Roethlein et al. BMC Biophysics (2015) 8:5 DOI 10.1186/s13628-015-0020-z

RESEARCH ARTICLE Open Access

A flexible approach to assess fluorescence decayfunctions in complex energy transfer systemsChristoph Roethlein1*, Markus S Miettinen2 and Zoya Ignatova1,3*

Abstract

Background: Time-correlated Förster resonance energy transfer (FRET) probes molecular distances with greateraccuracy than intensity-based calculation of FRET efficiency and provides a powerful tool to study biomolecularstructure and dynamics. Moreover, time-correlated photon count measurements bear additional information on thevariety of donor surroundings allowing more detailed differentiation between distinct structural geometries whichare typically inaccessible to general fitting solutions.

Results: Here we develop a new approach based on Monte Carlo simulations of time-correlated FRET events toestimate the time-correlated single photon counts (TCSPC) histograms in complex systems. This simulation solutionassesses the full statistics of time-correlated photon counts and distance distributions of fluorescently labeledbiomolecules. The simulations are consistent with the theoretical predictions of the dye behavior in FRET systemswith defined dye distances and measurements of randomly distributed dye solutions. We validate the simulationresults using a highly heterogeneous aggregation system and explore the conditions to use this tool in complexsystems.

Conclusion: This approach is powerful in distinguishing distance distributions in a wide variety of experimentalsetups, thus providing a versatile tool to accurately distinguish between different structural assemblies in highlycomplex systems.

Keywords: Time resolved FRET, Monte-Carlo simulations, Complex heterogeneous systems, Protein aggregation

BackgroundFRET is a powerful approach that is widely used toassess structural dynamics in biomolecules. FRET isbased on non-radiative energy transfer between two dyesdependent on their absolute distance [1,2] which pro-vides specific information about distances betweendye-labeled macromolecules and on structural character-istics of investigated systems [3-5]. For many years it hasbeen used to differentiate a small set of defined states,e.g., in protein folding studies [6] and co-localizationof biomolecules in cells [7]. The shape of distance-dependence distribution is increasingly exploited formore detailed analysis. For complex FRET systems, e.g.membranes or spherical vesicle surfaces, some analyticalsolutions have been developed to determine distance

* Correspondence: roethlein@outlook.de; zoya.ignatova@chemie.uni-hamburg.de1Biochemistry and Biology, University of Potsdam, Potsdam, Germany3Biochemistry and Molecular Biology, University of Hamburg, Hamburg,GermanyFull list of author information is available at the end of the article

© 2015 Roethlein et al.; licensee BioMed CentCommons Attribution License (http://creativecreproduction in any medium, provided the orDedication waiver (http://creativecommons.orunless otherwise stated.

distributions for various shapes [8-10]. However, it hasbeen increasingly recognized that more information canbe extracted from time-resolved fluorescence basedapproaches [3,11]. In heterogeneous biological systemswith complex distance distributions, monitoring FRETin a time-resolved manner provides ample informationon the variety of donor surroundings and allows detaileddifferentiation between distinct structural geometries.Various approaches which differ in their complexity andutility have been developed to extract different parametersfrom time-correlated FRET measurements. Monte Carlo-based methods have been applied to achieve higher levelquantification of FRET experiments; examples include ac-curate determination of distributions of membrane compo-nents and assessment of FRET efficiency in complexsystems with multiple transfer opportunities in large actin-filaments [12,13]. The ExiFRET tool (www.exifret.com)

ral. This is an Open Access article distributed under the terms of the Creativeommons.org/licenses/by/4.0), which permits unrestricted use, distribution, andiginal work is properly credited. The Creative Commons Public Domaing/publicdomain/zero/1.0/) applies to the data made available in this article,

http://www.exifret.commailto:roethlein@outlook.demailto:zoya.ignatova@chemie.uni-hamburg.dehttp://creativecommons.org/licenses/by/4.0http://creativecommons.org/publicdomain/zero/1.0/

Roethlein et al. BMC Biophysics (2015) 8:5 Page 2 of 10

developed by Corry and co-workers is capable of calcu-lating FRET efficiencies and ratios between donor andacceptor emission in a large variety of FRET-based mi-croscopy studies using the excluded circular volumesaround the dye labels [14]. However, this tool is limitedto only predicting the averaged FRET efficiency of asystem.Recent developments include Monte Carlo analysis of

