Kilian Evang
email: k.evang@rug.nl

Valerio Basile
email: v.basile@rug.nl

Grzegorz Chrupała
email: g.chrupala@uvt.nl

Johan Bos
email: johan.bos@rug.nl

Elephant: Sequence Labeling for Word and Sentence Segmentation

Tokenization is widely regarded as a solved problem due to the high accuracy that rulebased tokenizers achieve. But rule-based tokenizers are hard to maintain and their rules language specific. We show that highaccuracy word and sentence segmentation can be achieved by using supervised sequence labeling on the character level combined with unsupervised feature learning. We evaluated our method on three languages and obtained error rates of 0.27 ‰ (English), 0.35 ‰ (Dutch) and 0.76 ‰ (Italian) for our best models.

An Elephant in the Room

Tokenization, the task of segmenting a text into words and sentences, is often regarded as a solved problem in natural language processing [START_REF] Dridan | Tokenization: Returning to a long solved problem -a survey, contrastive experiment, recommendations, and toolkit[END_REF], probably because many corpora are already in tokenized format. But like an elephant in the living room, it is a problem that is impossible to overlook whenever new raw datasets need to be processed or when tokenization conventions are reconsidered. It is moreover an important problem, because any errors occurring early in the NLP pipeline affect further analysis negatively. And even though current tokenizers reach high performance, there are three issues that we feel haven't been addressed satisfactorily so far:

• Most tokenizers are rule-based and therefore hard to maintain and hard to adapt to new domains and new languages [START_REF] Silla | An analysis of sentence boundary detection systems for English and Portuguese documents[END_REF]);

• Word and sentence segmentation are often seen as separate tasks, but they obviously inform each other and it could be advantageous to view them as a combined task;

• Most tokenization methods provide no alignment between raw and tokenized text, which makes mapping the tokenized version back onto the actual source hard or impossible.

In short, we believe that regarding tokenization, there is still room for improvement, in particular on the methodological side of the task. We are particularly interested in the following questions: Can we use supervised learning to avoid hand-crafting rules? Can we use unsupervised feature learning to reduce feature engineering effort and boost performance? Can we use the same method across languages? Can we combine word and sentence boundary detection into one task?

Related Work

Usually the text segmentation task is split into word tokenization and sentence boundary detection. Rulebased systems for finding word and sentence boundaries often are variations on matching hand-coded regular expressions [START_REF] Grefenstette | Tokenization[END_REF][START_REF] Silla | An analysis of sentence boundary detection systems for English and Portuguese documents[END_REF][START_REF] Jurafsky | Speech and Language Processing. An Introduction to Natural Language Processing[END_REF][START_REF] Dridan | Tokenization: Returning to a long solved problem -a survey, contrastive experiment, recommendations, and toolkit[END_REF].

Several unsupervised systems have been proposed for sentence boundary detection. [START_REF] Kiss | Unsupervised multilingual sentence boundary detection[END_REF] present a language-independent, unsupervised approach and note that abbreviations form a major source of ambiguity in sentence boundary detection and use collocation detection to build a high-accuracy abbreviation detector. The resulting system reaches high accuracy, rivalling handcrafted rule-based and supervised systems. A similar system was proposed earlier by [START_REF] Mikheev | Periods, capitalized words, etc[END_REF].

Existing supervised learning approaches for sentence boundary detection use as features tokens preceding and following potential sentence boundary, part of speech, capitalization information and lists of abbreviations. Learning methods employed in these approaches include maximum entropy models [START_REF] Reynar | A maximum entropy approach to identifying sentence boundaries[END_REF] decision trees [START_REF] Michael | Some applications of tree-based modelling to speech and language[END_REF], and neural networks [START_REF] Palmer | Adaptive multilingual sentence boundary disambiguation[END_REF].

Closest to our work are approaches that present token and sentence splitters using conditional random fields [START_REF] Tomanek | Sentence and token splitting based on conditional random fields[END_REF][START_REF] Fares | Machine learning for high-quality tokenization -replicating variable tokenization schemes[END_REF]. However, these previous approaches consider tokens (i.e. character sequences) as basic units for labeling, whereas we consider single characters. As a consequence, labeling is more resource-intensive, but it also gives us more expressive power. In fact, our approach kills two birds with one stone, as it allows us to integrate token and sentence boundaries detection into one task.

Method

IOB Tokenization

IOB tagging is widely used in tasks identifying chunks of tokens. We use it to identify chunks of characters. Characters outside of tokens are labeled O, inside of tokens I. For characters at the beginning of tokens, we use S at sentence boundaries, otherwise T (for token). This scheme offers some nice features, like allowing for discontinuous tokens (e.g. hyphenated words at line breaks) and starting a new token in the middle of a typographic word if the tokenization scheme requires it, as e.g. in did|n't. An example is given in Figure 1.

It didn't matter if the faces were male, SIOTIITIIOTIIIIIOTIOTIIOTIIIIOTIIIOTIIITO female or those of children. Eighty-TIIIIIOTIOTIIIIOTIOTIIIIIIITOSIIIIIIO three percent of people in the 30-to-34 IIIIIOTIIIIIIOTIOTIIIIIOTIOTIIOTIIIIIIIO year old age range gave correct responses. TIIIOTIIOTIIOTIIIIOTIIIOTIIIIIIOTIIIIIIIIT

Datasets

In our experiments we use three datasets to compare our method for different languages and for different domains: manually checked English newswire texts taken from the Groningen Meaning Bank, GMB [START_REF] Valerio Basile | Developing a large semantically annotated corpus[END_REF], Dutch newswire texts, comprising two days from January 2000 extracted from the Twente News Corpus, TwNC [START_REF] Ordelman | TwNC: a multifaceted Dutch news corpus[END_REF], and a random sample of Italian texts from the PAIS À corpus [START_REF] Borghetti | I testi del web: una proposta di classificazione sulla base del corpus PAIS À[END_REF]. The data was converted into IOB format by inferring an alignment between the raw text and the segmented text.

Sequence labeling

We apply the Wapiti implementation [START_REF] Lavergne | Practical very large scale CRFs[END_REF] of Conditional Random Fields [START_REF] Lafferty | Conditional random fields: Probabilistic models for segmenting and labeling sequence data[END_REF], using as features the output label of each character, combined with 1) the character itself, 2) the output label on the previous character, 3) characters and/or their Unicode categories from context windows of varying sizes. For example, with a context size of 3, in Figure 1, features for the E in Eighty-three with the output label S would be E/S, O/S, /S, i/S, Space/S, Lowercase/S. The intuition is that the 31 existing Unicode categories can generalize across similar characters whereas character features can identify specific contexts such as abbreviations or contractions (e.g. didn't). The context window sizes we use are 0, 1, 3, 5, 7, 9, 11 and 13, centered around the focus character.

