Michael Atlan

Michel Gross

Pierre Desbiolles

Émilie Absil

Gilles Tessier

Ma Ïté Coppey-Moisan

Heterodyne holographic microscopy of gold particles

We report experimental results on heterodyne holographic microscopy of subwavelength-sized gold particles. The apparatus uses continuous green laser illumination of the metal beads in a total internal reflection configuration for dark-field operation. Detection of the scattered light at the illumination wavelength on a charge-coupled device array detector enables 3D localization of brownian particles in water. OCIS : 180.6900, 090.1995, 170.0180 Assessing microscopic processes by tracking optical labels has a broad range of applications in biology. In particular, monitoring biological phenomena such as cellular-level dynamics is a subject of growing interest. Fluorescent molecules are widely used in this aim, but the observation time of dyes is limited by photobleaching. Quantum dots offer a much better photostability, but they have the inconvenient to blink. Noble metal nanoparticles have the advantage of being perfectly photostable [1]. Originally, light scattered by small metallic particles has been detected in dark field [1] or total internal reflection [2] configuration. To improve the detection sensitivity, interferometric approaches have been introduced [3,4,5,6,7]. Common-path interference, initially achieved with a Nomarski interferometer [8], translates phase variations into intensity variations, and enables the detection of phase perturbation provoked by spatial [9] or photothermal [10] modulation. Scanning heterodyne detection of the photothermal modulation [3] enables an unmatched combination of sensitivity and selectivity suitable to discriminate particles smaller than 5 nm from their background. Photothermal imaging relies on a spatial scanning of the laser beams to track the index modulation in the neighborhood of the beads; the temporal noise in the incident light leads to spatial noise in the image acquired sequentially [7]. Wide-field detection schemes alleviate such issues, but their sensitivity in optical mixing configurations would not match heterodyne detection on single detectors levels. We propose here a wide-field, shot-noise limited, tunable CCD heterodyne detection technique able to achieve high-resolution, 3D microscopy of gold particles with a laser source at rates compatible with biological dynamics. The experimental setup is sketched in fig. 1(a).

The main laser beam (λ = 532 nm, field E L , 50 mW, single axial mode, CW) is split with a polarizing beam splitter (PBS) in local oscillator (LO) and illumination arms (fields E LO and E I ). A λ/2 waveplate (HWP), and neutral densities (A1, A2) allow control of the optical power in each arm. Both beams are frequencyshifted around 80 MHz by acousto-optical modulators (AOM) driven at frequencies ω AOM1 , ω AOM2 .The LO beam passes through a beam expander (BE) to form a plane wave which polarization angle is adjusted with a HWP to maximize the holographic modulation depth. The object is illuminated in dark field configuration by using total internal reflection (TIR). The scattered field E ≪ E LO passes through a microscope objective (MO, 60× magnification, NA = 0.9, air). Off-axis optical mixing (θ tilt) of E with E LO with a beam splitter (BS) results in a fringe pattern recorded with a CCD camera (PCO Pixelfly QE, 1392 × 1024 square pixels of 6.7 µm, frame rate ω S = 12 Hz). Proper frequency detuning ∆ω = ω AOM2 -ω AOM1 , and angular tilt θ enable accurate and sensitive phase-shifting off-axis holography [11,12]. The first samples observed are thin layers of gold beads (diameter d = 50 nm to 200 nm) immobilized in a polyvinyl alcohol (PVA) matrix spread by spin coating onto a glass slide. These slides are set onto a prism used to guide the illumination field E I and provoke TIR at the PVA-air interface. Microscope immersion oil between the prism and the slide enables refractive index matching. The evanescent wave locally frustrated by the beads is turned into a propagative scattered field E collected by the MO. To make E and E LO in phase opposition from one frame to the next, two-phase detection (∆ω = ω S /2) is performed. Exposure time is τ E = 50 ms, and holograms are obtained by making the difference of consecutive frames.The complex field E(x, y, z), in the beads plane z, is reconstructed by numerical Fresnel transform [11].

Fig. 2 shows the bead images (a,b,c), and the corresponding intensity profiles along x (d,e,f). The images are obtained by averaging (|E(x, y, z)| 2 ) over 4 sequences of 2 images. The beads diameters are 200 (a,d), 100 (b,e) and 50 nm (c,f). Under 50 nm the coherent parasitic light scattered by dust or surface roughness prevents the beads to be distinguished. Fig. 3 shows, for 200 nm beads, the x, y images (a,b,c) and the x profiles (d,e,f), at z = 0, 17, and 22 µm relative reconstruction distances. It also shows the x, z image (g) and the z profile (h), in the z = 0 to 47 µm range. Because the bead emitters are close to a plane dielectric interface [13], one observes an asymmetric signal in the z direction of ∼ 7 µm width at half maximum with rings in the x, y plane. Similar rings have been observed with quantum dot emitters [14,15]. Quantitative study of this effect [13,16], is out of the scope of this letter.

To illustrate the sensitivity of our holographic setup, we have displayed on Fig. 4 images ((a, ..., e) and (k, ..., o)) and profiles ((f, ..., j) and (p, ..., t)) of 200 nm beads at different illumination intensities. The illumination beam (|E I | 2 ) is attenuated by a factor 10 0 (a,f,k,p), 10 1 (b,g,l,q) ..., 10 4 (c,j,o,t). The LO beam is either off (direct imaging : (a, ..., j)), or on (holographic regime : (k, ..., t)). The beads are imaged with much better sensitivity with the LO beam, since the bead signal remains visible by reducing the illumination power over 4 orders of magnitude. Moreover, the signal is proportional to illumination (dashed line throughout Fig. 4(p) to (t)). We must note that, at high illumination level (p), the bead detection dynamic range is limited by the parasitic light background (arrow 1) which is several orders of magnitude larger than the holographic detection noise floor (arrow 2), obtained in a cut from the off-axis quietest region of the reconstructed image, away from the beads region. The noise floor (about 10 10 A.U.), which is related to shot noise of the LO beam, gives an absolute calibration of the bead peak (5 × 10 14 A.U. on (k)), since it corresponds to 1 photo electron (e) per pixel [12]. Without attenuation (k), the bead peak area is about 20×20 pixels (see (p)); it corresponds thus to ∼ 2 × 10 7 e. One watt of laser yields ∼ 10 19 photons per second. With 10 mW and 2τ E = 100 ms (2 frames to make an hologram), we get 10 16 photons. The Mie scattering cross section of 200 nm gold particles (refractive index : n = 0.39 -2.38j at 532 nm; with j 2 = -1) [17] is 0.1µm 2 . The total illumination area is ∼ 1mm 2 . Thus we get 10 9 scattered photons per bead. Since the bead is located at an air-glass interface, most of the light (85%) is scattered within the glass [18], and 1.5 × 10 8 photons are scattered forward within a 2π solid angle. The collection solid angle of a NA = 0.9 objective is ∼ π/4. We then get 4×10 7 photons on the CCD yielding an expected signal of 2 × 10 7 e (50% quantum efficiency at 532 nm), in agreement with the experiment.

To assess imaging performances in experimental conditions compatible with biological microscopy, we have performed axial sectioning of dynamic beads; we have imaged a suspension 200 nm beads in brownian motion in water (see Fig. 1(c)). A ∼ 30 µm-thick layer sample is realized within a slide / parafilm (TM) / coverslip stack. The parafilm layer was heated until melting to serve as waterproof spacer of ∼ 30 microns thickness. In this configuration, the top interface (coverslip-air) is the one where TIR occurs. The part of the illumination field E I not diffracted by the beads undergoes TIR while the scattered field E passes through the MO and is used for imaging. Since the exposure time is short : τ E = 1 ms, the LO power is increased to fill the CCD dynamic range. Over a time τ , brownian particles travel by a distance r(τ ) = (6D B τ ) 1/2 , where D B = 2.1 × 10 -12 m 2 .s -1 is the diffusion coefficient of a 200 nm particle in water. During one exposure, this travel is smaller than the particle size: r(1 ms) ∼ 110 nm. 200 nm beads thus appear to be quasi immobile, but travel from one image to the next by r(80 ms) ∼ 1 µm. This property is used for signal demodulation. The hologram is obtained by subtracting from the current recorded frame the time average of the 10 next consecutive frames. In this way the holographic information is recorded in one exposure τ E = 1 ms. To localize smaller beads, the exposure time might be shortened accordingly.

Fig. 5 shows a series of images (a,b,c,d) and profiles (e,g,f,h) calculated from a single hologram. Images (a) to (d) and profiles (c) to (h) are calculated at axial positions z (a,e), z + 10 µm (b,f),... z + 30 µm (d,h). A composite image (i) is made from reconstructed intensity holograms at axial positions (z) coded in color in the 30 µm range. The attached multimedia file shows the video rate motion of the particles in this composite image.

In this letter we have performed full field imaging of gold nano particles in 3D at video-rate with exposure times as short as 1 ms. Our holographic technique, which benefits from heterodyne gain, exhibits optimal detection sensitivity, and, since the signal can be easily calibrated, we have verified its agreement with the expected signal. In the future, better axial resolution might be obtained by using the phase of the holographic data. Lowcoherence light sources might be used to reduce the background light component and improve depth-sectioning. The background might also be reduced by a selective modulation (e.g. spatial or photothermal) of the particles.

The authors acknowledge support from the French Agence Nationale de la Recherche (ANR) and the Centre de compétence NanoSciences Île de France (C'nano IdF).

Contact author : atlan@lkb.ens.fr

FIG. 1 :

 1 FIG.1: Experimental setup (a). Evanescent wave illumination of static beads in a plane (b). Total internal reflection configuration for dark-field illumination of beads in a tridimensional environment (c). Acronyms defined in the text.

FIG. 2 :

 2 FIG. 2: Squared amplitude holograms |E| 2 in logarithmic arbitrary units. 200 nm beads (a), 100 nm beads (b), and 50 nm beads (c). Scale bar is 10 µm. Horizontal profile traces of |E| 2 at the beads positions, averaged over 4 pixels (d, e, f), in linear arbitrary units.

FIG. 4 :

 4 FIG. 4: Intensity images calculated from 4 × 2 camera frames data of a pair of 200 nm particles. (a) to (e) : direct image (LO beam is off). (k) to (o) : holography (LO beam is on). Scale bar is 5 µm. Illumination field power attenuation ranges from 10 0 to 10 4 . Bead profiles without (f) to (j), and with LO beam (p) to (t). The vertical axis range (A.U.) is the same for (f) to (j), and (p) to (t).

