Gabriele Cazzoli

Luca Dore

Cristina Puzzarini
email: cristina.puzzarini@unibo.it

Jürgen Gauss

The hyperfine structure in the rotational spectra of D 2 O: Lamb-dip measurements and quantum-chemical calculations 1

Keywords: D2O, rotational spectrum, hyperfine structure, Lamb-dip spectroscopy, quantum-chemical calculations D 2 O, rotational spectrum, hyperfine structure, Lamb-dip spectroscopy, quantum-chemical calculations

published or not. The documents may come

The hyperfine structure in the rotational spectra of D2O: Lamb-dip measurements and quantum-chemical calculations

Introduction

Water is a molecule of great importance in many areas of science. From a spectroscopic point of view, water vapor is an important constituent of the terrestrial atmosphere and plays a central role in the absorption of microwave as well as infrared radiation of the atmosphere [1][2][3]. In addition, it is of astrophysical relevance as corresponding infrared and/or microwave transitions have been observed in sunspots [4], in the atmosphere of an extrasolar planet [5], and in the interstellar medium [6]. Although water and its isotopic species have been the subject of many experimental as well as theoretical investigations, its spectroscopic characterization is still incomplete. Strong centrifugaldistortion effects significantly affect the water spectra and hamper both the experimental determination of the spectroscopic parameters required to accurately reproduce all the observed transitions and to accurately predict those not measured [7][8][9] as well as the theoretical evaluation of rotational and vibrational frequencies [10].

Focusing in the following on the doubly deuterated species, D 2 16 O, it should be noted that, although the rotational spectrum has been extensively investigated in selected frequency regions from 10 GHz up to 2.7 THz [9], to the best of our knowledge, no investigations have been reported with respect to its hyperfine structure in the millimeter-and submillimeter-wave ranges. For this reason, the Lamb-dip technique has been used in the present study to resolve the hyperfine structure of the rotational spectra as well as to provide transition frequencies with an accuracy of one order of magnitude, or even more, better than the standard techniques. The hyperfine structure of H 2 O, HDO and D 2 O has been investigated in the sixties by means of beam maser spectroscopy [11][12][13][14], but only incomplete information on the corresponding hyperfine parameters could be derived. Later on, in 1991, molecular beam electric resonance (MBER) measurements on the J = 1 1,1 and 1 10 rotational states provided values for the diagonal elements of the deuterium quadrupolecoupling tensor as well as for various combinations of the diagonal deuterium spin-rotation constants [15].

The present paper is organized as follows. In the next section the experimental details are described. Thereafter, as the experimental investigation has been supported by quantum-chemical calculations, all computational details are provided. Finally, the results concerning the hyperfine parameters are reported and discussed.

Experiment

Spin statistics

According to Bose-Einstein statistics, the presence of two equivalent deuterium nuclei leads to the existence of ortho and para species. As the deuterium nuclear spin, I D , is equal to 1, the total nuclear spin due to the coupling of Making use of the asymmetric-top notation, J Ka,Kc , the ortho transitions are those for which the sum K a + K c is even, whereas the para ones are those for which this sum is odd.

For both para-and ortho-D 2 O, with the exception of the case I D,tot = 0, the hyperfine structure (hfs) in the rotational spectrum is due to the interactions between the deuterium quadrupole moments Q and the electric-field gradients at the corresponding nuclei, interactions between the rotational angular momentum and the deuterium nuclear spin, and interactions between the nuclear spin momenta themselves. For both species, the various hyperfine components result from the ∆F =0,±1 selection rules, where F is the hyperfine quantum number due to the coupling scheme F = J + I tot . To a generic F ← F transition corresponds only the I D = 1 ← 1 transition in the case of para-D 2 O, while the three cases I D = 2 ← 2, 2 ← 0 and 0 ← 0 apply to the ortho species.

In the present study, seven ortho and six para rotational transitions have been recorded as described below. The former are

J = 6 2,4 ← 7 1,7 , 5 5,1 ← 6 4,2 , 5 1,5 ← 4 2,2 , 7 4,4 ← 6 5,1 , 5 2,4 ← 4 3,1 , 7 7,1 ← 8 6,2 , 2 1,1 ← 2 0,2 , while the latter are J = 6 1,6 ← 5 2,3 , 4 1,4 ← 3 2,1 , 5 5,0 ← 6 4,3 , 1 1,0 ← 1 0,1 , 7 2,5 ← 8 1,8
, and 7 7,0 ← 8 6,3 .

Figures 1 and2 provide examples for the ortho and para transitions, thereby displaying the hyperfine structure of the J = 2 1,1 ← 2 0,2 and J = 1 1,0 ← 1 0,1 rotational transitions, respectively. They furthermore demonstrate the resolution that can be achieved by our experimental set up. In both figures the experimental spectra are furthermore compared to the calculated (frequencymodulated as well as stick) spectra, which will be commented on later in the text.

Experimental details

Lamb-dip measurements have been performed employing a frequency-modulated, computer-controlled spectrometer in conjunction with a conventional freespace cell, as shown in Ref. [16]. To increase the sensitivity of the spectrometer as well as the Lamb-dip effect, the radiation path has been doubled [17]. A detailed description of the spectrometer is given in Ref. [18], whereas our experimental set up for performing sub-Doppler resolution spectroscopy has been described in previous papers (see, for example, Refs. [19][20][21]). Therefore, we only report the main details that are of relevance for the present investigation. The millimeter-and submillimeter-wave sources employed are either frequency multipliers driven by Gunn-diode oscillators or Gunn diodes themselves, and the 90-400 GHz frequency range has been considered. The frequency modu- lation is obtained by sine-wave modulating the 75 MHz local oscillator of the synchronization loop at 1.666 kHz. A liquid He-cooled InSb detector has been used, and its output has been processed by means of a Lock-in amplifier tuned at twice the modulation frequency. This leads, to a good approximation, to the recording of the second derivative of the natural line profile.

Measurements have been carried out at pressures of 0.5 mTorr, or even lower, employing a commercial sample of D 2 O. Such low values of working pressure have been chosen to minimize the dip widths as much as possible in order to improve the resolution and in order to avoid pressure frequency shifts. In addition, the source power has been adequately reduced to have narrow and non-distorted dips. The modulation depth used has been adjusted in the 8-20 kHz range according to the experimental conditions and the transition under consideration. All measurements have been carried out at room temperature.