Deep learning of features

Automatically learned word embeddings have been successfully used in NLP to reduce reliance on manual feature engineering and boost performance. We adapt this approach to the character level, and thus, in addition to hand-crafted features we use text representations induced in an unsupervised fashion from character strings. A complete discussion of our approach to learning text embeddings can be found in [START_REF] Chrupała | Text segmentation with character-level text embeddings[END_REF]. Here we provide a brief overview.

Our representations correspond to the activation of the hidden layer in a simple recurrent neural (SRN) network [START_REF] Elman | Finding structure in time[END_REF][START_REF] Elman | Distributed representations, simple recurrent networks, and grammatical structure[END_REF], implemented in a customized version of Mikolov (2010)'s RNNLM toolkit. The network is sequentially presented with a large amount of raw text and learns to predict the next character in the sequence. It uses the units in the hidden layer to store a generalized representation of the recent history. After training the network on large amounts on unlabeled text, we run it on the training and test data, and record the activation of the hidden layer at each position in the string as it tries to predict the next character. The vector of activations of the hidden layer provides additional features used to train and run the CRF. For each of the K = 10 most active units out of total J = 400 hidden units, we create features (f (1) . . . f (K)) defined as f (k) = 1 if s j(k) > 0.5 and f (k) = 0 otherwise, where s j (k) returns the activation of the k th most active unit. For training the SRN only raw text is necessary. We trained on the entire GMB 2.0.0 (2.5M characters), the portion of TwNC corresponding to January 2000 (43M characters) and a sample of the PAIS À corpus (39M characters).

Results and Evaluation

In order to evaluate the quality of the tokenization produced by our models we conducted several experiments with different combinations of features and context sizes. For these tests, the models are trained on an 80% portion of the data sets and tested on a 10% development set. Final results are obtained on a 10% test set. We report both absolute number of errors and error rates per thousand (‰).

Feature sets

We experiment with two kinds of features at the character level, namely Unicode categories (31 different ones), Unicode character codes, and a combination of them. Unicode categories are less sparse than the character codes (there are 88, 134, and 502 unique characters for English, Dutch and Italian, respectively), so the combination provide some generalization over just character codes. From these results we see that categories alone perform worse than only codes. For English there is no gain from the combination over using only character codes. For Dutch and Italian there is an improvement, although it is only significant for Italian (p = 0.480 and p = 0.005 respectively, binomial exact test). We use this feature combination in the experiments that follow. Note that these models are trained using a symmetrical context of 9 characters (four left and four right of the current character).

In the next section we show performance of models with different window sizes.

Context window

We run an experiment to evaluate how the size of the context in the training phase impacts the classification. In Table 4.2 we show the results for symmetrical windows ranging in size from 1 to 13.

SRN features

We also tested the automatically learned features derived from the activation of the hidden layer of an SRN language model, as explained in Section 3. We combined these features with character code and Unicode category features in windows of different sizes. The results of this test are shown in Table 4. The first row shows the performance of SRN features on their own. The following rows show the combination of SRN features with the basic feature sets of varying window size. It can be seen that augmenting the feature sets with SRN features results in large reductions of error rates. The Cat-Code-1-SRN setting has error rates comparable to Cat-Code-9.

The addition of SRN features to the two best previous models, Cat-Code-9 and Cat-Code-13, reduces the error rate by 83% resp. 81% for Dutch, and by 24% resp. 26% for Italian. All these differences are statistically significant according to the binomial test (p < 0.001). For English, there are too few errors to detect a statistically significant effect for Cat-Code-9 (p = 0.07), but for Cat-Code-13 we find p = 0.016. In a final step, we selected the best models based on the development sets (Cat-Code-7-SRN for English and Dutch, Cat-Code-11-SRN for Italian), and checked their performance on the final test set. This resulted in 10 errors (0.27 ‰) for English (GMB corpus), 199 errors (0.35 ‰) for Dutch (TwNC corpus), and 454 errors (0.76 ‰) for Italian (PAIS À corpus).

Discussion

It is interesting to examine what kind of errors the SRN features help avoid. In the English and Dutch datasets many errors are caused by failure to recognize personal titles and initials or misparsing of numbers. In the Italian data, a large fraction of errors is due to verbs with clitics, which are written as a single word, but treated as separate tokens. Table 5 shows examples of errors made by a simpler model that are fixed by adding SRN features. Table 6 shows the confusion matrices for the Cat-Code-7 and Cat-Code-7-SRN sets on the Dutch data. The mistake most improved by SRN features is T/I with 89% error reduction (see also Table 5). The is also the most common remaining mistake.

A comparison with other approaches is hard because of the difference in datasets and task definition (combined word/sentence segmentation). Here we just compare our results for sentence segmentation (sentence F 1 score) with Punkt, a state-of-the- art sentence boundary detection system [START_REF] Kiss | Unsupervised multilingual sentence boundary detection[END_REF]. With its standard distributed models, Punkt achieves 98.51% on our English test set, 98.87% on Dutch and 98.34% on Italian, compared with 100%, 99.54% and 99.51% for our system. Our system benefits here from its ability to adapt to a new domain with relatively little (but annotated) training data.

6 What Elephant?

Word and sentence segmentation can be recast as a combined tagging task. This way, tokenization is cast as a supervised learning task, causing a shift of labor from writing rules to manually correcting labels. Learning this task with CRF achieves high accuracy.1 Furthermore, our tagging method does not lose the connection between original text and tokens.

In future work, we plan to broaden the scope of this work to other steps in document preparation, such as normalization of punctuation, and their interaction with segmentation. We further plan to test our method on a wider range of datasets, allowing a more direct comparison with other approaches. Finally, we plan to explore the possibility of a statistical universal segmentation model for mutliple languages and domains.

In a famous scene with a live elephant on stage, the comedian Jimmy Durante was asked about it by a policeman and surprisedly answered: "What elephant?" We feel we can say the same now as far as tokenization is concerned.

Figure 1 :

 1 Figure 1: Example of IOB-labeled characters

Table 1 :

 1 Datasets characteristics.

	Name Language Domain	Sentences Tokens
	GMB English Newswire	2,886 64,443
	TNC Dutch	Newswire	49,537 860,637
	PAI Italian	Web/various	42,674 869,095

Table 2 :

 2 Error rates obtained with different feature sets. Cat stands for Unicode category, Code for Unicode character code, and Cat-Code for a union of these features.

