identical for all included datasets. Since the covariance matrix is unique

454

to the three-color FRET dataset, the inclusion of additional two-color FRET datasets increases the

455

robustness of the extracted correlation coefficients between distances. Furthermore, the Bayesian

456

framework applied here can naturally incorporate additional information (available e.g. from structural

457

methods such as X-ray crystallography, NMR, cryo-EM or SAXS) into the model by means of the prior

458

probability distribution of the model parameters (see Supplementary Note 9).

459

Application of 3C-PDA to triple labeled DNA



Page 14 of 39

460

As an experimental benchmark, we measured double stranded DNA labeled with three dyes as a static

461

model system. To see if we could quantitatively detect small changes in a three-color FRET system,

462

we arranged the dyes Alexa488, Alexa568 and Alexa647 such that significant FRET could occur

463

between all dye pairs. The green and red dyes were positioned on one strand of the double helix at a

464

distance of 27 bp, while the blue dye was positioned on the complementary strand between the two

465

acceptor dyes. Two constructs were designed where the position of the blue dye was shifted between

466

them by 3 bp, resulting in distances of 17 bp to the green dye and 10 bp to the red dye for construct 1

467

(DNA-Alexa1), and 14 bp to the green dye and 13 bp to the red dye for construct 2 (DNA-Alexa2)

468

(Figure 5A). While not directly apparent from the photon count distributions of DNA-Alexa1 (Figure

469

S18A), a second population with ~20% contribution was required to describe the data. DNA-Alexa2

470

more visibly showed a second population with a lower value for PRBG, accounting for ~15% of the

471

measured data (Figure S18B). Here, we focus the analysis on the main population (Table 1 and Figure

472

5B) and attribute the minor secondary population to a photophysical artifact of the dyes, discussed

473

further below. As expected, the distance between the green and red dye was unchanged for both

474

arrangements at 94.9 Å and 96.4 Å for DNA-Alexa1 and DNA-Alexa2, respectively. Changing the

475

position of the blue dye resulted in a distance change for RBG from 64.4 Å to 52.3 Å, and for RBR from

476

62.2 Å to 72.6 Å, corresponding to differences of ΔRBG= -12.1 Å and ΔRBR= 10.4 Å. To investigate

477

whether we could infer correct distances, we compared our result with expected distances as

478

determined by accessible volume (AV) simulations 67. When comparing the relative distance differences

479

between the two constructs, we found very good agreement between measurement and AV

480

simulations, which yielded ΔRBG= -13.5 Å and ΔRBR= 10.8 Å (Table 1). The absolute inter-dye distances

481

for RGR and RBG also matched reasonably well with the AV-derived distances, while a large deviation

482

was observed for RBR (𝑅exp–𝑅AV = 13/18 Å for DNA-Alexa1/2, Table S4). We separately analyzed

483

double-labeled subpopulations of the same measurement using 2C-PDA (Table S4 and Figure S19)

484

and performed measurements on DNA molecules carrying two of the three dyes (Table S4 and Figures

485

S20A-D). The results agreed well with the distances determined by 3C-PDA, showing that the deviation

486

of absolute distances from expected values for RBR is not an artifact of the 3C-PDA method (see

487

Supplementary Notes 10 and 11 for further discussion of potential photophysical and structural

488

artifacts).

489

We then tested another combination of fluorophores by arranging the dyes Atto488, Atto565 and

490

Atto647N first in a cascade geometry (Figure 5C). This arrangement (DNA-Atto1) is expected to show

491

a high FRET efficiency from the blue to green dye and the green to red dye, while the FRET efficiency

492

from the blue to red dye will be very low, similar to the situation studied using simulated datasets in the

493

previous section. This resulted in similar distributions of the three-color FRET efficiencies as observed

494

for the simulated data, including the occurrence of false correlations (Figure S21). A single population

495

was again not sufficient to describe the data. Therefore, a second population was included in our fit,

496

accounting for approximately 15% of the data (Figure S22A and Table S5). For the main population,

497

we found a large distance of 74 Å between the blue and red dye and shorter distances between the

498

green and red dye (46 Å) as well as the blue and green dye (61 Å) (Table 1 and Figure 5D). We then

499

switched the positions of the blue and green dye such that substantial energy transfer could occur 14 ACS Paragon Plus Environment

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500

between all dyes (DNA-Atto2). Again, inclusion of a small secondary population was necessary,

501

accounting for 20% of the data (Table S5 and Figure S22B). As expected, the distance between the

502

green and red dye of the main population was now similar to the distance between the blue and red

503

dye in the cascade arrangement (74 Å). Likewise, the distance between the blue and red dye in DNA-

504

Atto2 was close to the distance measured between the green and red dye in DNA-Atto1 (44 Å).

505

Surprisingly, we found a significant disagreement for the distance between the blue and green dye at

506

48 Å, which should be identical in both arrangements (Figure 5D). We analyzed separate

507

measurements of the same constructs carrying only two of the three dyes by 2C-PDA, which resulted

508

in similar distances for DNA-Atto2 as compared to the 3C-PDA results (Figure S20 E-H and Table S5).

