Applying Corrections in Single-Molecule FRET

Single-molecule Förster resonance energy transfer (smFRET) experiments can detect the distance between a donor and an acceptor fluorophore on the 3-10nm scale. In ratiometric smFRET experiments, the FRET efficiency is estimated from the ratio of acceptor and total signal (donor + acceptor). An excitation scheme involving two alternating lasers (ALEX) is often employed to discriminate between singly– and doubly-labeled populations thanks to a second ratiometric parameter, the stoichiometry S. Accurate FRET and S estimations requires applying three well-known correction factors: donor emission leakage into the acceptor channel, acceptor direct excitation by the donor excitation laser and the “gamma factor” (i.e. correction for the imbalance between donor and acceptor signals due to different fluorophore’s quantum yields and photon detection effciencies). Expressions to directly correct both raw FRET and S values have been reported in [1] in the context of freely-diffusing smFRET. Here we extend Lee et al. work providing several expressions for the direct excitation coeffcient and highlighting a clear interpretation in terms of physical parameters and experimental quantities. Moreover, we derive a more complete set of analytic expressions for correcting FRET and S. We aim to provide a clear and concise reference for different definitions of correction coeffcients and correction formulas valid for any smFRET experiment both in immobilized and freely-diffusing form.

Förster resonance energy transfer (FRET) is a Coulombic interaction between the dipoles of two uorophores, which results in the resonant and non-radiative transfer of excitation energy from a donor to an acceptor uorophore (and the energy transfer probability decreases with the sixth power of the distance). Donor de-excitation via FRET competes with the donor's intrinsic radiative and non-radiative de-excitation paths. erefore, in the presence of a nearby acceptor, the lifetime of the donor is reduced. e quantum yield, or e ciency, of the FRET process can be computed as [2]: where τ F RET is the D lifetime in presence of FRET and τ D is the intrinsic D lifetime without any acceptor nearby. Computing E following eq. 1 requires measuring the D excited-state lifetime, for example using a TCSPC setup. A simpler method of estimating E consists in measuring only the intensity of donor and acceptor uorescence (F D and F A respectively) and computing the FRET e ciency ratiometrically as: e previous eq. 1 and 2 require that the uorescence lifetimes or intensities be relative to a single specie. In biological samples, where almost inevitably multiple FRET populations are present, single-molecule FRET (smFRET) experiments allows identifying di erent sub-populations and, for each of them, estimating the FRET eciency [3]. e ratiometric approach of computing E is very common in smFRET owing to its modest hardware requirements (compared to TCSPC measurements) and has been extensively applied both to freely-di using and to surface-immobilized experiments. Unfortunately, unlike lifetime-based experiments, ratiometric FRET is a ected by three systematic errors (or biases) intrinsic to the way F A and F D are measured. e rst, a fraction of the donor emission spectrum almost inevitably falls in the acceptor detection band, causing spurious increase in acceptor-channel signal named "donor leakage". Additionally, the acceptor signal is contaminated by a fraction of uorescence due to direct excitation of the acceptor uorophore by the donor laser (ideally the acceptor should only be excited by the donor). Finally, the relative (detected) donor and acceptor uorescence intensity is biased because of the di erent uorescence quantum yields and photon detection e ciencies in the two detection channels (requiring the so-called "gamma factor" correction). ese biases are wellknown and expressions for their correction have been derived [1].
Contrary to ensemble measurements, single-molecule experiments can resolve di erent subpopulations and recover mean/peak FRET e ciencies of each single conformational or binding state (at least in cases where there are no conformations that interconvert much faster than di usion times). However, obtaining accurate mean FRET e ciencies also requires applying corrections for the aforementioned biases.
is paper extends the ratiometric FRET corrections reported in Lee et al. [1]. We de ne the acceptor direct excitation as a function of di erent observable. For each de nition we derive the direct excitation coe cient as a function of physical parameters and discuss its physical interpretation. We also derive a complete set of formulas for computing E or S as a function of the raw E and S as well as of the aforementioned correction factors. Note that the expression here presented are valid for any ratiometric smFRET or ALEX-smFRET experiment, being it immobilized or freely di using [2,3].