Analysis of the Spectra

As it is evident from Figures 1 and2, a large number of lines are actually responsible for the observed features. We note that, in addition to the hyperfine (hf) components, crossing resonances, the so-called ghost transitions (also known as crossover resonances or simply crossovers), are present. This effect is due to the saturation of overlapping Gaussian profiles of two or more transitions with a common rotational energy level [START_REF] Winton | Observations and Applications of the Lamb-dip in Millimeter-Wave Molecular Spectroscopy[END_REF][START_REF] Letokhov | Nonlinear Laser Spectroscopy[END_REF]. As already noted in previous investigations (see, for example, Refs. [19,20,[START_REF] Cazzoli | [END_REF]), a line-profile analysis is necessary for correctly retrieving the hf component frequencies [16]; this implies that ghost transitions must be taken into account in such an analysis. Since a detailed description of the applied procedure has been given in Ref. [START_REF] Cazzoli | [END_REF], we only remind the reader that it is necessary to make a prediction of the spectrum containing only the hf components (real spectrum), of the spectrum containing only the crossover resonances (ghost spectrum), and of the spectrum containing both hf components and crossing resonances. This procedure allows to understand how the real spectrum is modified by the presence of ghost transitions and to take them properly into account in the line-profile analysis. We furthermore note that the accurate prediction of the intensity for the ghost transitions (see Ref. [25]) is essential for correctly reproducing the observed spectra. Figures 12 and 2 provide representative examples, clearly demonstrating that ghost transitions cannot be neglected for a proper analysis. In fact, these figures show how the hfs changes when going from a situation in which only hf components are present to that in which both The line-profile analyses have been carried out iteratively by comparing in each step the graphical representation of the predicted spectra with the observed one. As starting point, the experimental values of nuclear quadrupole coupling constants from Ref. [15] and the theoretical predictions of spin-rotation and spin-spin interaction constants, computed as described in the following section, have been used. Already after the first step of the iterative analysis, almost all hf components and crossovers were properly assigned and the assignments were used to obtain a set of improved hyperfine parameters (by means of the fit described in the following). Based on these updated parameters, new predictions have been made and the procedure has been then repeated until a complete assignment of the recorded spectra and until close agreement between predicted and observed spectra.

The Hamiltonian used for fitting the retrieved frequencies is given as the following sum:

H = H ROT + H N QC + H SR + H SS , ( 1
)
where H ROT is the usual rotational part of the Hamiltonian operator (see, for example, Ref. [START_REF] Gordy | Microwave molecular spectra[END_REF]), H N QC is the nuclear quadrupole-coupling (NQC) Hamiltonian [START_REF] Gordy | Microwave molecular spectra[END_REF][START_REF] Townes | Microwave Spectroscopy[END_REF], H SR is the Hamiltonian describing the spin-rotation (SR) interaction [START_REF] Townes | Microwave Spectroscopy[END_REF][START_REF] Flygare | [END_REF], and H SS is the spin-spin (SS) Hamiltonian [START_REF] Townes | Microwave Spectroscopy[END_REF][START_REF] Flygare | [END_REF] accounting for the direct dipolar spin-spin interaction. While we do not go into detail with respect to H ROT , the other three Hamiltonians will be briefly addressed in the section describing our quantum-chemical calculations. The retrieved hf components as well as crossovers have been included in a least-squares fit in which each line frequency is weighted proportional to the inverse square of its experimental uncertainty. Ghost transitions have been considered as arithmetic mean of the frequencies of the two generating hf components. For blended features, the hf components and/or ghost transitions involved are weighted according to their intensities. The fit has been carried out with Pickett's SPCAT/SPFIT suite of programs [29] employing Watson's A-reduced Hamiltonian in the I r representation [START_REF] Watson | Vibrational Spectra and Structure[END_REF].

The retrieved hf components as well as ghost-transition frequencies, obtained by the line-profile analyses and as averages of sets of measurements, are reported in the supplementary material (Table S1) together with the residuals (observed -calculated differences) from the fit described above. Experimental uncertainties are usually assigned based on the standard deviations of the averages of the measured values, but in the present case a more conservative estimate of the experimental uncertainties, i.e., 0.7 kHz, has been chosen. This estimate results to be conservative even when compared to the rms error of the measurement residuals, which is 0.63 kHz. All computations have been carried out using the CC singles and doubles (CCSD) approach augmented by a perturbative treatment of triple excitations (CCSD(T)) [START_REF] Raghavachari | [END_REF] with the basis-set convergence monitored by using the aug-cc-pCVnZ (n=T,Q,5,6) [32][33][34][35][36] hierarchical series of correlation-consistent basis sets. All electrons have been included in the electron-correlation treatment.

The semi-experimental equilibrium structure (r(OH) = 0.9575 Å, (HOH)= 104.51 • [37]) has been used in the computations of the equilibrium values for the hyperfine parameters, while vibrational corrections have been computed at the optimized geometry corresponding to the level of theory considered. Second-order vibrational perturbation theory (VPT2) [START_REF] Mills | Molecular Spectroscopy: Modern Research[END_REF] has been used for evaluating the latter.

As the focus of the present investigation is on the determination of the nuclear quadrupole-coupling, spin-rotation, and spin-spin interaction constants, we will briefly summarize how these quantities are obtained by means of quantumchemical calculations.

For the deuterium nuclei, which are the only ones with a nuclear quadrupole moment, the interaction of the latter with the electric-field gradient at the corresponding nuclei is responsible for the NQC (for details, see Refs. [START_REF] Gordy | Microwave molecular spectra[END_REF][START_REF] Townes | Microwave Spectroscopy[END_REF][START_REF] Lucken | Nuclear Quadrupole Coupling Constants[END_REF]). This is expressed by the following Hamiltonian:

H N QC = 1 2 K -eQ K q K J I K (2I K -1)J(2J -1) 3(I K • J) 2 + 3 2 (I K • J) -I 2 K J 2 ,
(2) where eQ K and q K J denote the quadrupole moment and electric-field gradient of the K-th nucleus, respectively, and the sum runs over the two deuterium nuclei. The quantum-chemical computation of the NQC constants, denoted by χ, thus requires the determination of the corresponding electric-field gradients, which are first-order properties and accessible by means of analytic gradient techniques [START_REF] Watts | [END_REF]. The required nuclear qudrupole-moment value of deuterium, 0.002860 mbarn, has been taken from Ref. [41].