509

In DNA-Atto1, however, the dye Atto565 showed position-dependent quenching causing the observed

510

deviation for the distance RBG. The quenching likely occurs due to photoinduced electron transfer to

511

nearby guanine residues, as has previously been reported for rhodamine-based fluorophores68 (see

512

Supplementary Note 12 for a detailed discussion). By selective analysis of the unquenched population

513

using 2C-PDA, we could recover comparable values for RBG for both constructs, yielding a corrected

514

value for DNA-Atto1 of ~50 Å (black diamond in Figure 5D). Such a selective analysis was not possible

515

for the three-color FRET data because the fluorescence lifetime of Atto565 was already significantly

516

reduced due to the high FRET efficiency to the red dye (EGR > 0.9, 𝜏𝐺 < 1 ns), making it difficult to

517

separate quenched and unquenched populations.

518

In our model function, we assumed a Gaussian distribution of inter-dye distances. A Gaussian distance

519

distribution model has been standard for almost all PDA studies so far, even for systems that are

520

expected to be rigid such as dsDNA18,19. A detailed discussion of the origin of the broadening of the

521

FRET efficiency histograms beyond the shot-noise in the absence of conformational heterogeneity is

522

given in reference56 and is attributed to the existence of different photophysical states of the acceptor

523

leading to an apparent distribution width that is proportional to the mean inter-dye distance. The

524

observed distribution widths in the 3C-PDA of dsDNA are in the range of 8.4 ± 3.1% of the center

525

distances (averaged over all dye combination). This value is comparable to previously reported results

526

for two-color PDA analyses of dsDNA labeled with Alexa488 and Cy5, yielding a value of 7.6%56. Here,

527

we did not consider the covariances in the analysis of the DNA constructs. When artifacts due to dye

528

photophysics are the main contribution to the width of the observed distance distribution, e.g. the

529

existence of multiple photophysical states, 3C-PDA will reveal apparent false correlations. Considering,

530

for example, a quenched state of the red dye, the distances to both the green and the blue dye would

531

be affected, which would result in a false-positive covariance for these distances. While it is in principle

532

possible to use the off-diagonal elements of the covariance matrix to investigate the correlation between

533

distances, special care has to be taken when investigating the covariance matrix in terms of physically

534

relevant conformational broadening. Moreover, whereas artifacts originating from the presence of

535

multiple photophysical states can usually still be described by a single population with an increased

536

distribution width in 2C-PDA, 3C-PDA is more sensitive to these artifacts. In this case, the covariance

537

matrix of the distance distribution would be required to describe the correlated broadening of the

538

apparent distribution of distances. As such, it is not unexpected that two populations were required to

539

describe the three-color FRET data in all cases. Indeed, the minor secondary population observed in 15 ACS Paragon Plus Environment


Page 16 of 39

540

the DNA measurements most likely originates from photophysical artifacts such as spectral shifts of the

541

fluorophores reported for Atto647N69,70 and Alexa48871,72 or sticking interactions of the dyes to the DNA

542

surface56,73.

543

Toward quantitative three-color FRET in proteins

544

To test 3C-PDA on a protein system, we labeled the Hsp70 chaperone BiP (binding immunoglobulin

545

protein) with the dyes Atto488, Atto565 and Atto647N (Figure 6A). During its nucleotide-dependent

546

conformational cycle, BiP undergoes a large conformational change whereby the lid of the substrate

547

binding domain assumes an open conformation when ATP is bound but is closed in the ADP-bound

548

state23,29. By positioning the green and red dyes on the flexible lid and the substrate-binding domain

549

(SBD), we can monitor the state of the lid (open/closed). The blue dye is specifically attached to the

550

nucleotide-binding domain (NBD) through introduction of an unnatural amino acid. The labeling of the

551

green and red dye, however, was performed stochastically using two site-specifically introduced

552

cysteines, resulting in a random distribution of the green and red dye labels between the two possible

553

configurations BGR and BRG. Under the assumption that the environment of the dyes is similar at both

554

labeling positions, the measured distance between the green and red dyes should not be affected. In

555

3C-FRET, however, a superposition of the two configurations is observed. The model function can be

556

extended to account for the stochastic labeling using only one additional parameter, which is the relative

557

population of the two configurations BGR and BRG. Considering that a permutation of the positions of

558

green and red dye is equivalent to adding a second population with interchanged distances BG and

559

BR, the total likelihood per time bin is then given by:

560

𝑃({𝐼}) = 𝑓𝐵𝐺𝑅 ∗ 𝑃({𝐼}|𝑀(𝐑BGR)) + (1 ― 𝑓𝐵𝐺𝑅) ∗ 𝑃({𝐼}|𝑀(𝐑BRG))#(25)

561

where 𝑓𝐵𝐺𝑅 is the fraction of molecules with labeling scheme BGR. When the labeling positions are not

562

equally accessible for the two acceptor dyes, the resulting distribution will be uneven. In such cases,

563

where 𝑓𝐵𝐺𝑅 significantly differs from 0.5, it could be extracted from the fit as a free fit parameter. A more

564

robust analysis can be performed when 𝑓𝐵𝐺𝑅 is determined by other methods (e.g. by mass

565

spectrometry).