		Error rates per thousand (‰)
	Feature set	English	Dutch	Italian
	Cat-9	45 (1.40)	1,403 (2.87)	1,548 (2.67)
	Code-9	6 (0.19)	782 (1.60)	692 (1.20)
	Cat-Code-9	8 (0.25)	774 (1.58)	657 (1.14)

Table 3 :

 3 Using different context window sizes.

		Error rates per thousand (‰)
	Feature set	English	Dutch	Italian
	Cat-Code-1 273 (8.51) 4,924 (10.06) 9,108 (15.86)
	Cat-Code-3 118 (3.68)	3,525 (7.20)	2,013 (3.51)
	Cat-Code-5	20 (0.62)	930 (1.90)	788 (1.37)
	Cat-Code-7	10 (0.31)	778 (1.60)	667 (1.16)
	Cat-Code-9	8 (0.25)	774 (1.58)	657 (1.14)
	Cat-Code-11	9 (0.28)	761 (1.56)	692 (1.21)
	Cat-Code-13	8 (0.25)	751 (1.54)	670 (1.17)

Table 4 :

 4 Results obtained using different context window sizes and addition of SRN features.

		Error rates per thousand (‰)
	Feature set	English	Dutch	Italian
	SRN	24 (0.75) 276 (0.56) 738 (1.28)
	Cat-Code-1-SRN	7 (0.21) 212 (0.43) 549 (0.96)
	Cat-Code-3-SRN	4 (0.13) 165 (0.34) 507 (0.88)
	Cat-Code-5-SRN	3 (0.10) 136 (0.28) 476 (0.83)
	Cat-Code-7-SRN	1 (0.03) 111 (0.23) 497 (0.86)
	Cat-Code-9-SRN	2 (0.06) 135 (0.28) 497 (0.86)
	Cat-Code-11-SRN	2 (0.06) 132 (0.27) 468 (0.81)
	Cat-Code-13-SRN	1 (0.03) 142 (0.29) 496 (0.86)

Table 5 :

 5 Positive impact of SRN features.

	Ms. Hughes will joi
	Cat-Code-7 SIIOSIIIIIOTIIIOTII
	Cat-Code-7-SRN SIIOTIIIIIOTIIIOTII
	$ 3.9 trillion by t
	Cat-Code-7 TOTTIOTIIIIIIIOTIOT
	Cat-Code-7-SRN TOTIIOTIIIIIIIOTIOT
	bleek 0,4 procent
	Cat-Code-11 OTIIIIOTTIOTIIIIIIO
	Cat-Code-11-SRN OTIIIIOTIIOTIIIIIIO
	toebedeeld: 6,2. In
	Cat-Code-11 TIIIIIIIIITOTTITOSI
	Cat-Code-11-SRN TIIIIIIIIITOTIITOSI
	prof. Teulings het
	Cat-Code-11 TIIITOSIIIIIIIOTIIO
	Cat-Code-11-SRN TIIIIOTIIIIIIIOTIIO
	per costringerlo al
	Cat-Code-11 TIIOTIIIIIIIIIIIOTI
	Cat-Code-11-SRN TIIOTIIIIIIIIITIOTI

Table 6 :

 6 Confusion matrix for Dutch development set.

		Predicted, Cat-Code-7 Predicted, Cat-Code-7-SRN
	Gold	I	O	S	T	I	O	S	T
	I 328128	0	2 469 328546	0	0	53
	O	0 75234	0	0	0 75234	0	0
	S	4	0 4323	18	1	0 4332	12
	T	252	0 33 80828	35	0 10 81068

All software needed to replicate our experiments is available at http://gmb.let.rug.nl/elephant/ experiments.php


Alexander Kulesza

Steven Daly

Luke Macaleese

Rodolphe Antoine

Philippe Dugourd

Structural exploration and Förster theory modeling for the interpretation of gas-phase FRET measurements: Chromophore-grafted Amyloid-β peptides

Keywords: numbers: 33.50.-j, 87.10.Tf, 87.14.ef, 87.15.A-

come   L'archive ouverte pluridisciplinaire

I. INTRODUCTION

The investigation of protein structure and aggregation behavior is integral to understand protein function and dysfunction and in particular protein misfoldingrelated diseases 1 . For example, the first stages of how the amyloid plaques of Alzheimer patients are formed and what causes this aggregation to happen is not yet fully understood. To tackle this quest, global structuredetermination methodologies have to evolve that allow monitoring structural parameters of the precursor molecules and to observe misfolding at the molecular level. The gas phase provides a well-suited environment to implement such global strategy as the conditions for the experiments can be precisely defined and the overall structure of complex molecular edifices can be preserved using soft ionization techniques 2,3 .

Förster resonance energy transfer (FRET) is a phenomenon that allows the probing of structural parameters by means of optical techniques. In the picture of Förster Theory developed in the late forties 4 , the efficiency for a resonant energy transfer between the chromophores depends on their mutual distance as well as orientation of their transition dipole moments. FRET is usually measured by recording a decrease in the fluorescence of the donor chromophore, with the concurrent emergence of the acceptor's fluorescence or by changes in fluorescence decay times. The distance dependence is described by Förster's approximation, (equation 1) a) Electronic mail: philippe.dugourd@univ-lyon1.fr where the so-called critical Förster distance R 0 defines the distance at which 50% of the donor excitation is transferred to the acceptor moiety. It is routinely applied as a model for donor-acceptor distances on a length scale from 1 to 10 nm and when dipole-dipole (or convergent multipole-multipole) approximation for the electronic coupling can be employed appropriately for the donor-acceptor interaction 5 . E F RET = 1 1 + (r/R 0 ) 6 (1)

The energy transfer is dominated by the 1 r 6 distancedependence that imprinted the term "spectroscopic ruler" 6 on the technique. Thus, the interpretation of FRET results relies on the possibility to graft specific chromophores at well-defined sites on a molecule. Taking into account the flexibility of the chromophores and their linkers, the measured intensities provide information on the mutual distance of these specific sites 7-10 e.g. to monitor structural changes in biomolecules 6,[START_REF] Wieb Van Der Meer | Resonance Energy Transfer: Theory and Data[END_REF] .