The usual formulation for SR interaction is given in terms of a second-rank tensor C coupled with the rotational and nuclear spin momenta [42]:

H SR = K I K • C K • J , ( 3
)
with the sum running over all K nuclei of the molecule having nonzero nuclear spin (I ≥1/2, i.e., only the deuteriums in our case). Each element of the spinrotation tensor has an electronic and a nuclear contribution. While the latter only depends on the molecular geometry, the electronic contribution is evaluated as second derivative of the electronic energy with respect to the rotational angular momentum and the involved nuclear spin [START_REF] Flygare | [END_REF]. This is efficiently done using analytic second-derivative techniques [43][44][45][46]. Perturbation-dependent basis functions (often referred to as rotational London Atomic Orbitals [47]) have been employed to improve the basis-set convergence. It should be noted that we adopt a sign convention for spin-rotation constants opposite to that originally used by Flygare [42].

The dipolar interaction between two nuclear magnetic moments is described via the following Hamiltonian

H (KL) SS = I K • D KL • I L , ( 4
)
with the expression for the components of the dipolar spin-spin coupling tensor D KL given, for example, in Refs. [START_REF] Gordy | Microwave molecular spectra[END_REF][START_REF] Abragam | Principles of Nuclear Magnetism[END_REF][START_REF] Schmidt-Rohr | Multidimensional Solid State NMR and Polymers[END_REF]. It is seen that the dipolarcoupling tensor D KL is completely determined once the molecular geometry is known, thus not requiring any knowledge of the electronic structure of the molecule. However, this strictly applies only to the case of a rigid, nonvibrating molecule, while experimentally derived values correspond to those averaged over the vibrational ground state of the molecule. Quantum-chemical calculations are needed to compute the zero-point vibrational (ZPV) corrections.

ZPV corrections to electric-field gradient, spin-rotation and spin-spin constants have been obtained using the perturbative approach described in Ref. [START_REF] Auer | [END_REF] for NMR shielding tensors. This approach has already been employed in many cases to compute vibrational corrections for hyperfine parameters (see, for example, Refs. [51][52][53][54][55][56][57]). Going into details, the vibrational corrections are obtained by expanding the expectation value of the considered property over the vibrational wavefunction in a Taylor series around the equilibrium value with respect to normal-coordinate displacements and computing the required expectation values of the normal-coordinate displacements in a perturbative manner. Anharmonic effects, which contribute here in lowest order of perturbation theory, require the evaluation of the cubic force field. While the harmonic force field has been computed in a fully analytic manner [45], the cubic force field as well as the corresponding derivatives of the considered property have been obtained using finite-difference techniques as described in Refs. [43,58]. The vibrational corrections, defined as the difference between the vibrationally averaged and equilibrium values determined at the same computational level, have then been added to our best equilibrium results in order to derive theoretical best estimates for the ground-state data. The latter can be directly compared to the experimental values.

The CFour program package [START_REF] Cfour | Coupled Cluster techniques for Computational Chemistry)[END_REF] has been employed for all quantum-chemical calculations reported in the present work. The experimental hf parameters obtained from the least-squares fit described above are gathered in Table 1, where they are also compared with the corresponding best theoretical estimates as well as with previous experimental determinations [13,15]. As the knowledge of the unperturbed frequencies for the rotational transitions investigated in the present work may turn out to be useful for different purposes, they have been derived and are reported in Table 2. They can be considered as the recommended unperturbed frequency values.

Both sets of previous experimental data reported in Table 1 deserves to be commented on. With respect to Ref. [15], it should be noted that all diagonal components of the nuclear quadrupole-coupling tensor have been determined, while for the spin-rotation interaction tensor only the sums C aa +C bb and C aa +C cc have been reported. For all of these parameters good agreement with our determinations is observed, i.e., the differences are lower than 1-2 times the given uncertainties. Concerning Ref. [13], we note that the diagonal components of the spin-rotation tensor were obtained by simultaneously analysing the data available for various isotopic species of water, namely, H 2 O, HDO and D 2 O, thereby making use of the rigid-rotor as well as some other approximations (for details see Refs. [13,42]). Though such a procedure led to good description of the experimental data [11,12], the authors themselves cast some doubts on their own analysis in a subsequent paper [14]. Nevertheless, we note a perfect agreement with our experimental as well as calculated values. As a sort of conclusion we point out that, even though our results are less accurate than those obtained from MBER and beam maser spectroscopies, the present study reports for D 2 O the first, entirely experimental, determination of the diagonal deuterium spin-rotation tensor elements and of the direct spinspin coupling constant. In Refs. [12][13][14][15] the latter was taken into consideration only as calculated value.

Concerning the comparison with our theoretical predictions, we note good agreement with discrepancies of less than 0.1 kHz for spin-rotation and spinspin constants and less than 1 kHz for χ bb -χ cc , and of the order of 1 kHz for χ aa . Therefore, the present investigation is a further confirmation of the fact that spectroscopic parameters can be accurately determined by quantum-chemical calculations [START_REF] Puzzarini | [END_REF]. In the following our theoretical results are discussed in some further details.

In Tables 3 and4 complete lists of our computed values for the hyperfine parameters are given; in the former, the results concerning the nuclear quadrupole coupling are collected, while the latter reports those for spin-rotation and spin-spin interactions. The first conclusion that can be drawn is that in all cases the convergence of the equilibrium values as well as of the vibrational corrections with increasing basis-set size is monotonic and smooth. Furthermore, in almost all cases (i.e., except the vibrational corrections to C ab and D aa ) the absolute value decreases by enlarging the basis set. Going into further details, we note for the NQC constants that the equilibrium values can be considered converged at the CCSD(T)/aug-cc-pCVQZ level as the differences with the CCSD(T)/aug-cc-pCV6Z level are on average smaller than 1.5 kHz. These differences further reduce to less than 1 kHz when considering the CCSD(T)/aug-cc-pCV5Z level. In the present case, for which the NQC constants are quite small, the latter level of theory is at least required when aiming at quantitative predictions. As far as spin-rotation constants are concerned, the aug-cc-pCVQZ is already able to provide well converged results that differ from those obtained with the largest basis set only by about 1%, i.e., 0.01-0.04 kHz. In line with the results reported in Ref. [46], for D 2 O correlation effects beyond CCSD(T) are small and can be safely ignored.