566

Here, we performed 3C-PDA on BiP in the presence of ADP and ATP. The conformational cycle of BiP

567

has previously been studied by single-pair FRET using the identical labeling positions23,29. We

568

determined the degree of labeling for the individual dyes using absorption spectroscopy, yielding an

569

upper estimate for the triple-labeling efficiency of ~25 %. However, after filtering for photobleaching and

570

-blinking, the fraction of usable single-molecule events for 3C-PDA was reduced to less than 5% of all

571

detected molecules. For the ADP-state, the obtained proximity ratio histograms showed a broad

572

distribution with a peak at high proximity ratio GR and low proximity ratio BG and BR (Figure 6B). The

573

ATP-state, on the other hand, was characterized by a main population at low proximity ratio GR and

574

BR, whereas the proximity ratio BG showed a bimodal distribution (Figure S23). Inspection of the one-

575

dimensional projections does, however, not reveal the full complexity of the data, which is more evident

576

in the two-dimensional projections. As such, it is not straightforward to assign the number of populations

577

based on the one-dimensional projections alone. To fit the data, we assumed the occupancy of the two

578

cysteine labeling positions to be equal between the green and red dye (𝑓𝐵𝐺𝑅 = 0.5). At a minimum, we 16 ACS Paragon Plus Environment

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579

required three components to describe the data. However, since inclusion of a fourth component

580

increased the likelihood and decreased the BIC (Table S6), we fit the data using a total of 4 populations

581

(Figure 6B, S23 and Table S7). Without further knowledge about the distribution of the green and red

582

labels, the three-color FRET distances RBG and RBR cannot be assigned to physical distances in the

583

molecule. However, by reducing the analysis to the main population (colored blue in the figures), we

584

were able to assign the three-color FRET derived distances by simple comparison with the previously

585

determined values using single-pair FRET (Figure 6C)29. Reasonable agreement was obtained between

586

two-color and three-color FRET experiments for all distances, showing that we can follow the

587

conformational cycle of BiP with three-color FRET. To probe the effect of the stochastic labeling fraction

588

on the fit result, we additionally varied it from the used value of 𝑓𝐵𝐺𝑅 = 0.5. By fixing 𝑓𝐵𝐺𝑅 to values of

589

0.3 and 0.7, we observed small differences of the fit results with root-mean-square errors of the center

590

distances over all four population of 4 and 7 Å for the ADP-state, and 3 and 4 Å for the ATP-state.

591

Interestingly, by treating 𝑓𝐵𝐺𝑅 as a free fit parameter, we obtained similar estimates of 𝑓𝐵𝐺𝑅 = 0.26 and

592

0.25 for the ADP- and ATP-state, respectively, indicating that it may potentially be possible to extract

593

this parameter from the analysis. However, as it is unclear how reliable this estimate is, we have

594

preferred the unbiased assumption of 𝑓𝐵𝐺𝑅 = 0.5 for the reported results.

595

Previously, the observed two-color FRET efficiency distributions for the NBD-lid and NBD-SBD FRET

596

sensors (i.e. the distances between the blue/red and the blue/green labels in Figure 6A) were found to

597

be very similar23,29. Based on this observation, we also performed a simplified analysis using a four-

598

component model without accounting for the stochasticity of the fluorescent labeling (Figure S24).

599

Indeed, also the inferred distributions for RBG and RBR were very similar (Figure S25 and Table S8),

600

indicating that both distances may be used interchangeably to address the interdomain distance. The

601

inferred two-dimensional distribution of the SBD-lid (RGR) and SBD-NBD (here RBG) distances are

602

shown in Figure 6D (see Figure S25 for all distance pairs). From the main populations of the distance

603

distribution, we can confirm the picture obtained from the crystal structures (Figure 5A), showing that

604

the protein indeed undergoes a coordinated conformational change. In the presence of ADP, the lid-

605

domain is closed (RGR  50 Å, GR  5 Å), while the interdomain distance shows a broad distribution

606

indicating conformational heterogeneity (RBG  80 Å, BG  15 Å, Figure 5D, left). In the ATP-state the

607

lid opens up (RGR  89 Å, GR  10 Å), while the domains come into closed contact, resulting in a

608

narrower distribution width for the interdomain distance (RBG  56 Å, BG  5 Å, Figure 5D, right). The

609

simplified model thus allowed us to characterize the conformational space of BiP using three-color

610

FRET and investigate correlated distance changes from the inferred multidimensional distance

611

distributions. Due to the broadness of the resulting distance distribution and the use of four populations,

612

we have avoided the assignment of the populations to distinct conformational states of the system by

613

means of their center distances. Instead, we interpret the three-dimensional distance distribution as a

614

continuous distribution that describes the conformational landscape of BiP.