The use of multiple dye pairs allows for triangulation of biomolecules, which provides three-dimensional structural information 9,[START_REF] Woźniak | [END_REF][13][14][15][16] . Consequently, combining this phenomenon with the concept of single molecule detection [START_REF] Sisamakis | Single Molecule Tools, Part B: Super-Resolution, Particle Tracking, Multiparameter, and Force Based Methods[END_REF][18][19] and time-resolved experiments 20 allows for obtaining a unique glimpse into molecular structure as well as inter-and intra-molecular motions. While there exist numerous applications of FRET in the condensed phase [21][22][23][24] , gas-phase FRET measurements [25][26][27][28][29][30][31][32][33][34] suffer from difficulties in recording the fluorescence of a small number of ions in a mass spectrometer. The development of techniques capable of probing FRET in the gas phase with higher sensitivity is thus of high interest. This FRET detection problem has been recently overcome by combining photofragmentation action-spectroscopy with FRET ("action"-FRET) 35 , opening up the possibility to integrate gas-phase FRET into a global approach for structure determination of proteins and protein complexes [36][37][38] .

Extracting exact distances from an experimental calibration of FRET efficiencies alone is difficult because additional structural parameters like the orientation of the chromophore's transition dipole moments affect the excitation energy transfer efficiency significantly. In fact, experimental single-molecule FRET (smFRET) studies on DNA strands allowed to specify those orientations and showed a clear orientation-modulation 39,40 indicating that a negligence of the explicit orientational distribution in "fixed conformations" limits the predictive power. Even in systems where the dye orientation is not strongly restricted, a significant influence on the overall FRET-efficiency was suggested 41 .

Therefore, modeling efforts that access the structure of the chromophore-grafting sites and their distribution or temporal evolution are most attractive. These serve as the basis for the calculation of excitation energy transfer efficiencies and thus assign a detailed molecular picture to the experiments. The accuracy of FRET-efficiency prediction is limited by two major factors. Firstly, the Förster formula is straightforward in its use, but it relies on the validity of its underlying assumption of dipoledipole interaction. Excitation energy transfer rates based calculated with approaches beyond Förster Theory 5 can overcome this issue. For a recent review on the calculation of electronic coupling in excitation energy transfer see ref. 42 . In particular, nonadiabatic dynamics [START_REF] Mitrić | Multistate nonadiabatic dynamics on the fly in complex systems and its control by laser fieldes[END_REF][START_REF] Okuno | [END_REF] or semiclassical treatments 45,46 of the excited state dynamics could deliver accurately the excitation energy transfer efficiencies (though at higher computational costs). Secondly, the prediction's accuracy depends critically on the reliability of the structural exploration -being the underlying foundation for the subsequent numerical modeling. Notably, Hoefmüller et al. recently combined single molecule FRET experiments with the mutual dye orientation statistics obtained from Molecular Dynamics (MD) simulations. From the simulated timedependent mutual orientations and distances, FRET efficiencies were calculated and the full statistics of individual photon absorption, energy transfer, and photon emission events were obtained from subsequent Monte Carlo (MC) simulations of the FRET kinetics 47 . The technique is dedicated to time-coherent single-molecule studies in solution and its algorithms and implementation are part of a computational toolkit for the simulation of smFRET on the basis of molecular dynamics (MD) trajectory ensembles 48 . Dedicated to gas-phase FRET, Talbot et al. have combined gas-phase fluorescence measurements and MD-based numerical modeling of the critical Förster distance 27 . The qualitative insight from the calculations allowed them to interpret molecu-lar structures on basis of their modeled conformations. However, conventional MD may not overcome barriers between the guessed initial configuration and the global minimum and may therefore fail to achieve sampling of a representative structural basis. The resulting errors in the modeling of FRET-efficiencies can be qualitative in nature also using accurate methods for excitation energy transfer calculations and can be overcome only with techniques that enhance the exploration of conformational space.

Such an enhanced structural sampling of gas-phase chromophore-grafted peptides together with numerical modeling of FRET-efficiencies based on distances and orientations at the same time has not yet been reported. In this work, we devise a strategy that succeeds in the accurate numerical first-principles modeling of FRET efficiencies based on a thorough exploration and enhanced sampling of gas-phase structures. In brief, we exploit Replica-Exchange Molecular Dynamics 49 (REMD) for the sampling procedure that offer a convenient framework to gas-phase conformations and statistical analyses 50 . Taking into account the distance distributionss and chromophore-orientation distributions at a given temperature, we calculate averaged FRET-efficiencies according to Förster-Theory and finally compare the results with experimentally measured "action"-FRET efficiencies published elsewhere 51 . The paper is structured as follows: we first describe the model peptide and chromophores used throughout this paper, then we explain our modeling setup detailing the procedure to obtain structural ensembles by means of REMD. Finally, we report the numerical modeling of FRET efficiencies and compare them to their experimental counterparts.

II. RESULTS AND DISCUSSION

A. System and experimental gas-phase FRET In order to establish our strategy for converging "action-FRET" experiments and REMD ensemble based FRET modeling, we chose the Amyloid-β 12-28 fragment with the sequence V 12 HHQKLVFFAEDVGSNK 28 for which we recently published experimental data 52 . It has been shown that the naturally occurring mutations in the hydrophobic cluster (residues 17-22) are an important factor of Amyloid-β aggregation and which are probably indicating an enhanced risk of developing Alzheimer's disease [53][54][55] . We used the FRET-donoracceptor pair that we have already proven suitable for this purpose 35 . The acceptor chromophore is QSY7-C5 maleimide, a dark quencher absorbing and fragmenting efficiently at 545 nm in the gas phase 35,56 . The donor chromophore is 5-carboxyrhodamine 575 (Rh575) C5 maleimide, absorbing at 505 nm in the gas phase. Rh575 was also used as the donor chromophore by Talbot et al. 27 , and both the fluorescence excitation and emission spectra have been previously studied in detail 57 . Both chromophores exhibit a C5 linker chain and a terminal maleimide group that allows thiol-specific grafting under mild conditions. The grafting does not affect significantly the absorption and emission properties of the chromophores. As the target peptide must contain 2 thiol groups for the grafting, our analyte molecule exhibits capped N-and C-terminal cysteine residues (Ace-CV 12 HHQKLVFFAEDVGSNK 28 C-NH 2 ). The chemical formulas of the chromophores grafted onto the cysteine residues are depicted in figure 1. The QSY7-C5 maleimide and Rh575-C5-maleimide grafted cysteine residues will be denoted throughout the paper as QSY and RDM, respectively. Complementing the herein established theoretical technique, an experimental-theoretical study of the gas-phase structure of chromophore-grafted Amyloid-β peptides is published elsewhere 51 .