Concerning the vibrational corrections, as already outlined in Refs. [55,57], the VPT2 approach is suitable for their evaluation. Due to their small magnitude and due to the rather large basis-set demands, the larger aug-cc-pCVQZ basis is required in the case of the NQC constants to obtain converged values within 0.1 kHz, while due to the use of rotational London orbitals the use of the augcc-pCVTZ basis is sufficient in the case of the SR constants. In general, we note that for all hyperfine parameters the vibrational corrections are small though not negligible, as they range from 1% to 3%. Therefore, inclusion of vibrational corrections is mandatory for quantitative predictions.

Conclusion

Using the Lamb-dip technique, the hyperfine structures of thirteen rotational transitions of D 2 16 O have been resolved, and their analysis, guided by previous experimental data as well as quantum-chemical calculations, enabled for the first time the complete experimental determination of the corresponding hyperfine parameters. Good agreement with theoretical values obtained by means of high-level quantum-chemical calculations further demonstrates that computations can provide reliable reference values and therefore, if required, can support, guide or even replace experimental determinations.

Acknowledgments

This work has been supported by "PRIN 2007" funds (project "Trasferimenti di energia, carica e molecole in sistemi complessi") and by University Table 1. Comparison of the experimental and calculated hyperfine parameters of D 2 16 O. 16 O recorded at P = 0.5 mTorr (mod. depth = 12 kHz). Calculated spectra for real hf components (in red), crossovers (in green) as well as both real and ghost transitions (in blue) are also depicted. Stick spectra of real and ghost transitions are also shown.

Figure 2. The J = 1 1,0 ← 1 0,1 rotational transition (para) of D 2

16 O recorded at P = 0.5 mTorr (mod. depth = 6 kHz). Calculated spectra for real hf components (in red), crossovers (in green) as well as both real and ghost transitions (in blue) are also depicted. Stick spectra of real and ghost transitions are also shown.

12

Transition

Freq.

J K a K c J K a K c (MHz) 6

  deuterium spins, I D,tot = I D 1 + I D 2 , can take the values 2, 1 and 0. The states with I D,tot equal to 2 or 0 correspond to ortho-D 2 O, whereas the species with I D,tot equal to 1 is known as para-D 2 O.

  transitions are considered. In both figures the stick spectra are also depicted, showing the hf components and the crossovers responsible for the actual spectrum recorded.

6 Page

 6

9

 9

10 Page

 10 RFO funds) as well as in Mainz by the Deutsche Forschungsgemeinschaft and the Fonds der Chemischen Industrie.

  Figure 1. The J = 2 1,1 ← 2 0,2 rotational transition (ortho) of D 216 O recorded at P = 0.5 mTorr (mod. depth = 12 kHz). Calculated spectra for real hf components (in red), crossovers (in green) as well as both real and ghost transitions (in blue) are also depicted. Stick spectra of real and ghost transitions are also shown.

  Figure 1. The J = 2 1,1 ← 2 0,2 rotational transition (ortho) of D 216 O recorded at P = 0.5 mTorr (mod. depth = 12 kHz). Calculated spectra for real hf components (in red), crossovers (in green) as well as both real and ghost transitions (in blue) are also depicted. Stick spectra of real and ghost transitions are also shown.

20 Page 20 of 24 URLFig. 1 .Fig. 2 .

 202412 Fig. 1. The J = 2 1,1 ← 2 0,2 transition of D 16 2 O.

Table 2 .

 2 Unperturbed frequencies for the J , K a , K c ← J, K a , K c rotational transitions (MHz) of D 216 O considered in this work.

Table 3 .

 3 Basis-set convergence and vibrational corrections for the deuterium quadrupole-coupling constants of D 2 16 O (kHz). All reported calculations have been performed at the CCSD(T) level.

Table 4 .

 4 Basis-set convergence and vibrational corrections for the spin-rotation and spin-spin constants of D 2 16 O (kHz). All reported calculations have been performed at the CCSD(T) level.

	11

Page 12 of 24 URL: http://mc.manuscriptcentral.com/tandf/tmph

				Molecular Physics Molecular Physics
	Table 1
	Comparison of the experimental and calculated hyperfine parameters of D 2	16 O.
	Parameter			This work	MBER a	MASER b
	F o r F o r F o r F o r F o r (kHz) aa (D) d χ aa (D) χ K (kHz) (χ bb -χ cc )(D) (kHz) F o r		Experiment 152.55(88) -0.289(59) 198.03(76)	Theory c 153.7 198.6	(1991) 153.92(11) 197.16(11)	(1968)
	C aa (D)	P P P P P (kHz) P	-2.89(26)		-2.98	-2.99(1)
	C bb (D)	e e e e e (kHz) -2.43(18) e		-2.41	-2.36(2)
	C aa (D)+C bb (D) (kHz) C cc (D) (kHz)	e r e r e r e r e r -5.32(31) e -2.62(15) e r		-5.38 -2.62	-5.353(11)	-5.35(2) e -2.64(1)
	C aa (D)+C cc (D) (kHz) D aa (D-D) (kHz)		R e R e R e R e R e -5.51(30) e -1.57(41) R e	-5.60 -1.61	-5.56(11)	-5.63(1) e
	σ f χ 2 g	(kHz)		0.63 0.89	v i e v i e v i e v i e v i e v i e
						w w w w w w
						O n l O n l O n l O n l O n l O n l
						y y y y y y

a Ref.

[15]

. b Ref.

[13]

.

c Equilibrium value obtained at the CCSD(T)/aug-cc-pCV6Z level augmented by vibrational corrections obtained at the CCSD(T)/aug-cc-pCV5Z level.

d Dependence on K of χ aa .

e Derived values.

f RMS error.

g Dimensionless (weighted) standard deviation.

17 Page 17 of 24 URL: http://mc.manuscriptcentral.com/tandf/tmph

Table 2

 2 Unperturbed frequencies for the J , K a , K c ← J, K a , K c rotational transitions (MHz) of D 216 O considered in this work.

It should be noted that the small bump shown in Figure1at approximately 403562.1 MHz is due to poor baseline (accidentally resembling a very weak dip) and does not correspond to any spectral feature.