615

While we generally expected broad distributions for the three-color FRET data, we also observed large

616

conformational heterogeneity for the sensor between the substrate-binding domain and the lid, which

617

previously was found to adopt more defined conformations23,29. A possible explanation for the excess



Page 18 of 39

618

heterogeneity might be given by a higher contribution of contaminations. Through the burst-selection

619

criteria for triple-labeled molecules that showed no photo-bleaching, our analysis was performed on a

620

small fraction of generally less than 5% of all detected molecules. Thus, it is expected that fluorescent

621

impurities, which by chance pass the filtering process, may have a higher contribution compared to two-

622

color FRET experiments. Although BiP was still functional after the triple-labeling step with respect to

623

its ATPase activity (see Figure S26), we also cannot exclude that the labeling with three dyes might

624

slightly destabilize the structure of the protein.

625

Biomolecules generally often show complex conformational landscapes. Here, we required a total of

626

four populations to describe our data. Due to the sensitivity of 3C-PDA to correlations in the data, it is

627

expected that more states are necessary to fully describe the data as compared to two-color FRET

628

experiments. On the other hand, the sensitivity of 3C-PDA is also what makes it possible to reliably

629

extract more information from the data. Using the example of the Hsp70 chaperones, previous studies

630

suggested that the opening and closing of the lid correlates with the distance between substrate-binding

631

and nucleotide-binding domains74,75. However, each state of the lid (open/closed), might show multiple

632

states of the interdomain sensor. Thus, while two-color experiments on the SBD-lid sensor could be

633

well described by two states, and the interdomain sensor fit well to a broad distance distribution, three-

634

color experiments naturally require a larger number of states due to the multidimensional and correlated

635

information available. In summary, our data reveals that the conformational space of BiP is more

636

complex than previously observed. Here, we did not attempt to assign the different minor populations

637

to physical states of the system. Instead, we interpret the inferred distance distribution as a continuous

638

distribution of conformational states that the system populations in solution, which is approximated by

639

our choice of four Gaussian basis functions. The complexity encountered in the analysis of BiP also

640

shows that two-color FRET control measurements are indispensable for the correct interpretation of

641

three-color FRET experiments. In fact, these controls can be obtained from the measurement by filtering

642

for incompletely labeled molecules. Moreover, the two-color FRET information may be included in the

643

likelihood function in a global two- and three-color PDA approach, while complimentary structural

644

information from other methodologies can be accounted for in the prior distribution of the Bayesian

645

framework.

646

With respect to the Hsp70 chaperones, the question of the allosteric communication between the

647

nucleotide-binding and substrate-binding domains is a key question to understand their function74,76.

648

3C-PDA is a promising method to elucidate the coordination between substrate binding and nucleotide

649

hydrolysis within these proteins. Fundamentally, our results on the conformational changes of BiP are

650

limited by the lack of specific labeling for the two acceptor dyes in the experiment. We describe how

651

this limitation can be overcome when the labeling efficiencies of the acceptor positions are different and

652

known a priori. Promisingly, many novel approaches for the specific labeling of proteins have been

653

developed in the past years46-48, enabling the attachment of three or more fluorophores to single

654

proteins through a combination of orthogonal labeling approaches43,77. Moreover, it is worth noting that

655

3C-PDA is not limited to the study of intramolecular distances in single proteins. Rather, it has promising

656

applications in the study of interactions between proteins and/or nuclei acids, whereby the interaction

657

partners may be labeled with a single or two fluorophores each. 18 ACS Paragon Plus Environment

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658

Summary and Conclusions:

659

Here, we presented 3C-PDA, a statistical method for the analysis of single-molecule three-color FRET

660

burst analysis experiments. The method incorporates the underlying physical distance heterogeneity

661

into the model function, enabling a detailed description of the conformational space of biomolecules by

662

three distances simultaneously. The need to apply statistical methods to the analysis of three-color

663

FRET data arises due to the inherent noise that occurs when converting the limited number of photons

664

per molecule to three-color FRET efficiencies. We describe the raw data with a distance distribution

665

model that includes all needed experimental correction factors, thus presenting an effective way to de-

666

noise three-color FRET experiments and extract information about the underlying distance

667

heterogeneity. The experimental results can be directly interpreted in terms of distance changes,

668

allowing the three-color FRET information to be used to reveal coordinated conformational changes.

669

Extension of the presented framework to account for dynamic interconversion between distinct

670

conformational states will enable the study of complex dynamic processes within single biomolecules.

671

Code availability

672

Source code for three-color photon distribution analysis is available through the open-source software

673

package PAM, freely available under http://www.gitlab.com/PAM-PIE/PAM59. The module tcPDA.m is

674

tightly integrated with PAM, but also supports a text-based file format for direct loading of three-color

675

FRET datasets. To reduce the computation time, a GPU-accelerated implementation of the likelihood

676

function is available using the CUDA framework. On a NVIDIA GTX 1080 Ti, the evaluation of the

677

likelihood function requires approximately 100-200 ms for the datasets presented in this work.