FIG. 5 :

 5 FIG. 5: Axial exploration of a 200 nm gold beads suspension in water. |E| 2 images reconstructed 10 µm apart (a) to (d). Linear scale profiles averaged over 4 pixels (e) to (h). Composite image (i). Scale bar is 5 µm.


Duong-Hung Pham
email: dhpham@unistra.fr

Sylvain Meignen
email: sylvain.meignen@univ-grenoble-alpes.fr

Demodulation Algorithm Based on Higher Order Synchrosqueezing

Keywords: time-frequency analysis, AM-FM mode, multicomponent signals, synchrosqueezing techniques, demodulation

This paper addresses the problem of detecting and retrieving amplitude-and frequency-modulated (AM-FM) components or modes of a multicomponent signal from its timefrequency representation (TFR) corresponding to its short-time Fourier transform. For that purpose, we introduce a novel technique that combines a high order synchrosqueezing transform (FSSTN) with a demodulation procedure. Numerical results on a multicomponent signal, both in noise-free and noisy cases, show the benefits for mode reconstruction of the proposed approach over similar techniques that do not make use of demodulation.

I. INTRODUCTION

Multicomponent signals (MCSs), defined as sums of amplitude-and frequency-modulated (AM-FM) waves, have received great interest from the signal processing community over the last decades because of their ability to accurately represent non-stationary signals arising from a wide range of applications e.g., audio recordings, structural stability [START_REF] Costa | Noise and poise: Enhancement of postural complexity in the elderly with a stochastic-resonance-based therapy[END_REF], [START_REF] Cummings | Travelling waves in the occurrence of dengue haemorrhagic fever in Thailand[END_REF], or physiological signals [START_REF] Herry | Heart beat classification from single-lead ecg using the synchrosqueezing transform[END_REF]. Time-frequency (TF) representations (TFRs) play a central role for characterizing such signals, among which the continuous wavelet transform (CWT) and the short-time Fourier transform (STFT) are the most popular [START_REF] Flandrin | Time-frequency/time-scale analysis[END_REF]. However, the effectiveness of these transforms is constrained by the choice of window or wavelet. Indeed, the Heisenberg uncertainty principle limits both the adaptivity and the readability of these TFRs. To improve the readability, a reassignment method (RM) was proposed by Kodera et al. [START_REF] Kodera | Analysis of time-varying signals with small bt values[END_REF] in the 1970s and then further developed in [START_REF] Auger | Improving the readability of time-frequency and time-scale representations by the reassignment method[END_REF]. Unfortunately, the reassigned representation given by RM was not invertible: when applied to an MCS, it did not allow for easy retrieval of signal components. Alternatively, a technique using the same principle as RM but keeping the invertibility property, and known as the synchrosqueezing transform (SST), was developed in the wavelet framework [START_REF] Daubechies | A nonlinear squeezing of the continuous wavelet transform based on auditory nerve models[END_REF], and then theoretically studied in [START_REF] Daubechies | Synchrosqueezed wavelet transforms: an empirical mode decomposition-like tool[END_REF]. In essence, the retrieval procedure consists of estimating, on the TFR associated with the modulus of CWT, so-called ridges which are estimators of the instantaneous frequencies (IFs) of the modes, reassigning the transform in the vicinity of the detected ridges, and finally inverting the reassigned transform. Such an approach was then adapted to the TFR given by STFT in [START_REF] Thakur | Synchrosqueezing-based recovery of instantaneous frequency from nonuniform samples[END_REF], and is now known as the STFT-based synchrosqueezing transform (FSST). However, all these transforms suffer from a serious limitation which is that they cannot deal with modes containing strong frequency modulations, commonly encountered in real-world situations e.g., radar [START_REF] Skolnik | Radar Handbook, Technology and Engineering[END_REF], speech processing [START_REF] Pitton | Applications of positive time-frequency distributions to speech processing[END_REF], or gravitational waves [START_REF] Candes | Detecting highly oscillatory signals by chirplet path pursuit[END_REF], [START_REF] Abbott | Observation of gravitational waves from a binary black hole merger[END_REF]. To better handle this case, an extension of FSST known as the second-order FSST (FSST2) was proposed in [START_REF] Oberlin | The Fourier-based synchrosqueezing transform[END_REF] and its theoretical foundations settled in [START_REF] Behera | Theoretical analysis of the second-order synchrosqueezing transform[END_REF]. Moreover, this new transform was recently used in conjunction with a demodulation operator leading to better mode reconstruction results [START_REF] Meignen | On demodulation, ridge detection and synchrosqueezing for multicomponent signals[END_REF]. FSST2 was also further extended to better deal with modes with very fast oscillating phase and denoted by FSSTN with N = 3, 4,... [START_REF] Pham | High-order synchrosqueezing transform for multicomponent signals analysis -with an application to gravitational-wave signal[END_REF]. To investigate how the combination of FSSTN with a demodulation operator can be beneficial to mode reconstruction is the main goal of the present paper.

To this end, after having recalled some useful background associated with SST in Section II, we revisit the idea of the demodulation algorithm based on FSST (or FSST2) for mode reconstruction and then detail its variant based on FSSTN context in Section III. Finally, we provide, in Section IV, numerical simulations, involving both noise-free and noisy MCSs, to demonstrate the improvement brought by the proposed approach over some state-of-the-art methods.

II. BACKGROUND TO SYNCHROSQUEEZING TRANSFORMS

Prior to starting, we recall useful notation, definitions and basic elements related to synchrosqueezing transforms.

A. Basic Notation and Definitions

Consider a signal f ∈ L 1 (R), its Fourier transform corresponds to:

f (ξ) = F{f }(ξ) = R f (t)e -i2πξt dt, (1)
and its short-time Fourier transform (STFT) is defined using any sliding window g ∈ L ∞ (R) by:

V g f (t, ξ) = R f (τ )g(τ -t)e -2iπξ(τ -t) dτ = R f (t + τ )g(τ )e -2iπξτ dτ. (2)
If f, f , g and ĝ are all in L 1 (R), f can be reconstructed from its STFT as soon as g is non-zero at 0:

f (t) = 1 g(0) R V g f (t, ξ)dξ, (3)
where Z denotes the complex conjugate of Z.

In this paper, we will intensively study multicomponent signals (MCSs) defined as a superimposition of AM-FM components or modes:

f (t) = K k=1 f k (t) with f k (t) = A k (t)e i2πφ k (t) , (4)
for some finite K ∈ N, A k (t) and φ k (t) being respectively the instantaneous amplitude (IA) and frequency (IF) f k satisfying: A k (t) > 0, φ k (t) > 0 and φ k+1 (t) > φ k (t) for all t. Such a signal admits an ideal TF (ITF) representation defined as:

TI f (t, ω) = K k=1 A k (t)δ (ω -φ k (t)) , (5)
where δ denotes the Dirac distribution. In practice, the IF of the modes cannot be recovered by estimating the IF of f as is done in the theory of analytical signals, and to locate the modes in the TF plane is essential before computing their IFs: this is one of the goals of SST whose construction is recalled hereafter.

B. STFT-based synchrosqueezing transform

The STFT based synchrosqueezing (denoted by FSST in the sequel) starts with defining the local instantaneous frequency ωf , wherever V g f (t, ξ) = 0, by [START_REF] Auger | Improving the readability of time-frequency and time-scale representations by the reassignment method[END_REF]:

ω f (t, ξ) = ∂ arg V g f (t, ξ) ∂t = 1 2iπ ∂ t V g f (t, ξ) V g f (t, ξ) . (6)
The FSST of f with threshold γ is obtained by moving any coefficient V g f (t, ξ) with magnitude larger than γ to location (t, ωf (t, ξ)). One then define the synchrosqueezing transform as:

T γ f (t, ω) = |V g f (t,ξ)|>γ V g f (t, ξ)δ(ω -ω f (t, ξ))dξ. (7)
Any mode f k can then be reconstructed by summing FSST coefficients around the k th ridge, which amounts to modifying the synthesis formula (3) to select only the coefficients related to the k th mode, namely:

f k (t) ≈ 1 g(0) |ω-ϕ k (t)|<d T γ f (t, ω)dω, (8)
where ϕ k is an estimation of φ k , and parameter d is used to compensate for errors in IF estimation; it should be kept small enough to avoid mode-mixing, and its influence will be discussed later.

C. Second order synchrosqueezing transforms

The applicability of FSST is however restricted to a class of MCSs composed of slightly perturbed purely harmonic modes. To overcome this limitation, an extension of FSST, called second-order STFT-based synchrosqueezing transform (FSST2) [START_REF] Behera | Theoretical analysis of the second-order synchrosqueezing transform[END_REF], [START_REF] Oberlin | Second-order synchrosqueezing transform or invertible reassignment? Towards ideal time-frequency representations[END_REF], was introduced based on a more accurate IF estimate than ωf . More precisely, one first defines a secondorder local modulation operator used to compute the new IF estimate. To do so, one introduces complex reassignment operators ωf (t, ξ) =

∂tV g f (t,ξ) 2iπV g f (t,ξ) and tf (t, ξ) = t - ∂ ξ V g f (t,ξ) 2iπV g f (t,ξ) ,
and then defines a complex frequency modulation operator as [START_REF] Oberlin | Second-order synchrosqueezing transform or invertible reassignment? Towards ideal time-frequency representations[END_REF]:

qf (t, ξ) = ∂ t ωf (t, ξ) ∂ t tf (t, ξ) = ∂ t ∂tV g f (t,ξ) V g f (t,ξ) 2iπ -∂ t ∂ ξ V g f (t,ξ) V g f (t,ξ) . (9)
The second-order local modulation operator then corresponds to {q f (t, ξ)}, and the second order complex IF estimate of f is defined by:

ω[2] f (t, ξ) = ωf (t, ξ) + qf (t, ξ)(t -tf (t, ξ)) if ∂ t tf (t, ξ) = 0 ωf (t, ξ) otherwise, (10)
and one then puts

ω[2] f (t, ξ) = ω[2] f (t, ξ) . It was proven in [15] that ω[2] f (t, ξ) = φ (t)
, when f is a Gaussian modulated linear chirp. It is also worth mentioning here that qf (t, ξ) can be computed by means of five different STFTs. Finally, FSST2 is defined by replacing ωf (t, ξ) by ω [START_REF] Cummings | Travelling waves in the occurrence of dengue haemorrhagic fever in Thailand[END_REF] f (t, ξ) in [START_REF] Daubechies | A nonlinear squeezing of the continuous wavelet transform based on auditory nerve models[END_REF], to obtain the so-called T γ 2,f , and mode reconstruction is then performed by replacing T γ f by T γ 2,f in (8).