De nitions 2.1 Fluorescence intensities
We start by de ning the uorescence intensity signal as a function of the physical parameters. For surface-immobilized measurements the signal can be donor or acceptor counts acquired in a camera frame for a given uorophore. For freely-di using experiments the signal can be the counts detected in the donor and acceptor channel during a "burst" (i.e. a single molecule crossing the excitation volume). Following [1] we de ne: Eq. 3 and 5 are the detected quantities (e.g. counts, or camera intensity) a er background correction in the DexDem and AexAem photon streams respectively. e n a quantity (eq. 4) needs to be estimated correcting the measured counts n * a in the DexAem stream (see eq. 10). e factors I, σ, φ and η are, respectively, the excitation intensity, the absorption cross-section, the uorophore quantum yield and the photon detection e ciency. e label D ex (resp. A ex ) indicates a coe cient computed at the donor-laser excitation wavelength. D det (resp. A det ) indicated the donor detection band. Finally, in the σ coe cient the superscript D or A indicates the uorophore. In addition to these quantities, we need to introduce the correction coe cient γ and β which are de ned as follows: Brie y, γ makes the DexDem and DexAem signals commensurable (i.e. on the same scale) taking into account di erence in dyes quantum yields and photon detection e ciencies. Similarly, the β factor is used to make the total Dex signal commensurable with the AexAem signal by taking into account the di erences in excitation intensities (I Aex vs I Dex ) and in dyes absorption cross-sections (σ A Aex vs σ D Dex ). is expression of the β coe cient has been derived in [1] during the derivation of the ing procedure for γ-factor.
It is also useful to introduce the total corrected signal during D-excitation which we can de ne equivalently with one of the following expressions: e choice between eq. 8 and 9 is only ma er of convention. Finally, in a real experiment we cannot measure n a directly, instead we acquire a value n * a that is contaminated by donor leakage (Lk) and acceptor direct excitation (Dir). We de ne n * a and the correction terms as follows: Consistently with the n d and n aa de nitions (eq. 3 and 5), the quantity n * a is assumed already background corrected.

FRET and Stoichiometry
We start de ning the FRET e ciency E and the proximity ratios E P R and E R : E P R = n a n a + n d E = n a n a + γ n d where n d , n a are the donor and acceptor detected counts a er all the corrections (see eq. 3 and 4), while n * a are the acceptor counts with only background correction of eq. 10 (no leakage and direct excitation corrections).
Similarly, for the stoichiometric ratio we can have di erent de nitions depending on the degree on corrections that are applied: S P R = n d + n a n d + n a + n aa (17) S R (eq. 16) is the raw stoichiometry without any correction except for background (see de nition of n * a in eq. 10). S P R (eq. 17) is the stoichiometry corrected for leakage and direct excitation (see n a de nition in eq. 4). S γ (eq. 18) is the stoichiometric ratio corrected for leakage, direct excitation and γ (so that FRET populations have stoichiometry centered around a constant value, typically close to 0.5). S γβ (eq. 19) includes a β correction ensuring that FRET populations have stoichiometry centered around 0.5. Since β (eq. 7) is equal to the ratio n aa /(n a + γn d ) (eq. 4 and 9) it follows that: From eq. 20 follows that when β is known it is possible to compute the S γ value around which all FRET populations are distributed. As noted before, for S γβ this value is always 0.5.

De nition of direct excitation
e term Dir can be equivalently expressed as a fraction of any uorescence intensity components (i.e. counts in the donor or acceptor channel during donor or acceptor excitation). Here we present ve di erent de nitions and their physical interpretation.

De nition 1
De ning Dir as a function of n aa we have: e coe cient d AA can be computed from an acceptor-only population in ALEX measurement, because in this case eq. 10 becomes n * a = Dir . In terms of physical parameters, recalling eq. 12 and 5, we can express d AA as: Since computing Dir through d AA requires the spectroscopic quantity n aa (see eq. 5), it cannot be used in case of single-excitation measurements. In this case the de nitions in the next section can be used. Note that d AA is indicated as d in [1].

De nition 2
De ning Dir as a function of the "corrected total signal" as de ned in eq. 8 ((n a + γ n d )) results in: From eq. 23, it follows that: To derive the expression of d T as a function of physical parameters, consider the case of 100% FRET molecule. In this case, knowing that n d = 0 and recalling the expression of n a from eq. 4, we obtain: Noting that, for E < 1 the "corrected total signal" n a + γ n d (e.g. the corrected burst size in freely-di using measurements) will not change when γ is constant. erefore the previous expression is valid for any E.
In ALEX measurements is easier to estimate d AA from the data. erefore, expressing d T as a function of d AA allows to easily estimate the former coe cient from the data. From the de nitions of eq. 22, 25 and 7 we obtain: is relation follows from the de nition of β reported in the previous section and originally de ned in [1].