678

Supporting Information

679

Derivation of the formulas for three-color FRET efficiencies (Supplementary Note 1 and Figure S1),

680

expressions for the calculation of accurate experimental three-color FRET efficiencies (Supplementary

681

Note 2), calculation of emission probabilities under consideration of experimental correction factors

682

(Supplementary Note 3 and Figure S2), procedure to determine the direct excitation probabilities

683

(Supplementary Note 4), comparison of the likelihood approach to the chi-squared goodness-of-fit

684

estimator (Supplementary Note 5), procedure to correct for brightness differences (Supplementary Note

685

6 and Figure S3), description of the Monte Carlo approach to simulate FRET efficiency histograms

686

(Supplementary Note 7), outline of the fitting procedure (Supplementary Note 8), additional information

687

on Bayesian inference (Supplementary Note 9), assessment of photophysical and structural artefacts

688

in the DNA experiments (Supplementary Notes 10-12 and Figures S4-6), procedure for the calculation

689

of the static-FRET line in three-color FRET (Supplementary Note 13 and Figure S7), summary of 3C-

690

PDA of simulated datasets (Tables S1-2), overview of correction factors (Table S3), fit results of DNA

691

(Tables S4-5) and protein measurements (Tables S6-8), details on DNA sequences and labeling

692

positions, experimental setup, data analysis, protein labeling and simulations (Supplementary

693

Methods), schematic of the experimental setup (Figure S8), illustration of three-color pulsed-interleaved

694

excitation (Figure S9), instabilities when calculating FRET efficiencies from mean signal ratios (Figure

695

S10), sampling problem in three-color FRET (Figure S11), example for Bayesian inference with 3C-

696

PDA (Figure S12), shot-noise problem (Figure S13), precision and accuracy maps (Figure S14), 3C19 ACS Paragon Plus Environment


Page 20 of 39

697

PDA of simulations (Figures S15-17) and DNA-Alexa measurements (Figures S18), two-color PDA of

698

subpopulations of DNA-Alexa (Figure S19) and control measurements for DNA-Alexa and DNA-Atto

699

(Figure S20), FRET efficiency histograms of DNA-Atto measurements (Figure S21), 3C-PDA of DNA-

700

Atto (Figure S22) and BiP (Figure S23-24), distance distributions of the 3C-PDA of BiP (Figure S25),

701

ATPase activity of labeled BiP (Figure S26).

702

Acknowledgements

703

We thank Jelle Hendrix for helpful discussions and Johannes Meyer zum Alten Borgloh for help with

704

the implementation of the computational routines on GPUs. Our thanks belong to Edward Lemke, Swati

705

Tyagi, and Christine Koehler for the introduction to protein expression using non-natural amino acids.

706

We thank Mathias Rosam and Johannes Buchner for providing the BiP plasmid. L.V. acknowledges

707

funding by P-CUBE: The research leading to these results has received funding from the European

708

Community's Seventh Framework Programme (FP7/2007-2013) under grant agreement n° 227764 (P-

709

CUBE). We gratefully acknowledge the financial support by the Deutsche Forschungsgemeinschaft via

710

the Collaborative Research Consortium on Conformational Switches (SFB1035, Project A11), and by

711

the Ludwig-Maximilians-Universität through the Center for NanoScience (CeNS) and the BioImaging

712

Network (BIN).

713

Author contributions

714

A.B. developed and programmed the analysis method, built the experimental setup and performed the

715

measurements and analysis of DNA samples. L.V. expressed and labeled proteins and performed the

716

protein experiments. A.B. and L.V. analyzed the protein experiments. A.B. and D.C.L. wrote the

717

manuscript.

718

References:

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862 863 864 865 866 867 868 869 870 871 872 873 874 875 876 877 878 879 880 881 882 883 884 885 886 887 888 889 890 891 892 893 894 895 896 897 898 899 900 901 902 903 904 905 906 907 908 909 910 911 912 913 914 915 916 917 918 919


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Sikor, M.; Mapa, K.; Voithenberg, von, L. V.; Mokranjac, D.; Lamb, D. C. Real-Time Observation of the Conformational Dynamics of Mitochondrial Hsp70 by spFRET. The EMBO Journal 2013, 32 (11), 1639–1649. Kityk, R.; Kopp, J.; Sinning, I.; Mayer, M. P. Structure and Dynamics of the ATP-Bound Open Conformation of Hsp70 Chaperones. Molecular Cell 2012, 48 (6), 863–874. Lee, T. C.; Moran, C. R.; Cistrone, P. A.; Dawson, P. E.; Deniz, A. A. Site-Specific ThreeColor Labeling of Alpha-Synuclein via Conjugation to Uniquely Reactive Cysteines During Assembly by Native Chemical Ligation. Cell Chem Biol 2018, 25 (6), 797–.