B. Computational setup and structural sampling

Replica Exchange Molecular Dynamics (REMD) simulations 49 were conducted to obtain structural ensembles for the chromophore grafted Amyloid-β peptides. The peptide was described on basis of the the Amber99 58 force field, that, different to more recent versions, was parametrized against gas-phase ab-initio calculations. It was completed with the Generalized Amber Force Field (GAFF 59 ) parametrizing the chromophore grafted cysteine residues QSY and RDM. GAFF is designed to "interoperate" with the Amber force field and is suitable for a broad variety of organic molecules. The dye residues were constructed by first finding a low-energy gas-phase conformation of Ace-QSY-NH 2 and Ace-RDM-NH 2 molecules. For this purpose we ran Molecular Dy-namics (MD) with simulated annealing based on forces obtained with the semiempirical AM1 Hamiltonian 60 using a local code. To obtain the GAFF parameter sets we exploited AM1-bcc 61 calculations that reproduce HF/6-31G* RESP charges. We used the software acpype 62 in conjunction with Antechamber of the AmberTools version 14 suite of programs 63 for this process. From the parameter sets, we created nonstandard internal residues by removing either the Ace or the NH 2 cap and applying manually a small offset to the partial charges of the connecting atoms in order to re-establish the correct total charge of the residues. The residue libraries in Amber format can be obtained from the supplemental material. The interface between the GAFF and Amber99 residues was achieved by assigning Amber99 amino acid bond-, angle-and dihedral-parameters of the interfacing peptide bond.

All forcefield MD simulations and optimizations were carried out with the GROMACS suite of software in its version 4.6.5 [64][65][66] . Topologies of the chromophoregrafted Amyloid beta peptides in 3+, 4+, 5+ and 6+ charge states were constructed using the Leap program of the Ambertools distribution and converted to GROMACS topologies with acpype.

The Ace-CV 12 HHQKLVFFAEDVGSNK 28 C-NH 2 peptides were created in a full extended conformation and the cysteine residues mutated to QSY and RDM in both possible sequences. Additional to the fixed charges from the chromophores (amounting to +2), one, two or three protons were distributed over the LYS (in priority) and HIS residues (LYS 16, LYS 28, HIS 13, HIS 14) and by using the respective parameter set of protonated or neutral residues. For the 3+ and 5+ system, two different topologies resulting from the different possible locations of the proton were obtained. ASP and GLU residues were treated in their neutral form (ASH and GSH Amber residue types) since for relatively small peptides zwitterionic structures are generally less stable in the gas phase. For the MD simulations the systems were treated nonperiodically without solvent molecules or implicit solvation and with no cutoff for nonbonded interactions. For optimizations, the L-BFGS algorithm was employed until a potential energy convergence of 1 kJ/mol was achieved.

MD trajectories were propagated using the velocityverlet integrator with a timestep of 0.001 ps. Thermalization was achieved via coupling with an external heat bath in frame of the velocity-rescaling algorithm 67 that incorporates a stochastic term and thus provides the correct canonical ensemble. Replica exchange molecular dynamics simulations as implemented in GROMACS were conducted by propagating 20 trajectories in parallel. The temperatures of the replicas span a range of 220 K to 850 K (220, 236, 254, 272, 292, 314, 337, 362, 389, 417, 448, 481, 517, 555, 596, 640, 687, 737, 792, 850). After a 0.1 ns equilibration of the replica at their temperatures, sampling of the structures with REMD was begun. A REMD simulation was conducted with a length of 9.9 ns (9.9•10 6 steps) totaling to 10 ns of MD trajec-tory in all temperatures. Exchanges between adjacent replica were attempted every 200 integration steps. The chosen temperatures gave rise to exchange probabilities of 0.01 to 0.1 that ensured the mobility of the replicas through the whole temperature range. After a completed REMD sampling, the 220 K trajectory was quenched by extracting and optimizing every 1000 th step to the next local minimum. A new calculation cycle was started with the found lowest-energy (LE) structure as starting point. The convergence of the procedure was monitored by the energy of the optimized LE structure and the potential energy distribution function (PEDF) of the 220 K and 292 K ensemble. Selecting the quenched lowest-energy structure from the 220K trajectory as starting point for a new run turned out to achieve convergence of the overall sampling procedure more quickly than using the 292K trajectory. When a convergence (energy of optimized LE structure and stationarity of the PEDF) was achieved, the 292 K trajectory was used to represent the ensemble of structures under ambient conditions. In total, about 25 µs of MD trajectory were propagated.

In order to validate our force field parametrization, we tested it against electronic structure calculations with the PM7 Hamiltonian 68 . The found lowest-energy structures from the 220 K REMD cycles were reoptimized without constraints using the MOPAC2012 code [START_REF] Stewart | Mopac2012, stewart computational chemistry, colorado springs, co, usa[END_REF] . The root mean square deviations (RMSD) of all atoms (including hydrogens) of the corresponding optimized structures are reported in table I. These indicate an overall correspondence of the structures with RMSD value between 0.4 and 0.8 Å (the difference between the semiempirical versus force-field optimized structures is visualized for the QSY-Aβ-RDM sequence in figure SA1 in the supplemental material) and confirm the expected accuracy of the chromophore-distance prediction of about 1 Å.

The semiempirical energetics can also be used to quantify the importance of the different possible tautomers (different charge locations) for the 3+ and 5+ systems. The relative PM7 energies for the lowest-energy structures of QSY-Aβ-RDM as well as RDM-Aβ-QSY in its 3+ and 5+ charge states are given in table I. Here the label LYN denotes which of the lysine residues was changed to the neutral form, while HIP denotes the position of the protonated histidine residue in Amber notation. The tautomers differ by at least 0.29 eV, so that only the lowest-energy tautomer ensemble was selected for further processing.

C. Calculation of FRET-efficiencies

The critical Förster distance for a chromophore pair is calculated as

R 0 = 9ln(10) 128π 5 N A • κ 2 Φ D n 4 • J 1 6 , (2)
where Φ D is the quantum yield of the donor in the absence of the acceptor, J the spectral overlap integral, N A Avogadro's number, n the index of refraction being 1 in vacuum, and κ 2 is the orientation factor [START_REF] Sisamakis | Single Molecule Tools, Part B: Super-Resolution, Particle Tracking, Multiparameter, and Force Based Methods[END_REF][START_REF] Schuler | Protein Folding Protocols[END_REF]71 . We use a fluorescence quantum yield of 0.82 as known from Rh575 in ethanol 72 . We first evaluated the sensitivity of FRET-efficiencies in the expected range of chromophore separations as a function of the distance between the optical units of the chromophores and the orientational factor κ 2 utilizing a fixed set of spectral parameters of the two chromophores (parameters are given in table II). The optically active parts of the QSY7 (left part in figure 1) and Rh575 (right part in figure 1) are marked in blue. The geometrical center of these units has been taken to evaluate the chromophore separation. To use the nuclear coordinates to define the center of transition dipole moments is a common strategy (see e.g. Refs. 47,48 ).