FRET kinetic traces which are used to estimate FRETrates taking into account the full statistics of individualphoton absorption, energy transfer, and photon emissionevents [15], or to gain detailed insight into the dynamicsof labeled proteins switching between two states [16].The algorithms and tools developed in this context focuson the precise prediction of the behavior of single FRETpairs and exact distributions of distances and relativeorientations. However, they require molecular dynamicssimulations to precisely predict the distance distribu-tions and thus are limited to simple FRET systems thatare manageable with moderate computational resources.Neurodegenerative pathologies based in misfolding

and aggregation, such as Alzheimer's and Huntington'sdiseases, are growing in medical prevalence [17]. Aggre-gation is a multi-step, heterogeneous process [18,19] andposes challenges for structural and mechanistic insight.A set of methods utilizing distance-dependent FRET issuitable to assess aggregate structure. Yet, it is limited inapplication because of lack of uniform algorithms andtools to extract distances from FRET-based measure-ments in such heterogeneous systems. In a recent study,we developed Monte Carlo simulations to deconvolutecomplex fluorescent decay signatures observed in poly-glutamine fibril architecture [20]. As this approach isbased on first principles of underlying fluorescence phe-nomena, its utility is generalizable in distinguishingbetween distance distributions of fluorescently labeledentities in a wide variety of experimental setups. Here,we extended the broad application of this algorithm toother heterogeneous systems in an accessible and opensource platform. This method is applicable for predictingthe time-correlated fluorescent signal for systems withmultiple donors and acceptors within close proximity.We assessed the generalized conditions and assumptionsunder which the tool can be utilized without comprom-ising the simulation results. Furthermore, we carefullyinvestigated the impact of various assumptions that re-duced the required input and the number and complex-ity of control measurements. The algorithm is a versatiletool for assessment of complex structural assemblies andrequires considering detailed competitive transfer reac-tions between multiple donors and acceptors in combin-ation with other experimental models given by moleculardynamics simulations, x-ray diffraction patterns or NMRspectroscopy.

MethodsChoice of dyesWe first selected a set of dyes for optimal consistency ofsimulated and experimental donor TCSPC histogramsand to reflect the expected behavior in the simulations.Note that this approach allows also for calculating theacceptor TCSPC histograms; here, if not otherwisementioned, we focus primarily on the donor TCSPC his-tograms. Typically, the Förster radius of the FRET pairshould be chosen within a range of distances which areexpected between the donor and its nearest neighboringacceptor for the following reasons. Firstly, FRET effi-ciency and measured donor TCSPC histograms respondsensitively to distances ranging from one half to twotimes the Förster radius, thus increasing the influence ofsmall deviations between distances. For distances beyondtwice the Förster radius, the FRET rates are extremelylow with very little influence on time-dependent photonemission probability. Energy transfer for distances closerthan half of the Förster radius is extremely efficient, buthardly any donor photons will be collected. Typically,those distances do not influence the donor photondistributions, since donors with very high quenchingrates have negligible brightness compared to donorswithout acceptors at distances closer than half of theFörster radius. Additionally, since within the half of theFörster radius, the donor-to-acceptor transfer occursalmost instantaneously, no significant change in thedonor TCSPC (or even in the acceptor TCSPC) isexpected. Secondly, the factor describing the relativeorientation of donor emission and acceptor absorptiontransition dipoles (κ2) is usually taken as 2/3, which isbased on the assumption that the dyes have enough timeto randomize their orientation during the time thedonor stays excited [1]. Shortening the fluorescent decaytime by introducing acceptors in much closer proximitythan the half of the Förster radius renders this assump-tion about κ2 inaccurate. Thirdly, too many short rangedistances dramatically increase the simulation time byreducing the number of the collected photons becauseof the large competition rates for transfer deactivation ofthe donors. Furthermore, the fluorescence decay time ofthe donor should be sufficiently long during the mea-surements to provide enough time for the donors torandomize their orientation after excitation and limit thefraction of photons detected by a non-random distribu-tion of photon emission angles at the onset of signaldetection.The possibility of homo-FRET adds other aspect to

consider when selecting the dyes. Usually, homo-FRETdecreases the time needed for randomization of theorientation of the excited dyes and introduces randomnessin the emission direction of transition dipole moment ofthe donor. Furthermore, homo-FRET changes the position