D. Higher order synchrosqueezing transforms

Despite FSST2 definitely sharpens the TFR it is based on, it is proven to provide a truly sharp TFR only for perturbations of linear chirps with Gaussian modulated amplitudes. To handle signals containing more general types of AM-FM modes having non-negligible φ (k) k (t) for k ≥ 3, especially those with fast oscillating phase, one defines new SST operators based on third-or higher-order approximations of both amplitude and phase [START_REF] Pham | High-order synchrosqueezing transform for multicomponent signals analysis -with an application to gravitational-wave signal[END_REF]. To introduce the technique, we restrict ourselves to the STFT context but the technique presented hereafter could easily be extended to the CWT setting. Let f (τ ) = A(τ )e i2πφ(τ ) with A(τ ) (resp. φ(τ )) being equal to its L th -order (resp. N th -order) Taylor expansion for τ close to t, namely:

log(A(τ )) = L k=0 [log(A)] (k) (t) k! (τ -t) k and φ(τ ) = N k=0 φ (k) (t) k! (τ -t) k , (11)
where Z (k) (t) denotes the k th derivative of Z evaluated at t. Such a mode, with L ≤ N , can be written as:

f (τ ) = exp N k=0 1 k! [log(A)] (k) (t) + i2πφ (k) (t) (τ -t) k , (12)
since

[log(A)] (k) (t) = 0 if L + 1 ≤ k ≤ N . Its corresponding STFT reads: V g f (t, ξ) = R exp N k=0 1 k! [log(A)] (k) (t) + i2πφ (k) (t) τ k g(τ )e -i2πξτ dτ.
Taking the partial derivative of V g f (t, ξ) with respect to t and then dividing by i2πV g f (t, ξ), the local complex reassignment operator ωf (t, ξ) defined in Section II-C can be written, when V g f (t, ξ) = 0, as:

ωf (t, ξ) = N k=1 r k (t) V t k-1 g f (t, ξ) V g f (t, ξ) = 1 i2π [log(A)] (t) + φ (t) + N k=2 r k (t) V t k-1 g f (t, ξ) V g f (t, ξ) , (13)
where

r k (t) = 1 (k-1)! 1 i2π [log(A)] (k) (t) + φ (k) (t)
. It is clear that to get an exact IF estimate for the studied signal, one needs to subtract

   N k=2 r k (t) V t k-1 g f (t, ξ) V g f (t, ξ)    to {ω f (t, ξ)},
which requires the calculation of r k (t) for k = 2, . . . , N .

For that purpose, one derives a frequency modulation operator q[k,N] f (t, ξ), equal to r k (t) for the type of modes just introduced, and the definition of the N th -order IF estimate then follows [START_REF] Pham | High-order synchrosqueezing transform for multicomponent signals analysis -with an application to gravitational-wave signal[END_REF]:

ω[N] f (t, ξ) =              ωf (t, ξ) + N k=2 q[k,N] f (ξ, t) (-x k,1 (t, ξ)) , if V g f (t, ξ) = 0, and ∂ ξ x j,j-1 (t, ξ) = 0, 2 ≤ j ≤ N ωf (t, ξ) otherwise. with x k,1 (t, ξ) = V t k-1 g f (t,ξ) V g f (t,ξ) . ω[N] f (t, ξ) = {ω [N ]
f (t, ξ)} is then the desired IF estimate which is, by construction, exact for f satisfying [START_REF] Candes | Detecting highly oscillatory signals by chirplet path pursuit[END_REF]. As for FSST2, the N th -order FSST (FSSTN) is defined by replacing ωf (t, ξ) by ω[N] f (t, ξ) in ( 7) to obtain T γ N,f (t, ω) and the modes of the MCS can be reconstructed by replacing T γ f (t, ω) by T γ N,f (t, ω) in (8).

III. DEMODULATION ALGORITHM BASED ON HIGHER ORDER FSST FOR MODE RECONSTRUCTION

In this section, we recall the concept of the demodulation algorithm based on FSST (or FSST2) which was introduced in [START_REF] Meignen | On demodulation, ridge detection and synchrosqueezing for multicomponent signals[END_REF]. Then, we propose a variant of this algorithm that we coin DFSSTN in the sequel and show how the use of FSSTN results in better IF estimation and improved mode reconstruction by means of demodulation. The demodulation algorithm is based on a ridge extraction technique from the reassigned transform which we first recall.

A. Ridge Estimation

Any mode reconstruction technique based on the synchrosqueezing transform requires an estimate of the ridges (t, φ k (t)), for which, assuming knowledge of the number K of modes, we use the same algorithm as in [START_REF] Daubechies | Synchrosqueezed wavelet transforms: an empirical mode decomposition-like tool[END_REF] or [START_REF] Auger | Improving the readability of time-frequency and time-scale representations by the reassignment method[END_REF], and originally proposed in [START_REF] Carmona | Characterization of signals by the ridges of their wavelet transforms[END_REF]. This computes a local minimum of the functional

E f (ψ 1 , • • • , ψ K ) = K k=1 - R |T F f (t, ψ k (t))| 2 + λψ k (t) 2 + βψ k (t) 2 dt,
where T F f is one of the TF representations introduced above, i.e. FSST, FSST2, or FSSTN. In [START_REF] Meignen | On demodulation, ridge detection and synchrosqueezing for multicomponent signals[END_REF], it was shown that to use regularization parameters λ and β did not bring any improvement in terms of ridge detection (especially when synchrosqueezed transforms are considered), so we set them to zero.

B. Mode Reconstruction with Demodulation Based on Higher Order Synchrosqueezing (DFSSTN)

The previous section has provided us with a means to extract the ridges from the reassigned transforms, and these correspond to piecewise constant approximations of the IFs of the modes which are subsequently used in the mode reconstruction by putting ϕ k (t) = ψ k (t) for k = 1, • • • , K. In [START_REF] Meignen | On demodulation, ridge detection and synchrosqueezing for multicomponent signals[END_REF], these IFs estimates were used in a demodulation algorithm but the obtained results, though encouraging, were strongly dependent on the frequency resolution when computing STFT. Indeed, assume f is with finite length, typically defined on the interval

[0, T ], discretized into f ( nT R ) n=0,••• ,R-1 , and g supported on [-LT R , LT R ] with L < R/2
, the STFT of f is then computed as follows:

V g f (t, ξ) = LT R -LT R f (t + τ )g(τ )e -2iπτ ξ dτ ≈ T R L n=-L f t + nT R g nT R e -i2π nT R ξ , ( 14
)
from which we infer that, for 0 ≤ p ≤ R -1:

V g f qT R , pR M T ≈ T R L n=-L f (q + n)T R g nT R e -i2π np M ,
for some M ≥ 2L + 1. The last sum is computed by means of a discrete Fourier transform, and the frequency resolution equals R M T . This has the consequence that STFT is all the more compact that the frequency resolution is low, and that the IFs estimation based on crude ridges extraction is very inaccurate in that case.

We now propose to use, for the purpose of mode reconstruction using a demodulation algorithm, much more relevant IFs estimates than those based on crude ridges extraction. Indeed, while performing FSST, FSST2, or FSSTN, one computes ωf (t, ξ), ω [START_REF] Cummings | Travelling waves in the occurrence of dengue haemorrhagic fever in Thailand[END_REF] f (t, ξ), and ω[N] f (t, ξ), respectively. These, evaluated on the ridge associated with the kth mode, i.e. ωf (t, ϕ k (t)), ω [START_REF] Cummings | Travelling waves in the occurrence of dengue haemorrhagic fever in Thailand[END_REF] f (t, ϕ k (t)), and ω[N] f (t, ϕ k (t)), lead to much smoother IF estimates than those proposed in [START_REF] Meignen | On demodulation, ridge detection and synchrosqueezing for multicomponent signals[END_REF]. Based on the IFs estimates we use the demodulation procedure proposed in [START_REF] Meignen | On demodulation, ridge detection and synchrosqueezing for multicomponent signals[END_REF] (see Algorithm 1).

Algorithm 1 Demodulation based on FSSTN (DFSSTN)

Estimate the ridges

(ψ 1 , • • • , ψ K ) from FSSTN. for k = 1 to K do 1. Compute φk (t) = ω[N] f (t, ψ k (t)). 2. Compute f D,k (t) = f (t)e -i2π( t 0 φk (x)dx-ψ0t) . 3. From T γ f D,k , extract the ridge ψ D,k corresponding to mode k of f D,k , by considering single ridge detection in [ψ 0 -∆, ψ 0 + ∆].
4. Reconstruct the kth mode of f D,k and then multiply it by the inverse of demodulation operator to recover f k :

f k (t) ≈ |ω-ψ D,k (t)|<d T γ f D,k (t, ω
)dω e i2π( t 0 φk (x)dx-ψ0t) .

IV. RESULTS AND DISCUSSION

This section provides some numerical experiments on an MCS, in both noise-free and noisy cases, illustrating the benefits of using Algorithm 1, with N = 3 or 4, rather than other existing techniques including FSST2, FSST3, FSST4 [START_REF] Pham | High-order synchrosqueezing transform for multicomponent signals analysis -with an application to gravitational-wave signal[END_REF] and DFSST2 [START_REF] Meignen | On demodulation, ridge detection and synchrosqueezing for multicomponent signals[END_REF]. The MATLAB scripts generating all the figures of this paper can be downloaded from github.com/phamduonghung/EUSIPCO2019.

Let us first consider a synthetic MCS composed of two AM-FM modes defined as: 4 )e i2π(340t-2e -2(t-0.2) sin(14π(t-0.2))) , for t ∈ [0, 1]. Note that f 1 is a polynomial chirp fitting in the model defined by [START_REF] Pitton | Applications of positive time-frequency distributions to speech processing[END_REF] and f 2 is a damped-sine function containing very strong nonlinear sinusoidal frequency modulations. Such a signal is sampled at a rate R = 1024 Hz on [0, 1]. We use the

f (t) = f 1 (t) + f 2 (t) with f 1 (t) = e 2(1-t) 3 e i2π(50t+30t 3 -20(1-t) 4 ) and f 2 (t) = (1 + 7(1 -t)
L 1 -normalized Gaussian window g(t) = σ -1 e -π t 2 σ 2
to compute the STFT of f , where σ is the optimal value determined minimizing Rényi entropy [START_REF] Stanković | A measure of some time-frequency distributions concentration[END_REF].