Sergi Padilla-Parra

Nicolas Audugé

Maïté Coppey-Moisan
email: coppey.maite@ijm.univ-paris-diderot.fr

Marc Tramier

Dual color Fluorescence Lifetime Correlation Spectroscopy to quantify protein-protein interactions in live cell

Keywords: FCS, FCCS, FLCS, quantitative fluorescence microscopy, TCSPC

Dual color FCS is an interesting method to quantify protein interaction in living cells. But, when performing these experiments, one must compensate for a known spectral bleed through artifact that corrupts cross-correlation data. In this article, problems with crosstalk were overcome with an approach based on Fluorescence Lifetime Correlation Spectroscopy (FLCS). We show that FLCS applied to dual color EGFP and mCherry cross-correlation allows the determination of protein-protein interactions in living cells without the need of spectral bleed through calibration. The methodology was validated by using EGFP-mCherry tandem in comparison to co-expressed EGFP and mCherry in live cell. The dual color FLCS experimental procedure where the different laser intensities do not have to be controlled during experiment is really very helpful to study quantitatively protein interactions in live sample.

INTRODUCTION

Fluorescence microscopy techniques are commonly used to monitor protein-protein interactions in living cells taking advantage of the detection of Förster Resonance Energy Transfer (FRET) [START_REF] Förster | Intermolecular energy migration and fluorescence[END_REF]. Techniques based on the calculation of the lifetime such as Fluorescence Lifetime Imaging Microscopy (FLIM) [START_REF] Wouters | Fluorescence lifetime imaging of receptor tyrosine kinase activity in cells[END_REF][START_REF] Tramier | Picosecond-hetero-FRET microscopy to probe protein-protein interactions in live cells[END_REF][START_REF] Padilla-Parra | Quantitative FRET analysis by fast acquisition time domain FLIM at high spatial resolution in living cells[END_REF] or intensity based ratiometric approaches [START_REF] Gordon | Quantitative fluorescence resonance energy transfer measurements using fluorescence microscopy[END_REF][START_REF] Berney | FRET or no FRET: a quantitative comparison[END_REF][START_REF] Elangovan | Characterization of one-and two-photon excitation fluorescence resonance energy transfer microscopy[END_REF]Stockholm et al., 2004) have been widely used to detect FRET by the biological community over the last decade. The main limitation, however, to use FRET is the requirement of the close proximity of the two fluorescent tags. Dual color Fluorescence Correlation Spectroscopy (FCS) is an interesting alternative to deal with this limitation [START_REF] Bacia | Probing the endocytic pathway in live cells using dual-color fluorescence cross-correlation analysis[END_REF]. The main idea is to follow the codiffusion of two proteins labeled with differently colored fluorescent tags by analyzing the cross-correlation of temporal fluorescence intensity fluctuations. The most common couples of protein tags used (ECFP/EYFP and EGFP/mCherry), however, present sufficient overlap in their emission spectra to give rise to spectral bleed through, an artifact that corrupts the crosscorrelation data [START_REF] Bacia | Practical guidelines for dual-color fluorescence crosscorrelation spectroscopy[END_REF]. The contribution of the cross-talk from the shorter wavelengths emission species in the longer wavelengths channel is usually minimized by decreasing the intensity of the first laser. The cross-talk can also be calibrated and its amount retrieves during data analysis [START_REF] Bacia | Practical guidelines for dual-color fluorescence crosscorrelation spectroscopy[END_REF]. Recently, this approach was used in live cells and brought novel informations on a number of macromolecular complexes, as the MAP kinase complexes in yeast pheromone signaling [START_REF] Maeder | Spatial regulation of Fus3 MAP kinase activity through a reaction-diffusion mechanism in yeast pheromone signaling[END_REF], the dimerization of EGF receptors [START_REF] Liu | Investigation of the dimerization of proteins from the epidermal growth factor receptor family by single wavelength fluorescence cross-correlation spectroscopy[END_REF], the T cell receptor complexes [START_REF] Lillemeier | TCR and Lat are expressed on separate protein islands on T cell membranes and concatenate during activation[END_REF], the estrogen receptor interactions and dynamics [START_REF] Savatier | Estrogen receptor interactions and dynamics monitored in live cells by fluorescence cross-correlation spectroscopy[END_REF], and the virus RNA polymerase assembly [START_REF] Huet | Nuclear import and assembly of influenza A virus RNA polymerase studied in live cells by fluorescence crosscorrelation spectroscopy[END_REF]. But nonetheless at least 10% of crosstalk is always observed when working with green and red fluorescent proteins, that impedes the study of very low amounts of protein-protein interactions. The development of novel fluorescent proteins [START_REF] Kogure | A fluorescent variant of a protein from the stony coral Montipora facilitates dual-color single-laser fluorescence cross-correlation spectroscopy[END_REF], decreases the contribution of the spectral cross-talk to the cross-correlation, but the weak photo stability of some of them introduces other difficulties.

A way of getting rid of the bleed-through is the so-called switching method or pulsed interleaved excitation [START_REF] Müller | Pulsed interleaved excitation[END_REF][START_REF] Thews | Cross talk free fluorescence cross correlation spectroscopy[END_REF][START_REF] Takahashi | Crosstalk-free fluorescence cross-correlation spectroscopy by the switching method[END_REF]. In this method, two alternating laser beams are controlled by an acousto-optic modulator and the synchronized correlation signals in both detectors are calculated as cross-correlation functions with no cross-talk. A similar method was recently implemented in dual color scanning FCS with a conventional confocal microscope to quantify ligand-receptor interactions in cell membranes of living zebrafish embryos [START_REF] Ries | Modular scanning FCS quantifies receptor-ligand interactions in living multicellular organisms[END_REF].