Roethlein et al. BMC Biophysics (2015) 8:5 Page 3 of 10

of the excited donor which is usually fixed in the simu-lation. An appropriate choice of the dye will render thiseffect negligible. The Stokes shift should be sufficientlylarge to minimize the amount of homo-FRET in thefluorescent decay time measurements. The effects oftransferring the excited states between two neighboringdonors in either direction do not necessarily cancel outeach other. Therefore, the overlap of the absorption andemission spectra of the donor should be minimal tolimit the donor-donor transfers to distances close tozero in which both donors would almost have the sameenvironment.Here, we used Alexa 488 and Alexa 594 (Life tech-

nologies). The dyes were chosen because they implementthe described requirements and criteria for suitableexperimental setups, e.g. high photostability, extinctioncoefficient and quantum yield [21,22].

Generation of input filesTo read the information of the positions serving as ananchor point into the code, input files are createdaccording to the following scheme. The files contain the

Figure 1 Schematic flow chart depicting the simulation routine.

three Cartesian coordinates that define the dimensionsof the simulated volume, three binary values determiningthe periodicity of the system in each direction andthe number and Cartesian coordinates of donors andacceptors in the system. All coordinates are given innanometers.We randomly positioned our dyes according to the

probabilities derived from (i) specific dye concentrationswith randomly distributed dyes in the three dimensionalspace, or (ii) fixed labeled structures with known labelingsites according to the probabilities of labeling with do-nors or acceptors. Additionally, files with defined donor-acceptor distances in defined ratios were created tocompare the simulations to theoretically calculatedfluorescent decay time behavior.

Calculation of FRET ratesBy simulating of the possible deactivation processes foreach donor, all acceptors that can get possibly closerthan twice of the Förster radius, considering also thelength of the dye linkers, are taken into account. Thedistances r, the FRET rates kT and the deactivation

Roethlein et al. BMC Biophysics (2015) 8:5 Page 4 of 10

probabilities Pdeactivation are then calculated for everydonor (Figure 1, Step 1).

kT ¼ 1τD;0

� R0r

� �6ð1Þ

where τD,0 is the donor lifetime in absence of acceptorand R0 is the characteristic Förster radius, at which halfof the deactivation of electronically excited donor statesoccurs via energy transfer to the acceptor.

Pdeactivation ¼ 1− exp −X

iki � Δt

� �ð2Þ

where ki denotes the individual deactivation rates forthis particular donor by photon emission, FRET, or otherdeactivation processes and Δt is the length of the timeinterval. The periodicity of the volume is taken into ac-count to correctly determine the environment of everydonor. [For additional information see Additional file 1.]The linker distances can be chosen for the donor and

acceptor individually. In the simulation, the position ofthe dyes is changed from the coordinates of the labelingsite given in the input file by that distance in a randomdirection (Figure 1, Step 4). During a single excitationcycle the dye positions are assumed to be constant.However between two excitation events all dyes aremoving within the volume which is limited by the linkerindependent of their last assigned position. Note thatthis leads to rather complex distance distributions be-tween the dyes resulting in uneconomical computationtimes for analytical solutions. A short linker distance willminimize errors due to non-random orientations of thelinker and simplifications of the distance distribution ofthe dye to its original coordinates. In contrast, largerlinker sequences allow for larger inaccuracies in the pre-dicted distribution of dye positions. However, a severeviolation of the assumed conditions would be to restrictthe randomness of the dye orientation or its rotationalflexibility. This could be the case by a very short or verystiff linker sequence. Restricted dye rotation flexibilitywill result in false predictions of κ2 values and, at leastduring the early time steps, the time-dependent detec-tion probability of the emitted photons at the chosendetection angle. We recommend ensuring reasonabledye flexibility by confirming the dye randomization byanisotropy measurements [20]. Alternatively, by re-stricted dye-dipol mobility accurate molecular dynamics(MD) simulations should be performed to obtain theexact distribution of the relative angles in each systemindividually [16]. Note that MD simulations might needan extraordinary computational power and thus mightnot be applicable in complex systems.