In Figure 1, we depict the modulus of STFT and close-ups of reassigned representations given by FSST2, FSST3, FSST4, respectively. It is clear that all the studied techniques lead to relatively sharp TFRs for this mode (f 1 behaves locally as a Gaussian modulated linear chirp). In contrast, for f 2 , the higher the order of FSST the sharper the TFR; especially when the IF of the mode has a non negligible curvature φ (t). It was shown in [START_REF] Pham | High-order synchrosqueezing transform for multicomponent signals analysis -with an application to gravitational-wave signal[END_REF] that the better TF concentration when using FSSTN also meant better mode reconstruction performance. However, the latter can be further improved by using DFSSTN as shown hereafter, in noise-free and noisy situations.

A. Noise-Free Case

We display, in Figures 2 (a) and (b), the quality of retrieval processes associated with f 1 and f 2 , respectively, with respect to parameter d defined in (8), using FSST4, DFSST2 and DFSST4 computed either with M = R and M = R/8 = 128. It is worth noting that, in our simulations, the optimal σ as determined above leads to a filter length 2L + 1 = 123, and M = R/8 = 128 is close to the smallest value one can choose. Besides, the performance of mode reconstruction are evaluated by SNR out = 20 log 10 f k 2 / f k,r -f k 2 , where f k,r is the kth reconstructed mode and . 2 is the l 2 norm. We first remark from Figure 2 that a bigger d results in better reconstruction qualities, which is consistent with earlier research [START_REF] Meignen | On demodulation, ridge detection and synchrosqueezing for multicomponent signals[END_REF], [START_REF] Oberlin | Second-order synchrosqueezing transform or invertible reassignment? Towards ideal time-frequency representations[END_REF]. Moreover, it can be seen that the tested techniques computed with M = R/8 (low frequency resolution) perform much better than those calculated with higher frequency resolution M = R: a compact representation should clearly be favored. Finally, yet most importantly, for each M used, while DFSST4 produces quite similar results to DFSST2, these are much better than those obtained with FSST4 for f 1 . For mode f 2 , DFSST4, taking into account the curvature in the demodulation operator, outperforms all the other studied techniques.

B. Noisy Case

To further challenge the different tested techniques in the presence of noise, we consider a noisy signal defined f ζ (t) = f (t) + ζ(t), where ζ(t) is a complex white Gaussian process with variance Var ( {ζ(t)}) = Var ( {ζ(t)}) = σ 2 ζ , where {Z} and {Z} stand for the real and imaginary parts of complex number Z. Furthermore, the noise level is measured by: SNR in [dB] = 20 log 10 f 2 / f ζ -f 2 . Then, we again investigate the reconstruction qualities for modes f 1 and f 2 , for FSST4, DFSST2 and DFSST4, when M = R or M = R/8 = 128 and with d = 1. Note that these qualities are the means of the reconstruction accuracies obtained by repeating each simulation 100 times. As in the previous case, DFSST4 still exhibits the best mode reconstruction performance, as reported in Figure 3 (a) and (b). To conclude, our simulations plead in favor of using FSSTN, with a low frequency resolution, followed by a demodulation procedure (DFSSTN) to reconstruct the modes of an MCS rather than FSSTN alone or DFSST2.

V. CONCLUSION

In this paper, we have introduced a new technique for the retrieval of the modes of multicomponent signals from their time-frequency representation, based on high-order synchrosqueezing transforms (FSSTN) together with a demodulation procedure. Numerical experiments demonstrated the improvement brought by the proposed technique over state-ofthe-art methods for both noise-free and noisy signals. Future work should now be devoted to a more theoretical analysis of the proposed techniques in more general noisy situations, including both Gaussian and non-Gaussian noises, as was done in [START_REF] Thakur | The synchrosqueezing algorithm for time-varying spectral analysis: Robustness properties and new paleoclimate applications[END_REF], [START_REF] Yang | Statistical analysis of synchrosqueezed transforms[END_REF] for synchrosqueezing transforms.

Fig. 1 .

 1 Fig. 1. (a): modulus of the STFT of f with a small patch (delimited by a red rectangle); (b): zoom in FSST2 computed from the patched STFT shown in (a); from (c) to (d), same as (b) but for FSST3 and FSST4 respectively.

Fig. 2 .

 2 Fig. 2. Noise-free case: (a) reconstruction accuracy measured by ouput SNR with respect to d for mode f 1 using FSST4, DFSST2 and DFSST4 (with M = R and M = R/8); (b): same as (a) but for mode f 2 .

8 (b): mode f 2 Fig. 3 .

 823 Fig. 3. In a noise-free case: (a) reconstruction accuracy measured using output SNR with respect to different input SNR for f 1 using FSST4, DFSST2 and DFSST4 computed with both M = R and M = R/8 and choosing d = 1; (b): same as (a) but for f 2 .


Ivana Milic

Eva Griesser

Venukumar Vemula

Naoya Ieda

Hidehiko Nakagawa

Naoki Miyata

Jean-Marie Galano

Camille Oger

Thierry Durand

Maria Fedorova
email: maria.fedorova@bbz.uni-leipzig.de

Profiling and relative quantification of multiply nitrated and oxidized fatty acids

Keywords: Cardiomyocytes, Lipid nitration, Lipid peroxidation, Nitrated fatty acids, Nitrosative stress

The levels of nitro fatty acids (NO 2 -FA), such as nitroarachidonic, nitrolinoleic, nitrooleic, and dinitrooleic acids, are elevated under various inflammatory conditions, and this results in different anti-inflammatory effects. However, other multiply nitrated and nitro-oxidized FAs have not been studied so far. Owing to the low concentrations in vivo, NO 2 -FA analytics usually relies on targeted gas chromatography-tandem mass spectrometry (MS/MS) or liquid chromatography-MS/MS, and thus require standard compounds for method development. To overcome this limitation and increase the number and diversity of analytes, we performed in-depth mass spectrometry (MS) profiling of nitration products formed in vitro by incubating fatty acids with NO 2 BF 4 , and ONOO -. The modified fatty acids were used to develop a highly specific and sensitive multiple reaction monitoring LC-MS method for relative quantification of 42 different nitrated and oxidized species representing three different groups: singly nitrated, multiply nitrated, and nitro-oxidized fatty acids. The method was validated in in vitro nitration kinetic studies and in a cellular model of nitrosative stress. NO 2 -FA were quantified in lipid extracts from 3-morpholinosydnonimine-treated rat primary cardiomyocytes after 15, 30, and 70 min from stress onset. The relatively high levels of dinitrooleic, nitroarachidonic, hydroxynitrodocosapenataenoic, nitrodocosahexaenoic, hydroxynitrodocosahexaenoic, and dinitrodocosahexaenoic acids confirm the presence of multiply nitrated and nitro-oxidized fatty acids in biological systems for the first time. Thus, in vitro nitration was successfully used to establish a targeted LC-MS/MS method that was applied to complex biological samples for quantifying diverse NO 2 -FA.

Introduction

Nitric oxide (•NO) is a well-known mediator of various biological processes [START_REF] Pacher | Nitric oxide and peroxynitrite in health and disease[END_REF]. Besides being a potent vasodilator, •NO is generated by phagocytes as part of the innate immune response [START_REF] Wink | Nitric oxide and redox mechanisms in the immune response[END_REF] preventing platelet aggregation [START_REF] Wang | Mechanism of platelet inhibition by nitric oxide: in vivo phosphorylation of thromboxane receptor by cyclic GMP-dependent protein kinase[END_REF][START_REF] Riddell | Nitric oxide and platelet aggregation[END_REF] and leukocyte adhesion to the endothelium [START_REF] Kubes | Nitric oxide: an endogenous modulator of leukocyte adhesion[END_REF]. Under aerobic conditions, nitric oxide readily reacts with reactive oxygen species (ROS) yielding various reactive nitrogen species (RNS) [START_REF] Bloodsworth | Nitric oxide regulation of free radical-and enzyme-mediated lipid and lipoprotein oxidation[END_REF]. Whereas some RNS can act as oxidants and nitrosylating agents (NO + , N 2 O 3 , etc.), nitrogen dioxide radical (•NO 2 ), peroxynitrite anion (ONOO -), and nitronium cation (NO 2 + ) are known nitrating species. The concentrations of •NO and Electronic supplementary material The online version of this article (doi:10.1007/s00216-015-8766-3) contains supplementary material, which is available to authorized users. oxygen in the hydrophobic environment of cellular membranes and the lipid core of lipoproteins can be several times higher than in the cytoplasm, triggering elevated RNS production rates. In fact, around 90 % of •NO is auto-oxidized in these compartments [START_REF] Liu | Accelerated reaction of nitric oxide with O2 within the hydrophobic interior of biological membranes[END_REF][START_REF] Moller | Membrane Blens^effect: focusing the formation of reactive nitrogen oxides from the • NO/O 2 reaction[END_REF] and this can modify surrounding biomolecules. Free polyunsaturated fatty acids (PUFA) and their esters are the primary targets of RNS owing to their high susceptibility to oxidation and nitration.