The donor and acceptor spectra were approximated assuming a Gaussian lineshape around absorption and emission maximum νmax with the half-width of 1000 cm -1 . Equation 3 yields the absorption extinction coefficient ǫ A in L/mol -1 cm -1 from the oscillator strength f QSY e of the acceptor that we have obtained from a TDDFT calculation (see supplemental material), transition energies given in cm -1 . The calculated spectra were used to calculate the spectral overlap integral (equation 4 with F D (λ) being the normalized fluorescence intensity).

ǫ A (ν) = 2.175091 • 10 8 σ • f QSY e • e -2.772( ν-νmax σ ) 2 (3)
J(λ) = ∞ 0 F D (λ)ǫ A (λ)λ 4 dλ (4)
The numerical results for these chromophoreparameters are visualized in figure 2 (compare also table 2. Calculated FRET efficiency of the QSY7-Rh575 chromophore pair as a function of the chromophore separation and the orientational factor κ 2 (equation 1,2,3,4 with the spectral parameters of QSY7 ad RDM575 in table II). Color coding of EF RET values according to the legend on the right. The average value of an isotropic orientational distribution κ 2 iso = 2/3 is indicated as a white line.

II). The calculated critical Förster distance R 0 for this chromophore pair amounts 40.3 Å (equation 2). The influence of a slightly different fluorescence quantum yield (Φ D =0.9) on the final FRET-efficiency values is only marginal (R 0 [Φ D =0.9]=40.9 Å) as is a slightly different oscillator strength (R 0 [f e =0.95]=41.0 Å). Within a chromophore separation of 80 Å the FRET efficiency has dropped to nearly zero even when orientation would favor excitation energy transfer (high κ 2 ). On the other hand, the occurrence of high FRET-efficiencies can be substantially decreased by a perpendicular chromophore orientation leading to κ 2 approaching zero. Consequently, if a simple distinction between near and far is to be extended by a more accurate quantitative measure, an inclusion of the orientational effects is mandatory. In order to predict the average FRET-efficiencies for the chromophore-grafted Amyloid-β peptides, we have postprocessed the calculated structural samples in the 3+, 4+, 5+ and 6+ charge states and have computed explicitly their FRET-efficiencies. In the following, we compare two different strategies with respect to the structural basis for the FRET-efficiency calculation: a) single low-energy conformations and b) the inclusion of the full canonical ensembles. The results for the modeled FRETefficiencies in comparison with the offset-corrected exper- imental efficiencies are displayed in figure 3.

Figure 3a depicts the FRET efficiencies obtained for the lowest-energy structure (LE) with an assumed isotropic chromophore-orientation distribution (κ 2 = 2/3). The agreement of the efficiencies in figure 3a is qualitative, showing that already the single lowest-energy structure is indicative for the overall trend and may be used to classify chromophores as near or far . Here, thermal effects are not accounted for as the lowest-energy structure corresponds to a temperature of 0 K. This adds to the negligence of chromophore-orientational effects and explains the mediocre agreement.

There are two main strategies to improve the agreement. On the one hand, chromophore-orientations can be explicitly accounted for rather than assuming their isotropic distribution, and, on the other hand, a statistically representative thermal ensemble can be used as a structural basis. We included both effects by using explicit orientational and distance distributions in the 292 K ensemble and report the resulting E F RET values in figure 3b. Evidently, this strategy -which we discuss in detail in the following -reproduces the experimental values in good agreement.

The results from analyzing the thermal ensembles are displayed in figure 4 together with a visualization of a small ensemble of structures. The sampled distance distribution functions as a function of charge state are displayed in figure 4a. The two sequences QSY-Aβ-RDM versus RDM-Aβ-QSY in the 3+ charge state exhibit different chromophore separations indicated by the two distinct features of the distribution function. In the 6+ charge state, three different conformations of the two sequences appear to be sampled at 292 K. In conclusion, a single distance from only one single structure might be a poor representation of the structured and nonsymmetric distance distributions present at 292 K. The orientation factor κ 2 can be calculated from a structure according to equation 5 with the N-atoms of the chromophores defining the transition dipole moment vectors µ of the optically active units. Upon the sampled canonical ensembles of structures at 292 K we have calculated the κ 2 distributions shown in figure SA3 in the supplemental material.

κ 2 = µ D • µ A -3(µ D • r) • (µ A • r) (5)
From the distance distributions and the κ 2 distributions we calculated FRET-efficiency distributions in the 292 K ensembles as shown in figure 4b. These assume a very different functional form compared to the distance distributions. While for the 3+ and 6+, the distributions are centrered around a value near to 1 and near to 0, they are spread over a large interval for the 4+ and 5+ charge state. This behavior is originating from the broad shape of the orientational factor κ 2 distribution function with values ranging between 0 and 4.

From the κ 2 distribution functions, the potential impact of chromophore orientations on the FRETefficiencies can be deduced. For this reason, we quantified the deviation between the observed and the theoretical isotropic κ 2 distribution functions. The integral absolute deviation between the two functions (see supplemental FIG. 5. Correlation between chromophore distance and FRET efficiency for Rh575/QSY7-labeled Amyloid-β peptide fragments in the charge range from +3 to +6. Blue color map shows the normalized probability distribution for the occurrence of a specific EF RET , r pair in the samples (dark blue regions indicate a high probability). Discrete EF RET values were calculated for all sampled distance and chromophore orientationsof all charge states and both possible sequences RDM-Aβ-QSY and QSY-Aβ-RDM and smoothed by twovariate Gaussian functions. The ensemble-average EF RET ,r pairs for each charge state are given as green dots. In red, the analytic FRET-efficiency curve is given for a fixed orientation factor of κ 2 = 2/3. material, figure SA3) amounts to 0.7 for the 3+ compared to about 0.35 for 4+, 5+ and 6+. This high deviation for the 3+ charge clearly indicates a nonisotropic distribution. In the molecular picture, the narrow κ 2 distribution is associated with discrete conformations being fixed by intramolecular interactions. Notice, that the particular conformation adopted is not only imposed by the overall charge state, but also depends on the charge location (that there exists an interplay between charge distribution, conformation and FRET-efficiencies can be depicted from the different 3+ κ 2 distributions in figure SA4 of the supplemental material). In the 4+, 5+ and 6+ charge systems a quasi-free rotation of the optically active units can be concluded from the κ 2 distributions -where fixing interactions are absent. Overall, the observed decrease in E F RET with growing charge state is in agreement with the expectation that the Coulombrepulsion between the charges is the main driving force for the structural changes upon protonation.