Simulation routineAfter every necessary parameter is read or calculated,the experimental procedure is simulated (Figure 1, Step2-11). A random subset of donors is chosen by assigningrandom numbers to each donor and excited at a fixedtime point zero, assuming an infinitely short light pulsefor excitation (Figure 1, Step 3). To take into accountthe chronology of events, every time step observed inthe experiment is simulated for all dyes (Figure 1, Step6-11). After simulation for a time equal to the wholeexperimentally monitored time or when no exciteddonor has remained in the excitation cycle, the circuit isterminated. Optionally, the time dependence of theacceptor deactivation is also simulated over the wholetime interval. The excitation cycles are repeated untilreaching in one time interval the threshold of photonswhich is set by the user and are needed to collect forproper statistics (Figure 1, Step 2). Within each timestep of a given excitation cycle the donors and acceptorsare monitored in the given order (Figure 1).

Acceptor deactivationTo allow acceptors that had already received an energyquantum to become available for new FRET events, we as-sign a random probability to every electronically excitedacceptor using a random number generator (Figure 1,Step 6). If it is lower than the deactivation probabilitywithin a simulation time step, the acceptor is againavailable for energy transfers in analogy to Eq. 2 wherethe acceptor lifetime is the reciprocal sum of the deacti-vation processes. The number of deactivation events ofacceptors is summed up for each time interval.

Donor excitation cyclesTo every donor in the system a random value is assignedwith a random number generator. Donors with randomvalues below the excitation probability are excited. Weaccelerated the simulation by ensuring that there is atleast one donor excited in every excitation cycle. Exciteddonors are subsequently ordered by their randomlyassigned value, from the lowest to the highest, and in agiven excitation cycle simulated in that order to avoida possible bias due to a fixed order in the simulateddonors (Figure 1, Step 7). All acceptors within a distancetwice of the Förster radius to a donor, considering thelinker lengths of both donor and acceptor, are tested fortheir availability for FRET within this time interval. Theprobability to remain excited is then calculated for everydonor based on its current environment (Eq. 2) and iscompared with a random probability to remain excitedor not (Figure 1, Step 8). In case of deactivation, the typeof deactivation is determined by defining probabilityintervals from the ratio of the deactivation rates ofdifferent processes and by comparing to another random

Roethlein et al. BMC Biophysics (2015) 8:5 Page 5 of 10

probability. The deactivation process representing thatinterval is chosen for deactivation (Figure 1, Step 9). Theindividual deactivation processes are summed up foreach simulated time interval. It should be noted that allphotons emitted during a single excitation cycle arecounted for the given time interval. This is done becausetime-dependent variations of the photon detection prob-ability (at the detection angle) are assumed to be ofminor importance for the simulated systems; also inreality the frequency of photon counting should be lowenough to not significantly contribute and detect morethan one photon in an excitation cycle. If a donor isdeactivated via a FRET event, the corresponding ac-ceptor is excited and therefore temporally blocked foranother energy transfer (Figure 1, Step 11).

Output of the simulationsThe output consists of all donor identities that do nothave any acceptor in a relevant distance and includesalso a table containing the frequencies of all donor de-activation processes observed in each time interval. Thetotal number of acceptor deactivations is included in theoutput. The ratio of deactivation rates for each acceptorusually stays constant over time. Hence, the acceptordeactivation represents a qualitative TCSPC histogram,given the time-dependent excitation pattern of thatFRET system. Note that the acceptor decays are onlysimulated completely if the excitation cycles are not ter-minated after deactivation of all donors. To compare theacceptor signal with experimentally recorded acceptorphoton statistics and set them in relation to the observeddonor signal, assuming a known ratio of detection prob-ability by the two detection systems, only the quantumyield of the acceptor ΦA needs to be determined.

ΦA ¼ kFluorescenceXi

kið3Þ

where kFluorescence depicts the rate of deactivation byphoton emission. To compare with the experimentaldata, the simulated photon decay must be convolutedwith the instrument response functions of the day ofmeasurement for the given dye channel to implementthe time dependence of the detection probability densityof the instrumental setup.