In vivo formation of nitrated fatty acids (NO 2 -FA) was supported by their detection in human blood plasma [START_REF] Lima | Characterization of linoleic acid nitration in human blood plasma by mass spectrometry[END_REF][START_REF] Tsikas | Nitro-fatty acids occur in human plasma in the picomolar range: a targeted nitro-lipidomics GC-MS/MS study[END_REF][START_REF] Trettin | GC-MS/MS and LC-MS/MS studies on unlabelled and deuterium-labelled oleic acid (C18:1) reactions with peroxynitrite (O=N-O-O(-)) in buffer and hemolysate support the pM/nM-range of nitro-oleic acids in human plasma[END_REF][START_REF] Bonacci | Conjugated linoleic acid is a preferential substrate for fatty acid nitration[END_REF][START_REF] Tsikas | Specific GC-MS/MS stable-isotope dilution methodology for free 9-and 10-nitro-oleic acid in human plasma challenges previous LC-MS/MS reports[END_REF], membranes of red blood cells [START_REF] Baker | Red cell membrane and plasma linoleic acid nitration products: synthesis, clinical identification, and quantitation[END_REF], and urine [START_REF] Baker | Fatty acid transduction of nitric oxide signaling: multiple nitrated unsaturated fatty acid derivatives exist in human blood and urine and serve as endogenous peroxisome proliferator-activated receptor ligands[END_REF][START_REF] Salvatore | Characterization and quantification of endogenous fatty acid nitroalkene metabolites in human urine[END_REF]. Recent evidence indicates increased nitroalkene production rates in various inflammatory models, such as a 20-fold higher cholesteryl nitrolinoleate level in murine J774.1 macrophages activated by an inflammatory stimulus [START_REF] Ferreira | Macrophage activation induces formation of the anti-inflammatory lipid cholesteryl-nitrolinoleate[END_REF]. Increased levels of nitrolinoleic acid (LaNO 2 ) were detected in mitochondria during ischemic preconditioning, whereas the level of nitrooleic acid (OaNO 2 ) remained unchanged, indicating possible selectivity of fatty acid nitration [START_REF] Nadtochiy | Mitochondrial nitroalkene formation and mild uncoupling in ischaemic preconditioning: implications for cardioprotection[END_REF]. Under inflammatory conditions LaNO 2 , OaNO 2 , and nitroarachidonic acid (AaNO 2 ) posses anti-inflammatory activities [START_REF] Khoo | Electrophilic nitro-fatty acids: antiinflammatory mediators in the vascular compartment[END_REF][START_REF] Rubbo | Nitro-fatty acids: novel anti-inflammatory lipid mediators[END_REF][START_REF] Freeman | Ischia M (2008) Nitro-fatty acid formation and signaling[END_REF], including activation of peroxisome-proliferator-activated receptors [START_REF] Ferreira | Nitro-fatty acids as novel electrophilic ligands for peroxisome proliferator-activated receptors[END_REF] and nuclear factor E2 related factor 2 [START_REF] Li | Mechanism of chemical activation of Nrf2[END_REF][START_REF] Kansanen | Electrophilic nitrofatty acids activate NRF2 by a KEAP1 cysteine 151-independent mechanism[END_REF], induction of heme oxygenase 1 [START_REF] Wright | Fatty acid transduction of nitric oxide signaling: nitrolinoleic acid potently activates endothelial heme oxygenase 1 expression[END_REF], inhibition of nuclear factor κB, and inhibition of prostaglandin endoperoxide H synthase (PGHS) [START_REF] Trostchansky | Nitroarachidonic acid, a novel peroxidase inhibitor of prostaglandin endoperoxide H synthases 1 and 2[END_REF] and NADPH oxidase [START_REF] Gonzalez-Perilli | Nitroarachidonic acid prevents NADPH oxidase assembly and superoxide radical production in activated macrophages[END_REF]. OaNO 2 decreased superoxide production in activated macrophages and pulmonary artery smooth muscle cells, demonstrating protective effects in a hypoxia-induced murine model of pulmonary hypertension [START_REF] Klinke | Protective effects of 10-nitro-oleic acid in a hypoxia-induced murine model of pulmonary hypertension[END_REF]. Additionally, OaNO 2 reduced atherosclerotic lesion formation in apolipoprotein E deficient mice by inhibiting vascular smooth muscle cell proliferation [START_REF] Rudolph | Nitro-fatty acids reduce atherosclerosis in apolipoprotein E-deficient mice[END_REF]. Recently it was demonstrated that OaNO 2 effectively induces expression of heat shock proteins and antioxidant enzymes in keratinocytes [START_REF] Zheng | Regulation of keratinocyte expression of stress proteins and antioxidants by the electrophilic nitrofatty acids 9-and 10nitrooleic acid[END_REF]. Synthetic AaNO 2 revealed several new regulatory properties in cellular models that might be common for NO 2 -FA. Release of nitric oxide by AaNO 2 can induce cyclic GMP (cGMP)-dependent vasorelaxation and downregulate inducible nitric oxide synthase [START_REF] Trostchansky | Synthesis, isomer characterization, and antiinflammatory properties of nitroarachidonate[END_REF]. Additionally, AaNO 2 was shown to prevent NADPH oxidase assembly [START_REF] Gonzalez-Perilli | Nitroarachidonic acid prevents NADPH oxidase assembly and superoxide radical production in activated macrophages[END_REF], regulate protein kinase C [START_REF] Bonilla | Regulation of protein kinase C by nitroarachidonic acid: impact on human platelet activation[END_REF], and inhibit oxygenase activities of PGHS-1 and PGHS-2.

Despite increasing interest in the therapeutic potential of NO 2 -FA, their in vivo diversity has not been fully discovered. Currently, data are limited to LaNO 2 , OaNO 2 , and AaNO 2 , whereas other possible modifications-for example, polynitro-PUFA and hydroxynitro-PUFA-have rarely been addressed. In vivo, ROS and RNS are produced simultaneously, and it is likely that oxidation of PUFA can be followed by their nitration, and vice versa, leading to a higher diversity of modified PUFA than is currently assumed. Owing to their very low in vivo concentrations, ranging from high picomolar to low nanomolar concentrations, mass-spectrometric detection of NO 2 -FA is very challenging. Therefore, multiple reaction monitoring (MRM) is often used to achieve the required sensitivity. However, MRM allows only targeted detection, and is thus limited to known or expected compounds. It does not provide information on NO 2 -FA diversity. To overcome these limitations, we performed an in-depth profiling of PUFA [oleic acid (Oa), linoleic acid (La), conjugated La (cLa), arachidonic acid (Aa), and docosahexaenoic acid (Dha)] nitrated in vitro under low and high oxygen tension to monitor the kinetics of NO 2 -FA. On the basis of the diversity of NO 2 -FA identified in vitro, a highly specific and sensitive liquid chromatography (LC)-tandem mass spectrometry (MS/MS) MRM method was developed and applied for the detection and relative quantification of NO 2 -FA in a cardiomyocyte cell model of nitrosative stress.

Materials and methods

Chemicals

All unsaturated fatty acids, tert-butyl methyl ether (MTBE), lithium trifluoroacetate, and horse serum were purchased from Sigma-Aldrich (Taufkirchen, Germany). Acetonitrile, methanol, and ammonium formate [ultrapure LC/mass spectrometry (MS) grade] were obtained from Biosolve (Valkenswaard, Netherlands). Chloroform was purchased from Merck (Darmstadt, Germany). 2,3,5,6-Tetramethyl-4-(methylnitrosoamino)phenol (P-NAP) was synthesized according to Ieda et al. [START_REF] Ieda | Photocontrollable peroxynitrite generator based on N-methyl-Nnitrosoaminophenol for cellular application[END_REF]. C 21 -15-F 2t -IsoP, d 4 -10-F 4t -NeuroP, and d 4 -4-RS-4-F 4t -NeuroP were synthesized according to the published procedure [START_REF] Guy | Oxygenated metabolites of n-3 polyunsaturated fatty acids as potential oxidative stress biomarkers: total synthesis of 8-F 3t -IsoP, 10-F 4t -NeuroP and [D 4 ]-10-F 4t -NeuroP[END_REF][START_REF] Oger | The handy use of Brown's P2-Ni catalyst for a skipped diyne deuteration: application to the synthesis of a [D 4 ]labeled F 4t -neuroprostane[END_REF]. Dulbecco's modified Eagle's medium/Ham's F12 medium was purchased from PAA Laboratories (Cölbe, Germany). Fetal bovine serum, phosphate-buffered saline, L-glutamine, nonessential amino acids, sodium pyruvate, and antibiotic (penicillin/streptomycin) solutions were obtained from Life Technologies (Darmstadt, Germany). 3-Morpholinosydnonimine (SIN-1) was purchased from Enzo Life Sciences (Lörrach, Germany).

PUFA nitration by NO 2 BF 4

Oa, La, Aa, and Dha (100 g/L in chloroform) were nitrated in the presence of solid nitronium tetrafluoroborate (NO 2 BF 4 ; 100 mg, PUFA-NO 2 BF 4 , 1:2 molar ratio, 20 °C, 1 h, intense shaking). Nitration was quenched (0.5 mL of deionized water; vortexing for 30 s), and the lower organic phase was dried under a vacuum and reconstituted in chloroform (1 mL). Vials were sealed with Teflon caps and purged with nitrogen gas prior to storage at -20 °C. The kinetics of nitration was recorded for 2 h by taking aliquots (100 μL) 0.5, 15, 30, 45, 60, 90, and 120 min after addition of the nitrating reagent. Samples were analyzed immediately.

Nitration of cLa by P-NAP

An aliquot of cLa (75 μL; 8 mmol/L) was co-incubated with the photocontrollable peroxynitrite donor P-NAP (24 μL, 100 mmol/L in dimethylformamide) [START_REF] Ieda | Photocontrollable peroxynitrite generator based on N-methyl-Nnitrosoaminophenol for cellular application[END_REF]. The open polypropylene tube containing the sample was placed under a UV lamp (Dr. Hönle, Gräfelfing, Germany) and irradiated (20 min) with monochromatic UV-A light (350 nm) with an average flux of 105 W/m 2 . The kinetics of nitration was recorded for 2 h by taking aliquots (50 μL) 0, 3, 6, 9, 15, 30, 60, and 120 min after the reaction had started. Nitrated cLa was extracted using chloroform (100 μL), dried, and reconstituted in chloroform (50 μL). Samples were analyzed immediately.

Electrospray ionization LTQ Orbitrap MS of nitrated PUFA

Samples were diluted in a mixture of methanol and chloroform (2:1, v/v) containing ammonium formate (5 mmol/L) to obtain 10 μmol/L solutions [used for negative ion mode electrospray ionization (ESI) MS] or in a 5 mmol/L solution of lithium trifluoroacetate in methanol (positive ion mode ESI-MS). The samples were analyzed using a robotic nanoflow ion source (TriVersaNanoMate; AdvionBioSciences, Ithaca, NY, ZSA) equipped with a nanoelectrospray chip (-1.2-kV ionization voltage, 0.3-psi nitrogen backpressure for negative ion mode; 1.4-kV ionization voltage, 0.4-psi nitrogen backpressure for positive ion mode) coupled to an LTQ Orbitrap XL ETD mass spectrometer (Thermo Fischer Scientific, Bremen, Germany). The temperature of the transfer capillary was set to 200 °C and the tube lens voltage was set to 120 V. Mass spectra were acquired with the Orbitrap with a target mass resolution of 100,000 at m/z 400. Collisioninduced dissociation (CID) and higher-energy CID (HCD) relied on an isolation width of 1.5 u and a normalized collision energy of 30 % (CID) and 42 % (HCD). Recorded data were analyzed manually using Xcalibur (version 2.0.7, Thermo Fisher).