Here, we overcome the spectral cross-talk with an approach based on Fluorescence Lifetime Correlation Spectroscopy (FLCS). The combination of TCSPC and FCS (FLCS) (Bohmer et al., 2002;[START_REF] Kapusta | Fluorescence lifetime correlation spectroscopy[END_REF] takes advantage of time resolved photon detection by using a pulsed laser to differentiate the contribution of two different species to the FCS trace. Each fluorescent component coming from the sample can be resolved on the basis of its fluorescence decay signature. FLCS was already used in vitro to suppress scattered light and detector after-pulses [START_REF] Kapusta | Fluorescence lifetime correlation spectroscopy[END_REF], to study dynamic transition between two fluorescence lifetime states [START_REF] Gregor | Time-resolved methods in biophysics. 3. Fluorescence lifetime correlation spectroscopy[END_REF], to resolve different lifetime species [START_REF] Ray | Fluorescence lifetime correlation spectroscopic study of fluorophore-labeled silver nanoparticles[END_REF] and to measure precise concentration values as low as picomolar of fluorophore [START_REF] Rüttinger | On the resolution capabilities and limits of fluorescence lifetime correlation spectroscopy (FLCS) measurements[END_REF]. Recently, FLCS was employed in live cells to quantify the interaction of EGFR neutralizing antibody labeled with Alexa488 and EGFR-GFP by fluorescence lifetime cross-correlation measurement [START_REF] Chen | Fluorescence Lifetime Cross Correlation Spectroscopy Resolves EGFR and Antagonist Interaction in Live Cells[END_REF]. Here, we show that FLCS applied to dual color EGFP and mCherry cross-correlation allows the determination of protein-protein interactions in living cells without the need of the spectral bleed through calibration. The advantage of the methodology was established as being independent of the respective amount of fluorescence of both species by performing experiments with EGFP-mCherry tandem and co-expressed free GFP and mCherry at different concentrations in live cell.

MATERIAL AND METHODS

Plasmid constructs and cell culture

The plasmid coding the mCherry-EGFP tandem has been described elsewhere [START_REF] Tramier | Sensitivity of CFP/YFP and GFP/mCherry pairs to donor photobleaching on FRET determination by fluorescence lifetime imaging microscopy in living cells[END_REF]. He-La cells were cultured in Dulbecco's Modified Eagle Medium containing 10% fetal bovine serum (PAA Laboratories GmbH, Pasching, Austria). The cultures were incubated at 37 °C in a humidified atmosphere of 5% CO 2 . He-La cells were seeded on 32 mm round glass coverslip at a density of 2x10 5 cells. When cells were about 50 -70% of confluence, they were transfected by EGFP and mCherry expressing plasmids or by mCherry-EGFP tandem expressing plasmid with a total amount of 1 g of expression vectors using Nanofectin I (PAA). 24 hours after transfection, coverslips were mounted in an open observation chamber with special DMEM-F12 to prevent fluorescence from the medium (DMEM-F12 without Phenol red, B12 vitamin, Riboflavin and supplemented with 20mM Hepes and L-Glutamine from PAA). A microscope incubation chamber (PeCon GmbH) was used to perform live cell measurements at 37 °C.

Fluorescence Lifetime Correlation Spectroscopy microscopy

The configuration of the scanning confocal time-resolved microscope MicroTime 200 (Picoquant, Berlin, Germany) used in this work is presented Figure 1A. Briefly, two laser excitation sources (a pulsed diode laser operating at 470 nm (Picoquant) and a continuous DPSS laser operating at 561 nm (Coherent)) are focused at the center of the optical axis in the sample through a water objective with a high numerical aperture objective (60, NA 1.3) mounted on an inverted microscope (Olympus, Tokyo, Japan). The fluorescence emission passes through a double dichroic mirror (DM 470/571 nm) and is focused on a pinhole, and a second dichroic mirror (DM 550 nm) splits the radiation into two channels. Two avalanche photodiodes (SPADs 14, Perkin Elmer) in front of which are placed the emission filters 525 +/-25 nm and 593 +/-20 nm, constitute the green and red detectors, respectively. The subcellular position of the sampled volume is chosen by first acquiring a pre-scan image of the field of view. A piezo displacement of the objective in the range of 80 x 80 µm can be carried out (P-733 Piezo-positioner, PI). The TimeHarp 300 PC board (PicoQuant) is employed in a time-tagged time-resolved (TTTR) mode and is coupled to a PC that pilots the data acquisition and analysis using the Symphotime software (PicoQuant). Single photon counting Time-Tagged Time-Resolved (TTTR) trace was acquired during 60 seconds. After lifetime filter calculation, auto-and cross-correlation functions of both green and red channels are computed using the Symphotime software following Eq. 3, 4 and 5. Fit were carried out with a single species Brownian diffusion model to recover amplitudes (Gg, Gr and Ggr) and apparent diffusion coefficients (D g , D r and D gr ) The beam waist,  0 , was determined using the point spread function of the confocal system (260 nm in the green channel and 320 nm in the red channel). For the fit,  0 is fixed to 260 nm (the smallest volume) and the structural parameter, z/ 0 to 4 (corresponding to a volume of 0.4 fl).

RESULTS AND DISCUSSION

Dual color FLCS methodology

The use of a pulsed laser to perform FCS takes advantage of Time Correlated Single Photon Counting (TCSPC) to differentiate the contribution of two different fluorescence lifetime species to FCS file (FLCS, Bohmer et al., 2002) . Indeed, each single photon is recorded with its global arrival time and the delay time with respect to the corresponding laser pulse. While the delay time is evaluated in fluorescence lifetime related analysis, the global arrival time can be used to form a fluorescence intensity time trace (Time-Tagged Time-Resolved (TTTR) file), making all related analysis possible. Sorting the photon events of a TTTR file into subsequent time bins, allows FCS calculation, time trace TCSPC fitting, time-gated FCS. The FLCS method relies on the calculation of lifetime filters for each species having different fluorescence decay signature in TCSPC histogram. For each time-resolved channel, the lifetime filter value gives the statistical weight of the counted photon for its corresponding species. These different filters can be selected during auto-and cross-correlation process of the absolute time traces to take into account the different weighting of each individual photon.