Fluorescence decay time measurementsFluorescent decay times were recorded on a FL920 spec-trometer (Edinburgh Instruments) operated in a TCSPCmode. In the measurements we used a time window of50 ns and 1024 channels. Samples were excited at λex =450 nm using a SC-400-PP supercontinuum source

(Fianum) and the emission was collected at λem =525 nm using a polarizer set at magic angle position anda multichannel plate (Europhoton) as a detector. Therepetition rate of the excitation light source was set to10 MHz.

Quantum yield and Förster radius determinationUsing a photoluminescence (PL) quantum yield meas-urement system (C9920, Hamamatu Photonics), thequantum yield ΦD for the donor was determined to be0.6. Based on that, the Förster radius was determined tobe 5.4 nm (Eq. 4) assuming that the randomization ofthe dye orientation is much faster than the fluorescencedecay time. The refractive index was 1.33. The overlapintegral J (λ) was calculated from the recorded donoremission and acceptor absorption spectra determinedwith the QuantaMaster 40 (Photon Technology Inter-national) with Felix32 software. The Förster radius is afunction of κ2, describing the relative orientation of thedonor and acceptor transition dipoles, the quantum yieldof the donor ΦD, the refractive index n of the solutionand the overlap integral J (λ) between the donor emis-sion and the acceptor absorption spectra:

R0 ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi9� ln 10ð Þ � κ2 �ΦD � J λð Þ

128� π5 � NA � n46

sð4Þ

where NA is the Avogadro number.

Theoretical calculationsTo verify the consistency of our code with the theory ofFörster resonance energy transfer, we calculated theprobability for photon emission. For multiple distances,the resulting distributions were set in relation to theirweight according to the photons expected during thefirst time interval based on the probability to emit pho-tons and on the fraction of excited donors in the par-ticular surrounding.

NP tð Þ ¼X

if D;i �

dDi;emissiondt

¼X

if D;i � kemission � exp −

Xjkj � Δt

� �ð5Þ

where NP(t) is the amount of photons in dependenceof time, fD,i is the fraction of donors with surrounding i,kemission is the emission rate of the donor and the sum ofkj represents the combined deactivation rate constantfor all deactivation processes. The derivative dDi,emission/dt represents the deactivations by emission per time unitat a given time point. It cannot be assumed to beconstant due to photon decay events of donors with very

Roethlein et al. BMC Biophysics (2015) 8:5 Page 6 of 10

close acceptors at earlier times than resolved by a typicalexperimental time resolution. Therefore in the first timeinterval it was determined for 1000 equal subintervalsand subsequently averaged. [For additional informationsee Additional file 1.]

Code accessThe code is deposited in an open-access platform domain(http://figshare.com/) and has the accession number:1158992 (http://figshare.com/articles/FRET/1158992).

Results and discussionTime-correlated photon count measurements offer apowerful tool to investigate dye surroundings. TCSPC his-tograms respond sensitively to environmental changes,such as solvent polarity, pH or presence of heavy atoms.Additionally, atomic distance information can be extractedusing resonance energy transfer between donors andacceptors that are introduced at targeted labeling sites.However, when using a mathematical fitting function togain many parameters it is often impossible to implementrestrictions and the fitting will instead settle into differentminima that represent meaningless dye arrangements.Connecting the knowledge of the possible distancedistributions to fluorescent decay functions using MonteCarlo simulations provides a powerful way to bypass this

Figure 2 Reproduction of theoretically calculated fluorescent decay tenvironments. The acceptor distances were between 1 and 10 nm. B, C: Si(from A). Representative curves of simulations with one donor-acceptor dispair the distance was variable and the distances are displayed in the legenin two plots. The amount of the two donors in each simulation was equal. Novalues are depicted with symbols and the matching theoretical calculations a

limitation. It is also based on detailed mechanistic under-standing of the underlying physical processes that canproduce observed signals.