Page 25 of 39 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


930 931

Tables:

932

Table 1: Results of the 3C-PDA of DNA constructs. Distances determined by 3C-PDA of the DNA

933

constructs labeled with Alexa488, Alexa568 and Alexa647 (DNA-Alexa1/2) or with Atto488, Atto565

934

and Atto647N (DNA-Atto1/2). The inferred distances for the major population (>80% contribution) are

935

listed with associated inference uncertainties given as 95% confidence intervals. For DNA-Alexa

936

measurements, experimental distance changes, Δ𝑅(meas.), are compared to theoretical distance

937

changes as determined from accessible volume (AV) calculations, 𝛥𝑅(AV). For a complete list of all

938

inferred model parameters, comparison to two-color control constructs and theoretical distances

939

estimated from accessible volume calculations, see Table S4-5. RGR

RBG

RBR

σGR

σBG

σBR

DNA-Alexa1

94.9 ± 0.2

64.4 ± 0.2

62.2 ± 0.2

11.2 ± 0.2

6.1 ± 0.2

7.3 ± 0.2

DNA-Alexa2

96.4 ± 0.4

52.3 ± 0.2

72.6 ± 1.0

9.1 ± 0.4

3.3 ± 0.2

10.4 ± 0.6

Δ𝐑(meas.)

1.5

-12.1

10.4

-

-

-

𝚫𝐑(AV)

0

-13.5

10.8

-

-

-

DNA-Atto1

46.0 ± 0.2

60.5 ± 0.2

73.7 ± 2.1

3.2 ± 0.2

4.9 ± 0.2

2.6 ± 1.7

DNA-Atto2

74.1 ± 0.2

48.5 ± 0.2

44.1 ± 0.2

4.5 ± 0.2

2.9 ± 0.2

3.4 ± 0.2

940 941



942

Page 26 of 39

TOC Graphic

943 944


Page 27 of 39 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


945

Figures:

946

Figure 1:

947 948

Single-molecule three-color FRET. (A) In a solution-based three-color FRET experiment, single-

949

molecules are measured as they diffuse through the observation volume of a confocal microscope.

950

Possible transition pathways in the three-color FRET system after excitation of the blue dye are

951

schematically shown on a two-domain protein (DnaK, PDB: 2KHO). (B) Single-molecule events result

952

in bursts of fluorescence. Shown are the fluorescence time traces of the combined signal after excitation

953

of the blue, green and red dyes, which are alternatingly excited using pulsed-interleaved excitation (PIE,

954

see Figure S9). Coinciding bursts in all three channels, belonging to triple labeled molecules, are

955

indicated by arrows. (C) Scheme of the possible transition pathways in a three-color FRET system

956

following excitation of the donor dye (blue). Energy can be transferred to either acceptor A1 or A2 (green

957

and red, respectively) with transition probabilities E’DA1 and E’DA2. Acceptor A1 may further transfer the

958

energy to A2 with the FRET efficiency EA1A2. (D) Possible transition pathways following direct excitation

959

of acceptor A1. Excitation of A1 reduces the complexity of the system to independently determine the

960

FRET efficiency between the two acceptors EA1A2. Experimentally, blue and green excitation are

961

alternated on the nanosecond timescale using pulsed interleaved excitation.



962

Page 28 of 39

Figure 2: Likelihood function Model parameters Parameters of the distance distribution: Center distances and covariances

For numeric evaluation of integral over distance distribution.

Correction factors

963

Ideal transition probabilities At every sampling point, calculate the ideal FRET efficiencies and transition probabilities in the three-color FRET system.

,

given

Eq. 23

Eq. 21

Eq. 18-20

Incorporate experimental correction factors: • Crosstalk • Direct excitation • Detection efficiencies • Quantum yields Supplementary Note 3

Likelihood of observed signal the model parameters :

Distance grid

Eq. 1,4,5

Eq. 10-14 • Evaluated by integrating over distance distribution Eq. 21 • Account for background signal Eq. 15-17 • Summation over multiple species Eq. 22

Emission probabilities Probability to detect a photon in the different detection channels: Eq. 6-8 and Supplementary Note 3

964

Flowchart for the calculation of the likelihood function in 3C-PDA. The model defines the normal

965

distribution of distances through the center distances and the covariance matrix. From the model

966

parameters, a distance grid is constructed over the range of ± 2𝜎 for the numeric integration over the

967

distance distribution. At every sampling point of the three-dimensional distance grid, the ideal FRET

968

efficiencies are calculated, which are transformed into the transition probabilities (apparent FRET

969

efficiencies, eq. 4-5) in the three-color FRET system. From the ideal transition probabilities, the

970

emission probabilities are calculated under consideration of the experimental correction factors. The

971

emission probabilities are used for the evaluation of the likelihood function as the probability parameters

972

of the binomial and trinomial distributions, evaluating the distribution of the signal over the different color

973

channels. The total log-likelihood is obtained as the sum over all time bins.