By using two laser to excite simultaneously two spectrally different fluorophores, a pulsed laser (470 nm which can excite a green fluorescent species, i.e. EGFP) and a continuous laser (561 to excite a red fluorescent protein,i.e. mCherry), dual color FLCS provides a way to remove the spectral bleed through of the green species into the red detection channel (configuration of the system is presented in Fig. 1-A). The "green" photons detected in the red channel are distinguished from the "red" ones because they are time correlated to the pulsed laser (see the fluorescence decay histogram in Fig. 1-B top) in contrast to the "red" photons which do not harbor any fluorescence decay signature (see the constant emission histogram in Fig. 1-B, middle). When both fluorophores are excited together, the fluorescence decay into the red channel (Fig. 1-B bottom) corresponds to the spectral bleed through of the green species, although the non-correlated trace (constant histogram) is the signature of the red emission. In practice, "lifetime filter" functions, f r a (i) and f r b (i), where i correspond to the time-resolved channel (in ns), are calculated from the two components fluorescence decay analysis acquired in the red channel as described below. Shortly, the fluorescence decay histogram, Decay r (i) (Fig. 2-A, black curve), corresponding to the red channel detection of simultaneous pulsed 470 nm and CW 560 nm excitation of EGFP and mCherry fluorophores can be expressed as a linear combination of two normalized histograms:

Decay r (i) = w r a . a(i) + w r b . b(i) (1)
With a(i) = A(i) / A(i) and b(i) = B(i) / B(i), A(i) and B(i) corresponding to the histograms of the EGFP fluorescence decay species (Fig. 2-A, blue curve) and of the steady-state fluorescence of mCherry (Fig. 2-A, constant histogram in pink), respectively. w r a and w r b are the amplitudes of the photon count contribution of blue and pink histogram species. These parameters are recovered from the adjustment of the right part of Eq. 1 to the experimental decay (residue of the adjustment displayed in light blue in Fig 2 A).

The corresponding lifetime filters functions, f r a (i) and f r b (i) are defined as follow:

<  f r a (i) . Decay r (i) > = w r a and <  f r b (i) . Decay r (i) > = w r b

(2)

where the brackets denote averaging over infinite number of measurements. The numerical calculation of these filters is described in details elsewhere [START_REF] Böhmer | Time-resolved fluorescence correlation spectroscopy[END_REF]. In Fig. 2-B are shown the filter corresponding to the fluorescence decay species (in blue,f r a (i)) and this corresponding to the steady-state fluorescence (in pink, f r b (i)),. Note that, for photons tagged with a time-resolved value comprised between 2 and 10 ns (corresponding to the fluorescence decay duration) the respective statistical weight is larger for the fluorescence decay species, whereas it is larger for the steady-state fluorescence species for photons tagged with other time value.

The auto (Gg() , Gr()) and cross-correlation (Ggr()) functions are directly calculated by taking into account the contribution of each species with its filter, in both green and red channels: 4)

Gg() = < (f a g (i) . I i g (t)) (f a g (i) . I i g (t + )) > (3) Gr() = < (f b r (i) . I i r (t)) (f b r (i) . I i r (t + )) > (
Ggr() = < (f a g (i) . I i g (t)) (f b r (i) . I i r (t + )) > (5)
where I i g and I i r (t) are the fluorescence signal in the ith time-resolved channel at measurement time t in the green and red detection channels, respectively and f a g (i) is the lifetime filter of the fluorescence decay of EGFP in the green channel, f b r (i) being the filter of the steady-state fluorescence of mCherry in the red channel.

FLCS remove EGFP spectral bleed through into mCherry detection channel

The efficient removal of spectral bleed through from EGFP emission into the red detection channel by dual-color FLCS was validated on co-transfected cells expressing free EGFP and mCherry. The cross-correlation curves calculated without (black) and with the lifetime filters (blue), displayed in Fig. 3-A, show the efficiency of the use of FLCS to get rid of the green fluorescence detected into the red channel. In the present example, for a similar amount of green and red fluorescence, as revealed from the auto-correlation amplitudes of both species (green and red curves), the contribution of the cross-talk is about 25%. This value is similar to the amplitude of the cross-correlation measured when EGFP and mCherry are linked together in the tandem molecule (Fig. 3-B, black curve: without FLCS). However, applying the lifetime filters to the calculation of the cross-correlation completely abolishes the cross-correlation for free EGFP and mCherry and leads to a significant amount of cross-correlation amplitude for the tandem (Fig. 3-B, blue curve). Importantly, the correction of the spectral bleed through by FLCS is independent of the respective green and red fluorescence intensities, and thus to the percentage of spectral bleed through in the red detector channel. Indeed, we show that decreasing the 561 nm laser power without changing the power of the 470 nm pulsed laser, leads increase of the green bleed through in the red channel that is completely eliminated by using FLCS in cells expressing free EGFP and mCherry (Fig. 3-A, inset). In contrast, for cell expressing EGFP and mCherry in tandem, the residual cross-correlation amplitude after lifetime filters application does not change significantly (Figure 3B,inset). This result is important for the quantification of the amount of interacting proteins in living cells. Indeed, the quantity of spectral bleed through (and its repercussion to the cross-correlation amplitude) depends on both the relative concentration of green and red species (which is generally not controlled in live cells) and on the relative laser intensities used during the experiment. Concerning the latter, it is always more convenient to adapt green and red laser power during the experiment, for example to increase signal to noise ratio or to take care of the photobleaching. If we want to use a conventional methodology to get rid of spectral bleed through by subtracting its quantity to the measured cross-correlation amplitude [START_REF] Bacia | Practical guidelines for dual-color fluorescence crosscorrelation spectroscopy[END_REF], the two laser powers need to be defined before the experiment. In addition, the best would be to increase mCherry excitation in comparison to EGFP for minimizing the spectral bleed through but increasing the photobleaching of the mCherry for which the photostability is weak. Here, we show that the use of FLCS avoids all these difficulties and thus that this method is well adapted for the quantification of protein-protein interactions in living cells.

After bleed through correction, the apparent ratio of interacting EGFP and mCherry determined by FLCS in cells expressing the tandem molecule is only 0.30 ([gr]/[g] in Table 1). This low value is, however, expected due to the following aspects: (i) FRET occurs between the two fluorophores (EGFP and mCherry) in the tandem [START_REF] Tramier | Sensitivity of CFP/YFP and GFP/mCherry pairs to donor photobleaching on FRET determination by fluorescence lifetime imaging microscopy in living cells[END_REF] that concomitantly slightly decreases the amplitudes of red auto-correlation and the crosscorrelation resulting in the underestimation of the ratio [START_REF] Hom | Analysis of coupled bimolecular reaction kinetics and diffusion by two-color fluorescence correlation spectroscopy: enhanced resolution of kinetics by resonance energy transfer[END_REF]; (ii) a known spectroscopic heterogeneity of mCherry is expected [START_REF] Hilesheim | Dual-color photon counting histogram analysis of mRFP1 and EGFP in living cells[END_REF][START_REF] Padilla-Parra | Quantitative comparison of different fluorescent protein couples for fast FRET-FLIM acquisition[END_REF] resulting in the existence of a large fraction of dark protein (without matured chromophore) and thus of a large amount of apparent "free" EGFP. It was already shown by FLIM-FRET that the apparent maximum fraction of interacting donor (EGFP) in the mCherry-EGFP tandem is 0.45 [START_REF] Padilla-Parra | Quantitative comparison of different fluorescent protein couples for fast FRET-FLIM acquisition[END_REF]; (iii) the two color confocal volumes are slightly different relative to each other ( 0 of 260 nm in the green channel and of 320 nm in the red channel, see Material and Methods), this will result in a diminution of the cross-correlation amplitude [START_REF] Bacia | Practical guidelines for dual-color fluorescence crosscorrelation spectroscopy[END_REF].