Simulations predict the calculated fluorescent decaysTo evaluate the predictions made with our code, we firstperformed simulations with simple dye systems whosefluorescent decay function can be calculated (Eq. 5). Wecreated input files with one (Figure 2A) or two (Figure 2B,and C) distinct donor environments. The acceptor wassimulated for each donor environment at defined dis-tances with 0.5 nm step (Figure 2). Notably, for a singledonor environment all fluorescent decay functions werepredicted accurately and precisely (Figure 2A).For the two donor environments we simulated all

possible combinations of pairs from the single donorenvironment to ensure that the code will weight them ac-cording to the expected stochasticity (Figure 2B and C).Note, that even for fluorescent decays, where one of thedistances was changed by 0.5 nm, the predictions wereclearly distinct (Figure 2B and C). As long as the distancesdwelled between half and twice the Förster radius, thepredicted photon distributions mirrored the calculatedTCSPC histograms. However, if one of the distances wasclose to those boundaries, a slight change in the distanceresulted only in minor differences in the theoretically

imes. A: Simulation of time-dependent photon counts for single donormulation of fluorescent decay signals of equal amounts of FRET pairstance fixed at 4 nm (B) or 6 nm (C) while for the other donor-acceptord. For better representation of the 6-nm experiment the curves are splitacceptor denotes a simulation experiment with donor only. Simulated

s lines in the same color.

http://figshare.com/http://figshare.com/articles/FRET/1158992

Figure 3 Reproduction of the experimental fluorescent decay

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expected curves (for example Figure 2A, 10 nm and noacceptor); either the whole signal (for short distances) oralmost none of it (for large distances) was quenched.A fraction of donors with acceptors at a distance of

2 nm or less had very little influence since the amountof photons contributed by such donors was small.Therefore, the calculated decay functions were mainlyinfluenced by other donor environments. That led to aparadoxical situation: a slight increase of those donor-acceptor distances beyond 2 nm, which resulted in lessquenching, seemed to increase the quenching effect byenhancing the relative contribution to the total signal(Figure 2B and C). However, increasing the dye distancesled to less quenching of that donor fraction, whichresulted in longer fluorescent decay times (Figure 2C).

times. Fluorescent decay times of random 3D-distributions weremeasured by mixing fluorescent dyes in solution, and were separatelypredicted with our Monte Carlo simulations. Note that if the experimentalfluorescent decay reaches the noise level, the simulations will approachzero value, thus only experimental data points above the noise levelshould be considered. The experimental values are depicted withsymbols and the matching calculations as lines in the same color.

Random dye distribution in the 3D-spaceNext, we created a system in which the dye distancesfollow a distribution instead of being fixed to discretevalues. We therefore used a simple experimental systemcontaining Alexa488- and Alexa594-maleimides with de-fined concentrations. High dye concentrations were usedin order to observe significant quenching. As expected,increasing the acceptor concentration shortened thefluorescent decay times. An increase of the concentra-tion of the donor, however, did not have any effect, sincethe donor excitation rates were chosen to be very low.The presence of additional donors did not change thedonor environment as long as the excitation was suffi-ciently low to avoid competition for nearby acceptors(Figure 3). The TCSPC histograms were predicted bysimulating a hypothetical dye solution with the equalamounts of donor and acceptor per volume unit com-pared to the experiments which were randomly distrib-uted throughout the simulated volume.Transfer processes between the dyes outside FRET are

not considered in our code. The parameters and fluores-cent dye pair should be chosen accordingly to minimizethe influence of processes such as homo-FRET or Dexterenergy transfer. When the expected dye distances are inthe range of the Förster radius and the donor or ac-ceptor properties are suitably chosen, ignoring theseprocesses should not influence the simulation results.

Influence of the linker lengthSubsequently, we assessed the relevance of predictingthe correct change in distances for dyes attached to fixedpositions (Figure 4). Since the dyes should rotate freely,it is crucial to allow for some freedom in rotation andattach them via a linker sequence. Hence, it is alsocrucial to run the simulation by allowing for similar free-dom in the dye coordinates. We tested the impact ofchanging dye positions on single FRET pairs and on a

model structure of amyloid polyglutamine fibrils withmultiple FRET interactions [20].The dye positions were reassigned to a sphere around

the labeling position with different radii. Allowing forchanges in the dye positions altered the shape of thefluorescent decay function (Figure 4). For single donor-acceptor pairs, the average distance increased relative tofixed positions, and further increased with the linkerlength (Figure 4A). Interestingly, changes in the dyepositions became significant only when they reachedvalues closer to the Förster radius. For the simulatedFörster radius in the fibril environment of 4.7 nm, asignificant influence was only observed for changes indye distances above 1 nm. Convoluting the signal withthe instrument response function would make thosechanges even less significant.For complex FRET systems, e.g. amyloid fibrils [20],