De nition 3
De ning Dir as a function of the "corrected total signal" as de ned in eq. 9 (n a /γ + n d ) we have: e coe cient d T can be obtained from the d T expression noting that we simply divide the "corrected total signal" by γ: e coe cient d T is indicated as d in [1] (main text p. 2943 and SI). Note that the de nition of d given in eq. (27) of [1] has been derived for a E = 0 population (for which n d + n a /γ = n d ). However, by using the corrected total signal, it is possible to use the same coe cient to express the Dir contribution for any FRET population (and independently from E).

De nition 4
De ning Dir as a function of n d we have: e coe cient d D is a function of E as well as the physical parameters. Taking the ratio of the physical de nitions of Dir and n d we obtain: De nition 5 De ning Dir as a function of n a : e coe cient d A is a function of E as well as the physical parameters. Taking the ratio of the physical de nitions of Dir and n a we obtain:

Discussion of De nitions 1-5
De nitions 4 and 5 are inconvenient because the coe cient depends on E. De nition 3 does not depend on E but depends on γ, while De nition 2 depends only on the ratio of two absorption cross sections and is therefore the most general form. De nition 1 can only be used in an ALEX measurement but it is easy to t from the S value of the A-only population. So, for non-ALEX measurements, De nition 2 (d T ) gives the simplest and most general coe cient. It can be computed from datasheet values or from d AA estimated from an ALEX measurement using the same dyes pair and D-excitation wavelength (d T = β d AA ).
As physical interpretation, de nitions 2 and 3 are similar. In De nition 2, when E = 1, the "corrected total signal" is n a . When E < 1, the "corrected total signal" does not change (at the same excitation intensity, and xed γ) being the sum of acceptor and γ-corrected donor counts. Similar considerations hold for De nition 3 (starting from E = 0). Note that using eq. 25 to estimate Dir requires the knowledge of the corrected total signal of eq. 8 (including A-direct excitation correction). For practical purposes, using a signal only corrected for γ and leakage to compute Dir via eq. 25, is a very good approximation. Alternatively, using eq. 33 (see next section) it is possible to compute corrected E values without any approximation.

Correction formulas
We can expressing E as a function of E R and the three correction factors as follows: is expression is the same of eq. S9 in [1] when we replace d T γ with d . Similarly we can express S as a function of S R , but in this case the expression will also depend on E R in addition of the correction parameters: A similar formula has been reported in [1] (SI) expressing S as a function of S P R and E P R . Here the expression is simply expanded as a function of E R and S R , resulting in an explicit dependence on lk and d T . e derivation of these formulas only involves using algebraic manipulations of the E and S expression. To avoid trivial errors, these expression have been derived with computer-assisted algebra (CAS). We also provide text-based version of the formula (python syntax) that is tested and easy to copy and paste in most other text-based language. For derivation details see Appendix: Derivation of the formulas.

Conclusion
We have introduced ve de nitions of acceptor direct excitation as a function of different experimental observable, and discussed that out of the ve, two have the most useful in practice. In particular, eq. 25 can be used to correct for A-direct excitation even in single-laser measurements provided the coe cient d T can be estimated independently. Furthermore, eq. 33 and 34 allows to apply corrections to E and S values, only knowing the raw E and S and the correction factors. With eq. 33 and 34 it is possible to correct the ed E or S values as a last independent step of the analysis, without the need to modify (i.e. correct) the distributions prior ing. is is important because, from a statistical point of view, the t of the raw E and S peaks can provide more reliable estimates due to simpler modeling (e.g. using a Binomial distribution) which requires less assumptions. For example, methods such as shot-noise [4] and probability distribution analysis [5,6] and Gopich-Szabo likelihood analysis [7,8] can be directly applied to raw FRET distributions. Conversely, applying these methods to the corrected FRET distributions requires unnecessary complex statistical models which include the e ect of each correction factor. In practice, the bene t of a more complex model is dwarfed by the inaccuracies arising from the additional approximations (even implicit) and from reliance on estimated correction parameters in the model itself. Using eq. 33 and 34, instead, allow to decouple the correction of E and S values from the population-level statistical modeling, resulting in more robust models and more accurate estimates.