PUFA nitration kinetics using MRM

Analyses were performed with as ESI 4000 QTRAP system (AB Sciex Germany, Darmstadt, Germany) operated in a negative ion mode with an ionization voltage of -4.5 kV, an entrance potential of -10 V, and an ion source temperature of 600 °C. The normalized collision energy, declustering potential, and exit quadrupole potential were optimized for each Q1/Q3 transition. The MRM included the two most intense transitions of each analyte with a dwell time of 40 ms. Each sample (30 μL in ESI solution) was infused with methanol (0.2 mL/min) using an Waters Alliance 2790 highperformance LC instrument (Waters, Eschborn, Germany) and measured in triplicate. Data were processed using Analyst (version 1.6, AB Sciex).

Analytical characterization of nitrated and oxidized PUFA in cardiomyocytes

In vitro nitration mixtures of Oa, La, Aa, Dha, and cLa were used to determine the retention times of all products. The LC-MRM was performed with a nanoACQUITY UPLC system (Waters, Eschborn, Germany) operated in high flow mode using the mixer kit for 1 mm columns (Waters, Eschborn, Germany) coupled to an ESI triple-quadrupole linear ion trap MS instrument (4000 QTRAP, AB Sciex Germany, Darmstadt, Germany). Samples were separated on an ACQUITY UPLC HSS T3 C 18 column (internal diameter 1 mm, length 100 mm, pore size 100 Å, and particle size 1.8 μm) using a column temperature of 50 °C , a flow rate of 100 μL/min, and a linear 6-minute gradient from 30 % acetonitrile to 98 % acetonitrile in 0.1 % aqueous HCOOH. The electrospray voltage was set to -4.5 kV, the ion source temperature was set to 600 °C, the entrance potential was set to -10 V, and the dwell time was set to 40 ms. All previously optimized transition pairs (declustering potential, collision energy, and exit quadrupole potential) of each analyte were included to record their retention times (see Table S1), whereas only the most sensitive and retention-time-specific transitions of each analyte were included in the final MRM method used for the relative quantification of nitrated and oxidized PUFA in the cardiomyocyte nitrosative stress model.

Cardiomyocyte cell culture and nitrosative stress induction

Primary rat cardiomyocytes (Innoprot, Elexalde Derio, Spain) were cultured until 80 % confluence in Dulbecco's modified Eagle's medium/Ham's F12 medium supplemented with fetal bovine serum (20 %), horse serum (5 %), L-glutamine (2 mmol/L), nonessential amino acids (0.1 mmol/L), sodium pyruvate (3 mmol/L), and antibiotics (1 %) at 37 °C (95 % air and 5 % CO 2 atmosphere). The medium was replaced with serum-free medium 24 h before treatment with 3morpholinosydnonimine (SIN-1; 10 μmol/L, 15, 30, and 70 min). Untreated (control) and treated cells (around 1.5× 10 6 ) were collected and washed with phosphate-buffered saline (pH 7.4), and lipids were extracted using MTBE [START_REF] Matyash | Lipid extraction by methyl-tert-butyl ether for highthroughput lipidomics[END_REF]. Briefly, cell pellets were resuspended in 50 μL of ice-cold 0.1 % aqueous ammonium acetate containing 1 ng of internal standards (C 21 -15-F 2t -IsoP, d 4 -10-F 4t -NeuroP, d 4 -4-RS-4-F 4t -NeuroP; Table S1). After 15 min on ice, 375 μL of methanol and 1.25 mL of MTBE were added, and the mixture was kept on a rotary shaker for 1 h at 4 °C. Separation of phases was achieved by adding water (313 μL) and keeping the mixture on the rotary shaker for 10 min at 4 °C. After centrifugation (5 min, 4 °C, 1,000g) the upper, organic phase was collected and dried under a vacuum. The lower, aqueous phase was used to measure the protein concentration by a Bradford assay. Data were acquired and analyzed using Analyst (version 1.6).

Results and discussion

Mass-spectrometric analysis of fatty acids nitrated with NO 2 BF 4 Nitronium tetrafluoroborate (NO 2 BF 4 ) readily produces NO 2 + , which can nitrate PUFA by electrophilic substitution. Although nitration of fatty acids by NO 2 BF 4 results in an array of different nitration products and not just a few positional nitration isomers, we used this approach because of its simplicity and the relatively high yields of nitrated PUFA. After 1 h nitration of Oa, La, Aa, and Dha, the sample mixtures were directly analyzed by ESI LTQ Orbitrap MS. The high resolution (HR) and mass accuracy provided elemental compositions of the detected nitration products (Table 1), including singly or multiply nitrated derivatives and nitro-oxidized species (Fig. 1). Fluorinated NO 2 -FA, which were also observed as by-products, were not considered in the following study.

Nitration of Oa yielded ten signals corresponding to different reaction products of Oa (Fig. 1a). The signal of deprotonated Oa (m/z 281.25) was not observed, indicating that within 1 h of nitration, Oa was quantitatively consumed. Two low-intensity signals at m/z 295.23 and m/z 297.24 were assigned as keto-Oa (OaO) and hydroxy-Oa (OaOH), whereas the remaining eight signals, including the base peak, corresponded to nitrated or nitroslyated Oa. Detection of OaNO 2 (m/z 326.23), dinitro-Oa [Oa(NO 2 ) 2 ; m/z 371.22], and trinitro-Oa [Oa(NO 2 ) 3 ; m/z 416.20] showed that up to three NO 2 groups can be present although Oa contains only one double bond. Although electrophilic nitration is known to produce vinylnitro isomers, detection of Oa(NO 2 ) 3 after Oa nitration suggests the presence of allylic NO 2 groups. Signals with a mass shift of +29.99 u were assigned to nitroso-Oa (OaNO; m/z 310. La nitration resulted in nine reaction products (Fig. 1b), including oxidized forms, such as keto-La (LaO; m/z 293.21) and hydroxy-La (LaOH; m/z 295.23). Notably, signals of the oxidized La were more intense then the signals of oxidized Oa species. The peak corresponding to deprotonated ESI Orbitrap mass spectra of Aa (Fig. 1c) and Dha (Fig. 1d) nitration mixtures displayed the deprotonated fatty acid ions as base peaks (m/z 303. [START_REF] Li | Mechanism of chemical activation of Nrf2[END_REF] Electrophilic fatty acid nitration by nitronium ions under low oxygen tension resulted in a high diversity of modified Oa, La, Aa, and Dha. ESI Orbitrap MS spectra acquired after a reaction time of 1 h (Fig. 1) revealed different susceptibilities. To the best of our knowledge, this is the first report on in vitro nitration of Dha. Besides single-nitro fatty acids, the formation of polynitro fatty acids was observed for each fatty acid. Regardless of a higher unsaturation degree of Aa (C20:4) and Dha (C22:6), these two fatty acids were detected with up to three nitro groups introduced along the alkyl chain, whereas for La with two double bonds even a tetranitro derivative was present. Our data indicate lower susceptibilities of Aa and Dha toward nitration in general and especially the formation of multiple nitration derivatives compared with Oa and La. The relative signal intensities of the modified species decreased with increasing fatty acid chain lengths and the numbers of double bonds in PUFA. Additionally, the low oxygen tension conditions applied here minimized formation of oxidized fatty acids and nitrohydroxy fatty acids.

MS/MS of NO 2 -FA

The CID and HCD fragmentation behavior of nitrated PUFA was studied using HR-MS/MS in order to evaluate the diversity of positional isomers on the basis of previously published mechanisms [START_REF] Bonacci | Gas-phase fragmentation analysis of nitrofatty acids[END_REF] (see Figs. S1-S4, Scheme S1). Neutral loss of nitrous acid (HNO 2 ; -47 u) produced under the CID conditions was observed in all tandem mass spectra of NO 2 -FA, and served as a reporter ion for the presence of the nitro group. Although successful nitration of Oa, La, and Aa and analysis of the corresponding MS/MS spectra was previously reported, to the best of our knowledge, this is the first report of Dha nitration. Since in the MS/MS spectrum of deprotonated DhaNO 2 the parent ion dominates the spectrum, and fragmentation of the alkyl chain was poor, we used the lithiated ion of DhaNO 2 for the fragmentation. We relied on the fragmentation mechanism previously reported by Bonacci et al. [START_REF] Bonacci | Gas-phase fragmentation analysis of nitrofatty acids[END_REF] to predict a series of fragment ions of each nitro-Dha (DhaNO 2 ) isomer, and compared them with the recorded Fourier transform CID-MS/MS spectrum (Fig. 2). It was possible to match 12 product ions, including structure-specific ions for seven nitro-Dha isomers, including 11-NO 2 -Dha (m/z 161.08), 13-NO 2 -Dha (m/z 175.09), 14-NO 2 -Dha (m/z 201.11), 16-NO 2 -Dha (m/z 215.12), 17-NO 2 -Dha (m/z 241.14), 19-NO 2 -Dha (m/z 255. [START_REF] Salvatore | Characterization and quantification of endogenous fatty acid nitroalkene metabolites in human urine[END_REF]), and 20-NO 2 -Dha (m/z 281.17). Fragment ions containing a nitrogen atom were not detected.

Similarly, the tandem mass spectrum of [AaNO 2 +Li + ] + recorded using HCD fragmentation and detection in the Orbitrap mass analyzer (Fig. S2) confirmed the presence of five AaNO 2 vinyl isomers-namely, 9-NO 2 -Aa, 11-NO 2 -Aa, 12-NO 2 -Aa, 14-NO 2 -Aa, and 15-NO 2 -Aa (Fig. S2, Scheme S1). We did not observe fragment ions specific for 5-NO 2 -Aa, 6-NO 2 -Aa, and 8-NO 2 -Aa or 4-NO 2 -Dha, 5-NO 2 -Dha, 7-NO 2 -Dha, 8-NO 2 -Dha, and 10-NO 2 -Dha. However, it cannot be ruled out that isomers of nitro fatty acids with the NO 2 group localized at the first three vinyl carbons are formed, but with lower yields compared with the other isomers, and thus the corresponding ions will result in signal intensities below the limit of detection of the Orbitrap mass analyzer. Furthermore, the fragmentation pattern might be different for the isomers with the NO 2 group near the carboxy group. Nevertheless, Trostchansky et al. [START_REF] Trostchansky | Synthesis, isomer characterization, and antiinflammatory properties of nitroarachidonate[END_REF] performed in vitro nitration of Aa by sodium nitrite under acidic conditions, and also observed only four major isomers of nAaNO 2 -namely, 9-NO 2 -Aa, 12-NO 2 -Aa, 14-NO 2 -Aa, and 15-NO 2 -Aa.