where the acceptors move in any direction (towards andaway from donors), the influence was somewhat differ-ent (Figure 4B). Small isotropic changes in position in-creased the quenching behavior, most likely due to theshorter distances than the distance between labelingsites. The effects of larger and smaller distances do notcancel each other out and instead result in higher donorquenching which is determined by the shape of thedistance dependence of FRET efficiency.Additionally, FRET pairs labeled at positions closer

than the half of the Förster radius had higher contribu-tion to the signal. If some dyes change their position andinfluence the donor-acceptor distance growth, it willresult in a relative increase in the number of photonsdetected shortly after excitation. For larger linker length,

Figure 4 Influence of the linker length. Changes in TCSPC histograms were investigated as a function of different lengths of linker betweendyes and labeling positions. These differences were based on equal amounts of FRET pairs with either 3 nm or 6 nm distance between the labelingpositions (A) or distance positions of a polyglutamine fibril model (B) as described in [20]. Note that the time scales are different in order to betterresolve an important area of signals before reaching the background.

Figure 5 Influence of collected photon number on the stochasticaccuracy. Residuals of simulations based on 100; 1000, 10000 or 100000photons collected in the most frequently detected time interval.

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however, the average distance was dominated by anothereffect. The total volume accessible to the dyes increasedin non-periodic systems in all three dimensions, thus re-ducing the effective dye concentration and increasingthe average dye distances. At some point, the originaldistances between the dyes will become insignificant,since they are comparably small to the length of the flex-ible linker. This is a limitation that needs to be carefullyconsidered when designing a system to be assessed withthis approach.In sum, allowing for changes in the dyes positions can

influence and potentially interfere with the success ofthe simulations, albeit within a limited distance scale.However, by linker size small compared to the Försterradius, it had minor influence on complex dye systemswith high label densities. Therefore, more precise as-sumptions, such as effective ranges of linker-length ra-ther than fixed length defined by theoretical models ormolecular dynamics simulations, or excluded dye vol-umes based on angle restriction from the volume occu-pancy, are not necessary for most applications.

Photon detection limitTo investigate how much signal needs to be obtained forstochastically reliable sample size, we performed a simu-lation using only Alexa 488 labeled protein within ourpolyglutamine amyloid fibrils [20] and investigated theeffect of the number of collected photons. Plotting theresiduals in relation to the expected values as a functionof time revealed an unsatisfactory result in the early timeintervals for the low photon statistics (Figure 5). Com-pared with the theoretical fluorescent decay times, thephoton counts for single time intervals deviated by up to30% when small amounts of photons were collected. Inaddition, for those early time intervals the photoncounts seemed to be systematically underestimated. Anoutlier that collects many photons will terminate thesimulation prematurely. Thus, collecting substantially

more photons decreased the impact of those outliersand the simulations run sufficiently long. Thereby, devi-ations from the theoretical predictions were randomlydistributed around the expected values which collectedmore reliable amounts of photons. The residuals alsodecreased with higher photon counts. Importantly, thebias of underestimating the expected number of photonsvanished when collecting at least 10000 photons.

Excitement probabilityIn time correlated single-photon count measurements,the probability P to obtain a photon within an excitationcycle is typically very low. This is necessary because in agiven excitation cycle only one photon can be detected.If more photons would reach the detector within oneexcitation cycle, only the first photon would be counted,leading to a bias and thus to a misrepresentation of thetime-dependent photon distribution. The probability todetect more than one photon should be negligibly small.In our Monte Carlo simulation we can simultaneously

Figure 6 Influence of excitation probability on the simulations.Simulations are based on excitement probabilities between 0.5%and 100%.