For making the overall effect of thermal distributions on FRET-efficiencies more comprehensive, we visualize the E F RET versus distance correlation in figure 5. This 2-dimensional plot maps the calculated FRET-efficiencies onto the space of sampled distances. The ensembleaverage values E F RET depicted by the green dots correspond to the values reported in 3b. This 2D map demonstrates that there exists no one-to-one mapping of a single distance with a single FRET-efficiency. An unambiguous assignment of a single conformation of a molecule to an experimental E F RET value is therefore only possible when distance and κ 2 distributions are narrow and posess only one maximum. This limits the isotropic orientational distribution assumption (κ 2 =2/3) often made. Concerning the ensemble-average of chromophore separations at 292 K, however, the (κ 2 =2/3) assumption seems reasonable for the studied system. In conclusion, the knowledge of the explicit thermal structural ensembles is mandatory to perform a structural assignment to a FRET efficiency and to judge assumptions of orientational distributions. For this reason a global experimental-theoretical strategy is the most promising route towards the determination of biomolecular structures in the gas phase.

III. CONCLUSIONS AND OUTLOOK

In the presented work, we have performed a complete first-principles modeling of FRET-efficiencies of the chromophore-grafted Amyloid-β (12-28) fragment in the charge states 3+, 4+, 5+ and 6+. The modeling is based on an extensive exploration of the conformational space of twelve different systems (charge states, tautomers) by Replica-Exchange Molecular Dynamics (REMD) totaling to about 25 µs of MD trajectory. We compared the FRET-efficiencies calculated using Förster Theory with measured values from the gas-phase "action"-FRET experiments (Daly et al. 51 ). By analyzing the agreement between experiment and theory employing different structural representations, we found that an inclusion of distance distributions of the canonical ensemble outperforms calculations where only single low-energy conformations are considered. The former agrees very well with the experimental data. The average FRET-efficiencies of the systems studied here mainly depend on the average chromophore separations and orientational effects do not play an important role. For for the 3+ charge state however, we have identified fixed chromophore orientations with a reduced rotational freedom which leads to clear non-isotropic κ 2 distribution. The fact that the extent of conformational fixing is depending on the particular system, suggests an explicit inclusion of chromophore orientation distributions into FRET modeling as best practice. Including thermal and conformational effects, we obtain excellent correspondence of theory and experiment, thereby assigning structural parameters such as the interchromophore distance distributions to the experimental signal. An "elongation" of the peptide and the resulting drop in FRET efficiency with increasing charge can therefore be interpreted. The achieved agreement encourages us to use the structural model, parametrization and sampling scheme for further investigations. The simulation protocol can be easily adapted to other peptides charge states and protonation patterns. It has to be noticed, however, that the bottleneck for the numerical accuracy of the procedure is a complete exploration of the molecule's conformational space. In view of the increasing conformational freedom of bigger systems, the presented strategy might be extended with biasing and steering algorithms [START_REF] Calvo | Supplemental Material for: Structural exploration and Förster theory modeling for the interpretation of gas-phase FRET measurements: Chromophore-grafted Amyloid-β peptides Alexander Kulesza[END_REF] .

IV. ACKNOWLEDGEMENT

We gratefully acknowledge computer time granted by the PSMN (Pôle Scientifique de Modélisation Numérique) computing center of ENS de Lyon. A. K acknowledges the German Science Foundation DFG for funding in frame of the Research Fellowship Ku 3251/1-1. The research leading to these results has received funding from the European Research Council under the European Union's Seventh Framework Program (FP7/2007-2013 Grant agreement N320659).

See supplemental material at [URL will be inserted by AIP] for an overlay of structures optimized with our force-field parametrization and the semiempirical PM7 level of theory; a detailed analysis of the influence of thermal effects for the distance and orientational distributions on the predicted FRET-efficiency; numerical values of the calculated FRET-efficiencies; inter-chromophore distance and orientational factor distributions of chromophore-grafted Amyloid-β 12-28; orientational factor distributions for a different tautomer (charge location) of the 3+ chromophore-grafted Amyloid-β 12-28; details of DFT/TDDFT calculations to obtain the QSY7 absorption oscillator strength and nonstandard GAFF/AMBER99 residue libraries in Amber format for Ac-QSY, Ac-RDM, QSY-NH 2 and RDM-NH 2 residues. Figure SA2 shows the influence of distance distributions and orientational distributions on the FRET-efficiency predictions. SA2a) and SA2b) correspond to figure 3a and 3b in the main text (taking the lowest-energy optimized structure and isotropic orientational distribution / explicit thermal distance and orientational distributions) Figure SA2b) shows the FRET-efficiency values from the lowest-energy optimized structures but with the explicit κ 2 value for the structure.

Assessment of parametrization

Influence of thermal and orientational effects for FRET modeling

Although the results for the 4+ and 5+ appear to better fit the experiment than the calculation in figure 3a/SA2a, a significant error cancellation might be present (QSY-Aβ-RDM and RDM-Aβ-QSY FRET efficiencies differ by up to 0.

2) The deviation of the 6+ charge state from the experiment is characteristic for a situation in which such cancellation does not take place. Therefore, we regard calculating FRET efficiencies based on a single e.g. LE structure but accounting its explicit chromophore-orientation inferior to using κ 2 =2/3. In SA2c) a thermal ensemble of distances corresponding to 292 K is used for the calculation. Here, the width and shape of the distance distribution function is accounted for in its influence on the FRET-efficiencies. Chromophore-orientations have been assumed to be isotropically distributed. This treatment proves to be rather accurate for the systems studied here, as a very good quantitative agreement with the experiment is achieved. Comparing figure SA2c with SA3d (figure 3b in the main text) indicates only a slight improvement of the numerical results when also explicit orientational distributions are included. Although the 3+ charge state displays fixed conformations with narrow κ 2 distribution, it does not greatly affect the averaged FRET-efficiencies. The low chromophore distance is responsible for FRET-efficiencies that are very near to one, irrespective of their orientation. It has to be pointed out that for a different system, conformational fixing could coincide with separations in the sensitive region (e.g . if conformational fixing would be existent in the 5+ charge state of our system) which would lead to a breakdown of the good agreement with the strategy of figure SA3c. Differences in the sampled κ 2 distribution functions from isotropic limit are marked in gray.