974


Page 29 of 39 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

975


Figure 3:

976 977

Stochastic noise in three-color FRET. (A-C) Three-color FRET efficiency distributions simulated for

978

an ideal system with the respective two-color FRET efficiencies as listed in (A). The input distances are

979

RBG = 40 Å, RBR = 60 Å and RBG = 40 Å using a Förster distance of R0 = 50 Å and a distribution width

980

of 𝜎 = 2 Å for all dye pairs. FRET efficiencies EBG and EBR show broad and skewed distributions, where

981

values well below zero and above one are observed for the FRET efficiency EBR. Red dashed lines

982

indicate the expected values of EBG = 0.8 in (A) and EBR = 0.25 in (B). The two-dimensional plot in (C)

983

shows inherent correlations between the two FRET efficiencies due to photon shot noise (see also

984

Figure S13). Data is shown as grey bars in (A/B) and as a surface plot in (C). Fits are shown as black

985

lines. In (C), the surface used to represent the data is colored according to the weighted residuals. (D-

986

I) Representation of the result of 3C-PDA of the dataset in proximity ratio parameter space. The data is

987

processed into signal fractions after blue excitation (PRBG and PRBR) and after green excitation (PRGR).

988

Shown are the two-dimensional (D-F) and one-dimensional (G-I) marginal distributions of the three-

989

dimensional frequency distribution. In the two-dimensional projections, the surface representing the

990

data is colored according to the weighted residuals (wres). In the one-dimensional projections, the data

991

is shown as grey bars and the 3C-PDA fit as black lines. Weighted residuals are plotted above.

992



993

Page 30 of 39

Figure 4:

994 995

Characterizing the performance of 3C-PDA. (A-B) Map of the distribution of signal after blue

996

excitation on the three-color channels (A: red detection channel, B: green detection channel) at a

997

constant distance between the green and red dye of 40 Å (R0 = 50 Å for all dye pairs). Error bars

998

represent the uncertainty at the sampled distance given by the 95% confidence intervals. For better

999

visualization, the absolute uncertainties are multiplied by a factor of 4. Regions where the change in

1000

signal is low with respect to distance changes correspond to large uncertainties. See Figure S14 for

1001

precision and accuracy maps at different values for RGR. (C) Uncertainty of extracted distances given

1002

by 95% confidence intervals as a function of the number of time bins used for the analysis. Fitted

1003

distance (D), relative error (E) and relative uncertainty (F) as a function of signal-to-noise ratio. To guide

1004

the eye, dashed lines, given by power law fits to the data, are shown in D-F.

1005


Page 31 of 39 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

1006


Figure 5:

1007 1008

3C-PDA of triple-labeled DNA. (A,C) Structural models of the DNA-Alexa (A) and DNA-Atto (C)

1009

constructs. The accessible volumes of (A) the dyes Alexa488 (cyan: DNA-Alexa1, blue: DNA-Alexa2),

1010

Alexa568 (green) and Alexa647 (red) and (C) the dyes Atto488 (blue), Atto565 (green) and Atto647N

1011

(red) are shown as clouds. Spheres with the respective colors indicate the mean positions of the dyes.

1012

In the DNA-Atto2 construct, the positions of the blue and green dyes are switched. (B,D) Fit results for

1013

DNA-Alexa (B) and DNA-Atto (D) measurements. The error bars are given by the fitted distribution

1014

widths. (B) Cyan diamonds show the results with the blue dye in the left position (DNA-Alexa1, cyan

1015

cloud in the structure above), whereas blue squares indicate the blue dye in the right position (DNA-

1016

Alexa2, blue cloud). The change of the position of the blue dye by 3 bp results in an anticorrelated

1017

change of the distances RBG and RBR. (D) White diamonds show the results of DNA-Atto1,

1018

corresponding to the arrangement of dyes as shown in the inset. Gray squares indicated the

1019

arrangement where the blue and green dye positions have been exchanged (DNA-Atto2). Switching of

1020

the positions of blue and green dye should not affect the recovered distances. This holds true for

1021

distances R2 and R3. For the distance R1, however, a large difference is observed. This deviation

1022

originates from dynamic quenching of Atto565 in construct DNA-Atto1 (see main text and

1023

Supplementary Note 12). By accounting for the quenching in the analysis, the correct distance can still

1024

be recovered for R1 in DNA-Atto1 (black diamond).

1025



Page 32 of 39

1026

Figure 6:

1027 1028

3C-PDA of the heat shock protein BiP. (A) Structural representations of the Hsp70 chaperone BiP in

1029

the ADP-bound state (left, based on the crystal structure of the homolog DnaK, PDB: 2KHO) and in the

1030

ATP-bound state (right, PDB: 5E84). The nucleotide-binding, substrate-binding and lid-domain (NBD,

1031

SBD and lid) are color-coded in blue, green and red respectively. Dye positions are indicated by clouds

1032

that represent the possible positions determined using accessible volume calculations. The lid of the

1033

SBD is closed in the ADP-bound state but opens up in the ATP-bound state. Black arrows indicate the

1034

movement of domains during the nucleotide-dependent conformational cycle. The distance between

1035

the SBD and NBD is larger in the ADP-bound state, while the two domains directly contact each other

1036

in the ATP-bound state (grey arrows). (B) 3C-PDA of BiP in the ADP state. Shown are the projections