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monitor several photons in the same excitation cycle.Given that the typical small fraction of excited donors inthe experimental setup slows down the simulation, it isreasonable to increase the probability for donor excita-tion artificially, although the correct value can be easilycalculated using:

Pexcitation ¼ I � Δt � AEP � ND � 1−10εD�l�ND

V

� �ð6Þ

with I – irradiance, Δt – the time interval of the exci-tation light pulse, A – the surface area of the light pulse,EP – the energy of a single photon, ND – the number ofdonors in the simulated volume V, εD – the extinctioncoefficient of the donor, and l – the length of the hypo-thetically excited volume in direction of the light beam.Despite the capability of detecting multiple photonswithout any bias, the simulations will be affected byincreasing fractions of excited donors. An increase ofthe amount of excited donors also results in an in-creased amount of electronically excited acceptors dur-ing an excitation cycle. In situations where two donorsin close proximity competing for the same acceptor aretoo frequent, they could induce error in the predictionof the donor TCSPC histograms. Thus, we addressed atwhich excitation probability such competition wouldalter the time-dependent profile of the photon emission.For this purpose, we simulated the structure of poly-glutamine fibrils composed of 20% of donor-labeledmonomers and 20% of acceptor-labeled monomers. Thesimulated fluorescent decay function was dramaticallyinfluenced with increasing excitation probabilities (Figure 6).Changing the excitation probability to 1%, which isorders of magnitude above a typical experimental exci-tation probability, had a minor influence on the fluores-cent decay. It should be noted that this boundary canslightly shift at different dye concentrations and for anew system repeating the test should be considered.Moreover, moving the excitation probability to a max-imum that does not alter the fluorescent decay time isadvisable, since it results in substantially shorter simula-tion times.

ConclusionsIn conclusion, our approach is a step forward in exploit-ing the power of FRET to determine distances at themolecular level. The Monte Carlo-based simulation toolthat we developed to predict the time-correlated singlephoton counts correctly assesses the donor behavior andallows for evaluation of precise structural models incomplex FRET systems. Different key input parameterscan change the predictions and reliability of the simula-tions. For a stochastically meaningful simulation at least

10,000 photons should be collected in the most fre-quently observed time interval. The simulation time canbe shortened by artificially increasing the excitationprobability of the donors in the simulated system. Forbulky dye systems whose dimensions largely exceed therelevant distance for FRET, including a flexible linkerflattens the distance distribution and alters the resultingphoton decay. For isolated donor-acceptor pairs, increas-ing the linker length systematically reduces the quench-ing as previously observed [16]. Taking into account theabove mentioned restrictions and appropriate choices ofparameter this tool bears a broad utility to validatestructural models for a variety of complex system. Im-portantly, this tool is also capable of studying dye sys-tems with dimensions not suitable for detailed moleculardynamics simulations.

Ethics statementThis research does not involve human subjects, humanmaterial, or human data.

Additional file

Additional file 1: Additional information on the derivations ofequations 2 and 5.

AbbreviationsFRET: Förster resonance energy transfer; TCSPC: Time-correlated singlephoton counts; MD: Molecular dynamics.

Competing interestsThe authors declare that they have no competing interests.

Authors’ contributionsCR and ZI conceived research, CR developed the algorithm and performedthe experiments, MSM contributed to the development of the algorithm andperformed some tests, CR and ZI wrote the paper. All authors read andapproved the final manuscript.

http://www.biomedcentral.com/content/supplementary/s13628-015-0020-z-s1.pdf

Roethlein et al. BMC Biophysics (2015) 8:5 Page 10 of 10

AcknowledgementsWe thank Kathlen Brennenstuhl and Michael Kumke for their help withrecording the fluorescent decay trajectories and adjuvant discussions, andRobert Smock for his comments on the manuscript.

Author details1Biochemistry and Biology, University of Potsdam, Potsdam, Germany.2Theoretical Physics, Free University Berlin, Berlin, Germany. 3Biochemistryand Molecular Biology, University of Hamburg, Hamburg, Germany.

Received: 12 November 2014 Accepted: 23 March 2015

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AbstractBackgroundResultsConclusion

BackgroundMethodsChoice of dyesGeneration of input filesCalculation of FRET ratesSimulation routineAcceptor deactivationDonor excitation cyclesOutput of the simulationsFluorescence decay time measurementsQuantum yield and Förster radius determinationTheoretical calculationsCode access

Results and discussionSimulations predict the calculated fluorescent decaysRandom dye distribution in the 3D-spaceInfluence of the linker lengthPhoton detection limitExcitement probability

ConclusionsEthics statement

Additional fileAbbreviationsCompeting interestsAuthors’ contributionsAcknowledgementsAuthor detailsReferences