Analysis of distance distribution and chromophore orientation distribution

Analysis of orientational factors as a function of charge distribution

Calculation of QSY7 absorption oscillator strength

The absorption oscillator strength was estimated from a TDDFT calculation of the S1 state of a QSY7 dye. We obtained a low-energy conformation of the molecule by running simulated annealing coupled to molecular dynamics where we used the semiempirical AM1 Hamiltonian [1] for the forces. This structure was optimized at the B3LYP/6-31G** level of density functional theory [2-5] and applying Grimme's D3BJ dispersion correction [6]. The S1←S0 transition properties were calculated with the CAM-B3LYP functional [7] and using the 6-311++G** basis set. All calculations were performed using Gaussian09 [8].

Nonstandard residue parameter files in Amber format AC-QSY-Nterm.lib -library for the N-terminal QSY residue AC-RDM-Nterm.lib -library for the N-terminal RDM residue SupplementalMaterial.CHROMOPHORES-AMBERLIB.frcmod -additional parameters from the Antechamber procedure and used Amber-GAFF interface QSY-NH2-Cterm.lib -library for the C-terminal QSY residue RDM-NH2-Cterm.lib -library for the C-terminal RDM residue

FIG. 1 .

 1 FIG. 1. Chemical formulas of the QSY7-C5 maleimide and Rhodamine 575-C5 maleimide chromophores grafted to a Accysteine-NH2 peptide used for the parametrization of the nonstandard aminoacid residues (denoted as RDM and QSY residues in the following).

FIG. 3 .

 3 FIG. 3. Explicitly calculated average FRET efficiencies EF RET as a function of charge state: In red the experimental offset-corrected FRET-efficiency is given. EF RET is calculated using a) the chromophore-distance of the lowest energy optimized structure rLE and κ 2 = 2/3 from an isotropic orientation distribution b) explicit distribution of chromophore-distances as well as orientational factors [r,κ 2 ] 292K . Experimental values are taken from ref. 51 .

FIG. 4 .

 4 FIG.4. Normalized probability distribution functions of a) chromophore-distances and b) FRET-efficiencies EF RET in the samples of Rh575/QSY7-labeled Amyloid-β peptide fragments (smoothed by Gaussians with the half-widths of 1 Å and 0.01 for r and EF RET ) in the charge states 3+, 4+, 5+ and +6 (from top to bottom). On top, a superposition of 5 structures (blue and red: Rh575 and QSY7 chromophores, green: Amyloid-β 12-28) from the 4+ ensemble visualizes the termal effect.

Figure SA1 :

 SA1 Figure SA1: Overlay of optimized structures of QSY-Aβ-RDM. The force-field optimized structure is given in yellow, while the PM7 optimized structure is given color-coded for the atom type.

Figure SA2 :

 SA2 Figure SA2: Explicitly calculated average FRET efficiencies E ⟨ FRET as a function of charge state: ⟩ In red the experimental offset-corrected FRET-efficiency is given. E ⟨ FRET is calculated using a) the ⟩ chromophore-distance of the lowest energy optimized structure r LE and κ 2 =2/3 b) the chromophoredistance of the lowest energy structure r LE and the actual value of κ 2 for the lowest-energy structure, c) the average from the thermal distribution of chromophore-distances [r] ⟨ 292K and κ ⟩ 2 =2/3 d) explicit distribution of chromophore-distances as well as orientational factors [r,κ 2 ] 292K . For values see table SA1.Experimental values are taken from S. Daly et al., to be published (2015).

Figure SA3 :

 SA3 Figure SA3: Normalized probability distribution functions for chromophore-chromophore distance r (left) and orientational factor κ 2 (right) in the structure samples (292 K) of chromophore-grafted Amyloid-β fragments in the charge states 3+ to 6+ (smoothing by Gaussian functions with halfwidths of 1 Å and 0.05). A 50/50 mixture of the two sequences QSY-Aβ-RDM and RDM-Aβ-QSY is assumed. The red curve in the κ 2 depicts ideal isotropic distribution of the transition dipole vectors. Differences in the sampled κ 2 distribution functions from isotropic limit are marked in gray.

Figure SA4 :

 SA4 Figure SA4: Normalized probability distribution functions for orientational factors κ 2 in the converged thermal ensembles for chromophore-grafted Amyloid-β in the charge state 3+ compared to isotropic distribution (green). The black curve P(κ 2 ) A is shown for the lowest-energy proton location where QSY-Aβ-RDM (LYN 28) and RDM-Aβ-QSY (LYN 16) are present (see main text and table 2) while P(κ 2 ) B denotes the configurations QSY-Aβ-RDM (LYN 16) and RDM-Aβ-QSY (LYN 28), which are higher in energy by 0.37 and 0.29 eV, respectively, and therefore not expected to be significantly populated at 293 K. Notice that P(κ 2 ) A is identical to the one shown in figureSA3(upper part).

TABLE I

 I

	. Semiempirical calculations: RMSD values of
	atomic coordinates for force-field compared to PM7-optimized
	structures, relative PM7 energies ∆E of different PM7-
	optimized tautomers for 3+ and 5+ charge states
	System	RMSD [ Å] ∆E [eV]
	QSY-Aβ-RDM
	3+ (LYN 16)	0.41	0.37
	3+ (LYN 28)	0.65	0.00
	4+	0.78
	5+ (HIP 13)	0.75	0.83
	5+ (HIP 14)	0.65	0.00
	6+	0.59
	RDM-Aβ-QSY
	3+ (LYN 16)	0.53	0.00
	3+ (LYN 28)	0.59	0.29
	4+	0.58
	5+ (HIP 13)	0.79	0.38
	5+ (HIP 14)	0.48	0.00
	6+	0.59

TABLE II .

 II Spectral properties of the QSY7 and RDM chromophores used for FRET-modeling variable value taken from

	λ RDM em	505 nm gas-phase action spectrum 35
	λ QSY abs ΦD	539 nm gas-phase action spectrum 35 0.82 Ref. 72
	f QSY e	0.85	this work (TDDFT calculation)
	R calc 0	40.3 Å this work
	FIG.

Table SA1 :

 SA1 Calculated average FRET efficiencies E ⟨ FRET as a function of charge state using a) the ⟩ chromophore-distance of the lowest energy optimized structure r LE and κ 2 =2/3, b) the chromophoredistance of the lowest energy structure r LE and the actual value of κ 2 for the lowest-energy structure, c) the average from the thermal distribution of chromophore-distances[r]

					⟨ 292K and κ ⟩	2 =2/3 d)
	explicit distribution of chromophore-distances as well as orientational factors [r,κ 2 ] 292K .
		a)	b)	c)	d)
	3+	1.00	0.96	1.00	0.96
	4+	0.51	0.25	0.37	0.29
	5+	0.27	0.14	0.16	0.17
	6+	0.15	0.00	0.10	0.06