Further, in vitro nitrated PUFA mixtures were used to optimize MRM parameters, including precursor to product ion transitions in a series of direct infusion experiments (Table S1). This in vitro approach allowed us to specify 91 different transition pairs for the simultaneous relative quantification of 42 nitrated, oxidized, and nitro-oxidized PUFA, which was further validated as part of the nitration kinetics study.

Kinetics of NO 2 + -mediated nitration of Oa, La, Aa, and Dha

The kinetics of PUFA nitration by NO 2 + was monitored by MRM-based ESI-MS/MS. Altogether, we were able to monitor eight, nine, eight, and five products of Oa, La, Aa, and Dha nitration and oxidation, respectively (see Figs S5-S8). Except for OaO, all monitored oxidized fatty acids (i.e., LaO, LaOH, AaO, AaOH, and DhaO) were formed immediately, and then their quantities decreased constantly, most likely due to further nitration by NO 2 + ions. Among the singly nitrated derivatives, OaNO 2 was formed the fastest, with a mild increase over the following 60 min. The quantities of LaNO 2 and AaNO 2 increased several fold until the maxima were reached after 30 and 15 min, respectively, whereas the quantity of DhaNO 2 increased over the whole reaction period. Nitro fatty acids via further reactions with NO 2 + ions yield multiply nitrated fatty acids, which was observed for Oa, La, Aa, and Dha. The nitration kinetics revealed that Aa and Dha with longer alkyl chains and more double bonds than Oa and La are less reactive toward nitronium ions. Maximal levels of Oa(NO 2 ) 2 and Aa(NO 2 ) 2 were reached after 60 min, and the level of La(NO 2 ) 2 gradually increased over the reaction time course. The kinetics of Dha(NO 2 ) 2 formation suggests that the product is formed at the reaction onset and quantity decreases in the first 60 min. Finally, nitro-oxidized products of Oa and La (OaNO 2 OH and LaNO 2 OH) showed an overall increase in the detected levels, whereas maximal levels of nitro-oxidized Aa and Dha (AaNO 2 OH, DhaNO 2 OH) were observed at the reaction onset, suggesting that the formed AaOH and DhaOH are readily consumed in further reactions with NO 2 + ions.

HR-MS detection of cLa nitration by P-NAP

Use of the UV-inducible peroxynitrite donor P-NAP was recently introduced. It releases NO, which reacts under aerobic conditions with O 2

•-, producing a peroxynitrite anion (ONOO -) [START_REF] Ieda | Photocontrollable peroxynitrite generator based on N-methyl-Nnitrosoaminophenol for cellular application[END_REF], which forms •NO 2 , an RNS capable of nitrating fatty acids in vivo or lipid radicals yielding nitro fatty acids and hydroxynitro fatty acids. Thus, we used P-NAP as an ONOO -donor in aqueous solutions to investigate fatty acid nitration for biologically more relevant conditions. As a recent study identified cLa as a better substrate for nitration than bisallylic La in vitro and in vivo [START_REF] Bonacci | Conjugated linoleic acid is a preferential substrate for fatty acid nitration[END_REF], we selected cLa as the substrate for P-NAP nitration (Fig. 3a). UV irradiation will induce peroxynitrite generation from P-NAP and additionally produce ROS, thus providing both ROS and RNS simultaneously, mimicking closely pathophysiological conditions. Signals corresponding to nitro-cLa (cLaNO 2 ; m/z 324.22) and dinitro-cLa [cLa(NO 2 ) 2 ; m/z 369.20] confirmed that P-NAP triggers free-radical-mediated nitration of cLa. Although La(NO 2 ) 2 is present with considerably lower relative signal intensities than after nitration of La by NO 2 BF 4 , the formation of La(NO 2 ) 2 by P-NAP suggests that multiple nitration derivatives can be formed via reaction with •NO 2 . Free-radical-mediated nitration of cLa under high oxygen tension yielded additionally four oxidation products-namely, keto-cLa (cLaO; m/z 293.21), hydroxy-cLa (cLaOH; m/z 295.23), hydroperoxy-cLa (cLaOOH; m/z 311.22), and hydroperoxyoctadecenoic acid (OaOOH; m/z 313.23). Among all the oxidation products, these showed the highest relative intensities. Moreover, several derivatives of cLa resulted from simultaneous oxidation and nitration, including hydroxynitro-cLa (cLaNO 2 OH; m/z 340.21), hydroxynitrooctadecenoic acid (OaNO 2 OH; m/z 342.23), hydroperoxynitrooctadecaenoic acid (OaNO 2 OOH; m/z 358.22), and hydroxydinitro-cLa [cLa(NO 2 ) 2 OH; m/z 385.20]. Additionally, nitroso derivatives were present after free-radical-mediated cLa nitration-namely, nitroso-cLa (cLaNO; m/z 308.22) and nitronitrosooctadecenoic acid (OaNO 2 NO; m/z 355.22)-indicating that nitroso derivatives are most likely formed from nitro fatty acids, independently of the fatty acid nitration mechanism. cLa nitration mixtures were used for further optimization of the MRM method (Table S1).

The kinetics of nitration of cLa by P-NAP was monitored using specific MRM transitions of four oxidation, four nitration, and four nitro-oxidation products (Fig. 3b). UV irradiation of P-NAP readily releases ONOO -with tenfold increased levels within 15 min [START_REF] Ieda | Photocontrollable peroxynitrite generator based on N-methyl-Nnitrosoaminophenol for cellular application[END_REF]. Thus, cLa nitration was monitored for shorter time intervals within the first 15 min (0, 3, 6, 9, and 15 min) and then for 30, 60, and 120 min of the nitration reaction. Conditions of higher oxygen tension favored fatty acid oxidation compared with the hydrophobic conditions of NO 2 BF 4 nitration, yielding more oxidized cLaNO 2 derivatives. The concentrations of cLaOOH, cLaOH, cLaO, and OaOOH increased over the whole reaction period. Two main mechanisms have been proposed for free-radical-induced nitration of fatty acids [START_REF] Woodcock | Nitrated fatty acids: synthesis and measurement[END_REF]. One suggests the abstraction of a bisallylic hydrogen followed by the reaction with •NO 2 , producing vinylnitro isomers. The second mechanism predicts a homolytic attack of •NO 2 at the C=C-bond, yielding a βnitroalkyl radical that can lose a hydrogen atom and form nitroalkenes. Owing to the absence of bisallylic hydrogens in cLaNO 2 , this product is most likely formed by the second mechanism. The kinetics of cLaNO 2 formation showed that maximal amounts of cLaNO 2 were reached after 6 min and then the amounts decreased rapidly until 15 min, reaching stable levels after 60 min. Indeed, this fast decrease between 6 and 15 min can be explained by the homolytic attack of a second •NO 2 responsible for the increased levels of cLa(NO 2 ) 2 (maximum at 15 min). Similarly, cLaNO 2 can also react with •OH to form cLaNO 2 OH and cLa(NO 2 ) 2 OH. Additionally, nitroalkyl radicals formed by the homolytic attack of a second •NO 2 can produce a nitronitrite derivative, hydrolysis of which can further result in hydroxynitroalkenes, such as OaNO 2 OH and OaNO 2 OOH. The formation rates of cLaNO 2 OH and OaNO 2 OH are clearly different. Whereas the OaNO 2 OH levels increase fast within the first 6 min, reaching relatively stable values, the cLaNO 2 OH and cLa(NO 2 ) 2 OH levels increase gradually over the full reaction period. The possibility that oxidized cLa is nitrated was also considered. The half-lives of •OH and •OOH (10 -9 s) [START_REF] Halliwell | Free radicals in biology and medicine, 3rd edn[END_REF] are significantly shorter than the half-life of •NO 2 (10 -5 -10 -4 s) [START_REF] Huie | The reaction kinetics of NO 2[END_REF]. Additionally, the reaction rates of an olefinic double bond with •OH and •OOH (10 8 M -1 s -1 ) [START_REF] Halliwell | Free radicals in biology and medicine, 3rd edn[END_REF] are much faster than with •NO 2 (10 6 -10 7 M -1 s -1 ) [START_REF] Huie | The reaction kinetics of NO 2[END_REF]. Thus, it appears very likely that oxidation occurs first and nitration follows. Both processes can produce cLaNO 2 OH and cLa(NO 2 ) 2 , although their individual contributions remain to be defined. Hydroperoxy and hydroxy derivatives of NO 2 -FA accumulate over time, and thus might provide a better Bchemical fingerprint^of nitrosative stress than NO 2 -FA itself. Finally, we observed formation of nitroso derivatives, including cLaNO and OaNO 2 NO, reaching the highest concentrations after 30 min before decreasing afterward.

The kinetics of electrophilic (NO 2

+

) and free radical (•NO 2 ) driven fatty acid nitrations are determined by the reaction conditions. Nevertheless, both electrophilic and free-radicaldriven nitration of fatty acids successfully produced a wide array of products. Fatty acids nitrated with NO 2 + under low oxygen tension yielded low levels of oxidized fatty acids, whereas high oxygen tension continuously produced oxidized fatty acids. Similar kinetics were observed for singly nitrated fatty acids for both nitration reactions, indicating that regardless of the nature of the RNS and the chemical environment, NO 2 -FA do not represent the final products, but are prone to further nitration and oxidation reactions.

NO 2 -FA in the cardiomyocyte nitrosative stress model

Following successful application of the new MRM method for simultaneous detection of nitrated and nitro-oxidized PUFA in kinetics experiments, we used this approach for relative quantification of modified PUFA in a cellular model of nitrosative stress using LC-MS coupling. Retention-time mapping was performed using each in vitro fatty acid nitration mixture separately and in combinations. Oxidation products with the same elemental composition (same m/z values) derived from different precursors, such as OaO and LaOH (m/z 295.2) have different functional groups (keto and hydroxy), and reversedphase high-performance LC separation of these fatty acids results in different elution times of isobaric analytes, which allowed us to perform structure-specific retention-time mapping. Rat primary cardiomyocytes were treated with the peroxynitrite donor SIN-1 (10 μM) for 15, 30, and 70 min, and the levels of modified PUFA in lipid extracts were compared with those in untreated cells. The most sensitive MRM transitions optimized for each modified fatty acid were used (Table S1). Under the CID conditions, nitro fatty acids exhibit strong neutral loss of nitrous acid. Thus, for each nitration and oxidation/nitration derivative, the product ion of deprotonated nitrous acid (NO 2 -; m/z 46), and the product ions of single or multiple neutral losses of HNO 2 were monitored. These transitions were, without exception, more sensitive than any isomer-specific ones. Moreover, a fast reversed-phase LC gradient was designed, providing higher sensitivities owing to co-elution of different isomers. The specificity of the LC-MS method was ensured by compound-specific retention-time mapping using mixtures of in vitro nitrated fatty acids. Thus, identification of the free and modified fatty acids in the cell model was based on the retention time and quadrupole 1/quadrupole 3 specific transitions for each compound.