1037

of the three-dimensional histogram of signal fractions PRGR, PRBG and PRBR. Two-dimensional

1038

projections are displayed as surface plots in the top row. The surface is colored according to the

1039

weighted residuals as indicated by the color bar. In the bottom row, one-dimensional projections (grey

1040

bars) are plotted together with the fit result. Weighted residuals are shown above the plots. The

1041

individual components of the fit are shown in blue, red, yellow and purple in the one-dimensional

1042

projections. Additionally, the second population originating from stochastic labeling is shown in darker

1043

color (dark blue, dark red, dark yellow, dark purple) in the one-dimensional projections of PRBG and

1044

PRBR. (C) Comparison of distances extracted from two-color FRET experiments of BiP (dark grey),

1045

taken from Rosam et al.29, and determined using 3C-PDA (light grey). D) Two-dimensional distance

1046

distributions extracted from the measurements of BiP in the presence of ADP (left) and ATP (right)

1047

using 3C-PDA without correcting for the stochastic labeling. The distance between the green and red

1048

fluorophores, RGR, reports on the distance between the lid-domain and the substrate-binding domain.

1049

The distance between the blue and green fluorophores, RBG, monitors the distance between the

1050

substrate-binding and the nucleotide-binding domain. The 3C-PDA confirms the picture obtained from

1051

the crystal structures (A): In the presence of ADP, the lid is closed and the interdomain distance shows

1052

a broad distribution, indicative of conformational heterogeneity; in the presence of ATP, the lid opens

1053

and the distribution width of the interdomain distance is narrower, showing a more defined


Page 33 of 39 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


1054

conformation. See Figure S24 for the associated 3C-PDA fits and Figure S25 for a complete display of

1055

the inferred distance distribution.


A

C

B

Page 34 of 39

100 kHz

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


500 ms

D

Iex

D E’DA1

A1

D E’DA2

EA1A2

Iex

A2

A1 ACS Paragon Plus Environment

EA1A2

A2

Page 35 of 39

Model parameters

1 2 3Parameters of the distance distribution: 4Center distances and covariances 5 ) = +%" , +%# , +"# 6 7 . . . -%" -%"/%# -%"/"# 8 9 . . . -%# -%#/"# , = -%#/%" 10 . . . 11 -"#/%" -"#/%# -"# 12 13 Eq. 18-20 14 15 16 17 18 19 20 21 Incorporate experimental 22 correction factors: 23 24 • Crosstalk 25 • Direct excitation 26 • Detection efficiencies 27 • Quantum yields 28 Supplementary Note 3 29 30 31 32 33 34 35 36 37 38 39 40 41

Correction factors


Distance grid For numeric evaluation of integral over distance distribution.

Likelihood function Likelihood of observed signal {2} given the model parameters 4:

log ℒ = 9 log : 2 |4 Eq. 23

: {2} = < 2"" , 2"# |0"# = … ? 2%% , 2%" , 2%# |0%% , 0%" , 0%#

Eq. 21

Ideal transition probabilities At every sampling point, calculate the ideal FRET efficiencies and transition probabilities in the three-color FRET system.

( Environment ( Paragon !"# ; !%" ,ACS !%# → !Plus %" , !%# Eq. 1,4,5

Eq. 10-14 • Evaluated by integrating over distance distribution Eq. 21 • Account for background signal Eq. 15-17 • Summation over multiple species Eq. 22

Emission probabilities Probability to detect a photon in the different detection channels:

0%% , 0%" , 0%# , 0"#

Eq. 6-8 and Supplementary Note 3

6000 5000

Occurrence

1 4000 2 3000 3 4 2000 5 6 1000 7 0 0 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38

B

EBG = 0.8 EBR = 0.25 EGR = 0.8

1500

C


Page 36 of 39 w

2

1000 100

0

50

500

-2

0 0

0.2

0.4

res

150

Occurrence

A

0.6

0.8

0 -1

1

FRET Efficiency BG

2 1

0

1

FRET Efficiency BR

0.5

2

FRET Efficiency BG

D

E

F

G

H

I

ACS Paragon Plus Environment

1

0 -1 FRET Efficiency BR

D


E

fraction of green photons

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20

B

fraction of red photons

A Page 37 of 39

C

F


A

D

RBG

RBR

80

Page 38 of 39

R2

R1

70

RGR

Distance [Å]

B

R3

Distance [Å]

1 2 3 4 110 5 6 100 7 8 90 9 80 10 11 70 12 13 60 14 15 50 16 17 18

C


60 50


RBG

RBR

RGR

40

R1

R2

R3

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49

ADP

ATP

NBD

C


SBD lid

80

60

2C 3C

ADP

ATP

100 80

Distance [Å]

A

Page 39 of 39

60

40

40

20

B

0

20

SBD-lid

NBD-SBD

NBD-lid

0

SBD-lid

NBD-SBD

NBD-lid

D ATP

[Å]

ADP

[Å]


[Å]

[Å]

Quantitative Single-Molecule Three-Color FÃ¶rster Resonance Energy

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