The auto-and cross-correlation curves obtained in cells expressing free EGFP and mCherry or the mCherry-EGFP tandem are fitted with a single species Brownian diffusion model for quantitative analysis and the results are summarized in Table 1. First, fit values confirm that using FLCS the amplitude of cross-correlation is canceled for freely diffusing EGFP and mCherry. This result is important for quantification since the spectral bleed through contribution (0.25 +/-0.12) is similar to the contribution of the cross-correlation amplitude for the mCherry-EGFP tandem (0.30 +/-0.15 using FLCS). Secondly, the diffusion coefficients of EGFP and mCherry determined by fit are in good agreement with other values reported in the literature [START_REF] Brock | Rapid characterization of green fluorescent protein fusion proteins on the molecular and cellular level by fluorescence correlation spectroscopy[END_REF][START_REF] Ruan | Cellular characterization of adenylate kinase and its isoform: two-photon excitation fluorescence imaging and fluorescence correlation spectroscopy[END_REF]Haustein et Schwille, 2003) regardless the different ways of calculating auto-and cross-correlation curves (with or without FLCS): i.e. 20 µm 2 s -1 for the monomeric fluorescent protein at 20°C in cells. Dual color FLCS does not introduce any distortion in auto-correlation curves and the fitted diffusion coefficients can be quantitatively interpreted.

CONCLUSION

We show that dual color FLCS is capable of eliminating artifacts such as the cross-talk effect. Moreover, we stress the potentiality of the technique when applied to quantify protein-protein interactions in vivo. In this method, the cross-correlation amplitude does not depend on both the relative concentration of green and red species and on the relative laser intensities used during the experiment. In addition, the diffusion coefficients determined by fit are insensitive to lifetime filtering. The dual color FLCS experimental procedure where the different laser intensities do not have to be controlled during experiment is really very helpful to study quantitatively protein interactions in live sample. Experimental auto-and cross-correlation curves (dots) and fitting curves with a single species Brownian diffusion model (line) are presented for green auto-correlation (green), red autocorrelation (red), conventional cross-correlation (black) and lifetime filter cross-correlation (blue) for a representative cell expressing EGFP and mCherry in separate diffusion (A) and expressing EGFP and mCherry in co-diffusion (B). Inset in A and B are the conventional cross-correlation (solid line) and the lifetime filter cross-correlation (dotted line) for three consecutive measurements on the same focal volume by decreasing 561 nm laser power without changing the power of the 470 nm laser (from grey to black).

LEGEND TO FIGURES

Figure 1 :

 1 Figure 1: Dual color FLCS instrumentation.A: Two lasers (470 nm pulsed and 561 nm CW) are conducted into epi-fluorescence microscope for live sample measurement. The confocal fluorescence selected by a pinhole is splited to green and red channels. Single photon counting detectors (SPADs) are used in timetagged time-resolved (TTTR) mode. B: Fluorescence time-correlation histograms of GFP and mCherry fluorescence detected in the green (left) and red (right) channels by using 470 nm pulsed laser alone (upper), 561 nm CW laser (middle) and both (lower).

Figure 2 :

 2 Figure 2: Fluorescence lifetime filters. A: The experimental fluorescence time-correlated histogram (black) is a linear combination of the fluorescence decay (blue) and the background (pink). The residual curve corresponds to the error between experimental curve and linear combination. B: Corresponding fluorescence lifetime filters of the fluorescence decay (blue) and the background calculated by Symphotime software (Picoquant, Berlin). These curves are respectively noted f a (i) and f b (i) in the text.

Figure 3 :

 3 Figure 3: Dual color FLCS in live cell expressing EGFP and mCherry alone and in tandem.Experimental auto-and cross-correlation curves (dots) and fitting curves with a single species Brownian diffusion model (line) are presented for green auto-correlation (green), red autocorrelation (red), conventional cross-correlation (black) and lifetime filter cross-correlation (blue) for a representative cell expressing EGFP and mCherry in separate diffusion (A) and expressing EGFP and mCherry in co-diffusion (B). Inset in A and B are the conventional cross-correlation (solid line) and the lifetime filter cross-correlation (dotted line) for three consecutive measurements on the same focal volume by decreasing 561 nm laser power without changing the power of the 470 nm laser (from grey to black).

  photons

Table 1 : Diffusion time and cross-correlation ratio of EGFP-mCherry tandem and co- expressed separately

 1 µm 2 .s -1 . b, calculated directly from G 0gr /G 0rr .Values +/-sd from five independent experiments.

			D g	a	D r	a	D gr	a	[gr]/[g] b
	mCherry-EGFP	without FLCS	11.6 +/-2.9	10.7 +/-3.4	11.9 +/-3.7	0.56 +/-0.10
	tandem	with FLCS	11.4 +/-2.6	8.5 +/-2.6	9.0 +/-3.0	0.30 +/-0.15
	Co-expressed	without FLCS	20.4 +/-2.8	24.9 +/-5.1	17.5 +/-5.2	0.25 +/-0.12
	EGFP and mCherry	with FLCS	20.3 +/-3.0	23.6 +/-2.7	-		0
	a, in

Acknowledgement

This work was performed at the Imaging facilities center of the Institut Jacques Monod and was supported by the Fondation pour la Recherche Médicale, the Région Ile de France (Soutien aux Equipes Scientifiques pour l'Acquisition de Moyens Expérimentaux), the Groupement Entreprises Françaises Lutte Contre Cancer, the Centre National de la Recherche Scientifique (Action Concertée Incitative Biologie Cellulaire, Moléculaire et Structurale), the Association pour la Recherche sur le Cancer, and the Association Nationale pour le Recherche.