The stability of endogenous NO 2 -FA is largely unknown. Owing to the absence of authentic isotopically labeled nitro fatty acid standards, we were unable to estimate compound stability or determine the limits of detection. Although artificial oxidation cannot be excluded, it is generally believed that the MTBE-methanol hydrophobic environment used here for lipid extraction in combination with low temperatures (4 °C) minimizes artificial oxidation of lipids. Nitro fatty acids were shown to be stabile under hydrophobic conditions, and organic extractions offer the highest recoveries of nitro fatty acids from samples [START_REF] Woodcock | Nitrated fatty acids: synthesis and measurement[END_REF]. Work by Baker et al. [START_REF] Baker | Fatty acid transduction of nitric oxide signaling: multiple nitrated unsaturated fatty acid derivatives exist in human blood and urine and serve as endogenous peroxisome proliferator-activated receptor ligands[END_REF] indicates the level of OaNO 2 decays by 10 % when it is stored in methanol for 1 month, while 40 % loss was observed in phosphate buffer after 2 h. LaNO 2 showed higher instability, probably owing to the higher unsaturation degree. This indicates that AaNO 2 and DhaNO 2 might decay even faster than OaNO 2 and LaNO 2 .

Apart from Oa, La, Aa and Dha, we detected and relatively quantified four oxidized fatty acids (OaOH, LaOH, LaOOH, AaOH), four NO 2 -FA [Oa(NO 2 ) 2 , AaNO 2 , DhaNO 2 , Dha(NO 2 ) 2 ] and two oxidized/nitrated fatty acids [DhaNO 2 OH, Dpa(NO 2 ) 2 OH] (Table 2, Figs. 4,S9). Induction of stress in the cardiomyocytes resulted in an increase in free fatty acid concentrations. After 15 min from stress onset, the levels of free La and Dha mildly increased (161 % and 155 % relative to the control for La and Dha, respectively; see Fig. S9), whereas approximately doubled levels of Oa and Aa (193 % for Oa; 183 % for Aa) were detected. An increase in the concentration of free fatty acids can be connected to elevated activity of phospholipase A 2 (PLA 2 ). Several studies showed that as a response to cell membrane lipid peroxidation, catalytic activity of PLA 2 is enhanced, leading to increases in the levels of free fatty acids and lysophospholipids [START_REF] Jezek | Mitochondrial phospholipase A2 activated by reactive oxygen species in heart mitochondria induces mild uncoupling[END_REF][START_REF] Rashba-Step | Phospholipid peroxidation induces cytosolic phospholipase A2 activity: membrane effects versus enzyme phosphorylation[END_REF]. An increase in the levels of oxidized fatty acids in the first 30 min from stress onset was observed for La (168 % and 150 % for LaOH and LaOOH, respectively) and Aa (169 % for AaOH), whereas the highest increase was detected for OaOH (234 %). The increases in the levels of OaOH, LaOH, LaOOH, and AaOH in the first 30 min can be accounted for by direct PUFA oxidation as well as increased activities of PLA 2 that could lead to the hydrolysis of hydroxylated and peroxidized fatty acids from phospholipids.

The relatively fast LC gradient resulted in co-elution of the single-nitro fatty acid derivatives, whereas multiply nitrated fatty acids and nitro-oxidized fatty acids appeared as multiple signals in the in vitro nitration mixtures. In the SIN-1-treated cardiomyocytes, the Oa(NO 2 ) 2 -specific transition resulted in multiple signals at 5.82, 7.47, 7.98, 8.72, and 9.49 min (Fig. 4a). Using the retention-time mapping based on the in vitro NO 2 + nitration of Oa, we were able to assign the peaks at 7.47 and 7.98 min to two Oa(NO 2 ) 2 isomers. After SIN-1 treatment, the levels of Oa(NO 2 ) 2 increased twofold compared with the levels in the untreated cells, reaching the maximal levels after 30 min from stress onset (199 %; Fig. 4a) and remained relatively stable after 70 min. AaNO 2 (retention time 8.33 min; Fig. 4b) and DhaNO 2 (retention time 8.53 min; Fig. 4c) reached maximal values after 30 min (148 % for AaNO 2 ; 230 % for DhaNO 2 ) of peroxynitrite treatment, followed by a mild decrease in the amounts detected.

Four signals (6.99, 7.65, 8.09, and 8.56 min) were detected for the Dha(NO 2 ) 2 -specific transition for in vitro nitrated Dha (Fig. 4e). In the cardiomyocytes treated with SIN-1 for 15 min, two Dha(NO 2 ) 2 peaks with similar intensities were observed at 8.20 and 8.66 min, and were assigned to two different Dha(NO 2 ) 2 isomers. The levels of Dha(NO 2 ) 2 were significantly increased in the cardiomyocytes under the stress conditions compared with untreated cells, reaching the maximum at 30 min (371 %; Fig. 4e).

For in vitro nitrated Dha, multiple signals of DhaNO 2 OH (6.47, 7.23, and 8.09 min; Fig. 2d) and Dpa(NO 2 ) 2 OH (6.22, 6.86, 7.54, 8.17, and 9.50 min; Fig. 4f) were detected. However, in cardiomyocyte lipid extracts, only the single peak for each compound was present. Thus, DhaNO 2 OH was eluted at 7.30 min and Dpa(NO 2 ) 2 OH was eluted at 8.05 min. The concentration of DhaNO 2 OH in SIN-1-treated cells mildly increased within the first 15 min (131 %; Fig. 4d) and remained stable afterward. On the other hand, the levels of Dpa(NO 2 ) 2 OH significantly increased (345 %; Fig. 4f) at 15 min, decreased at 30 min (210 %), and again increased at 70 min (369 %).

Overall, treatment of primary cardiomyocytes with the peroxynitrite donor resulted in increased amounts of oxidized fatty acids (approximately 150 % compared with the untreated cells). The singly nitrated fatty acids-namely, AaNO 2 and DhaNO 2 -were detected with maximal levels at 30 min, followed by a mild concentration decrease at 70 min. The relative increase in the level of DhaNO 2 (250 %) at 30 min of SIN-1 treatment was approximately three times higher than that of AaNO 2 (148 %). The levels of Oa(NO 2 ) 2 doubled after 30 min (199 %) and remained relatively stable afterward. Thus SIN-1 treatment of cardiomyocytes had the highest impact on the Dha nitration, resulting not only in pure nitration, but also in increased levels of nitration/oxidation derivatives. The highest and the fastest concentration increase of Dpa(NO 2 ) 2 OH (245 %) and Dha(NO 2 ) 2 (232 %) in SIN-1treated cells compared with the untreated cells was detected after 15 min of stress induction. These results indicate that multiply nitrated and nitro-oxidized derivatives of fatty acids, including Dha, are produced in cells even under basal conditions, and the levels are significantly elevated in the peroxynitrite stress model. To the best of our knowledge, this is the first report of the endogenous formation of AaNO 2 , and Dha nitration products such as DhaNO 2 , Dha(NO 2 ) 2 , DhaNO 2 OH, and Dpa(NO 2 ) 2 OH. The presence of the endogenously formed NO 2 -FA was reported for the first time more than a decade ago [START_REF] Lima | Characterization of linoleic acid nitration in human blood plasma by mass spectrometry[END_REF]. Several LC-MS/MS and gas chromatography-MS/MS studies have been directed toward quantification of LaNO 2 and OaNO 2 in human blood plasma, and reported values between 300 pmol/L [START_REF] Lima | Characterization of linoleic acid nitration in human blood plasma by mass spectrometry[END_REF][START_REF] Tsikas | Nitro-fatty acids occur in human plasma in the picomolar range: a targeted nitro-lipidomics GC-MS/MS study[END_REF][START_REF] Trettin | GC-MS/MS and LC-MS/MS studies on unlabelled and deuterium-labelled oleic acid (C18:1) reactions with peroxynitrite (O=N-O-O(-)) in buffer and hemolysate support the pM/nM-range of nitro-oleic acids in human plasma[END_REF][START_REF] Tsikas | Specific GC-MS/MS stable-isotope dilution methodology for free 9-and 10-nitro-oleic acid in human plasma challenges previous LC-MS/MS reports[END_REF] and 600 nmol/L [START_REF] Baker | Red cell membrane and plasma linoleic acid nitration products: synthesis, clinical identification, and quantitation[END_REF][START_REF] Baker | Fatty acid transduction of nitric oxide signaling: multiple nitrated unsaturated fatty acid derivatives exist in human blood and urine and serve as endogenous peroxisome proliferator-activated receptor ligands[END_REF], whereas AaNO 2 remained undetected [START_REF] Tsikas | Nitro-fatty acids occur in human plasma in the picomolar range: a targeted nitro-lipidomics GC-MS/MS study[END_REF]. Recently, Bonnaci et al. [START_REF] Bonacci | Conjugated linoleic acid is a preferential substrate for fatty acid nitration[END_REF] showed that dietary cLa rather than bisallylic La is a preferential substrate for the nitration by ONOO -and • NO 2 , with endogenous levels 0.3 to 1.3 nmol/L in healthy human plasma. In the present study, we did not observe detectable levels of LaNO 2 , cLaNO 2 , and OaNO 2 in control and SIN-1treated rat cardiomyocytes, most probably owing to the low concentrations and/or absence of cLa as a nitration substrate in the cell culture medium. Additionally, nitroalkylation of proteins and glutathione by the electrophilic NO 2 -FA and fatty acid metabolism can reduce endogenous levels of free NO 2 -FA [START_REF] Tsikas | Oxidized and nitrated oleic acid in biological systems: analysis by GC-MS/MS and LC-MS/ MS, and biological significance[END_REF]. Over the past two decades various techniques have been designed for the enrichment of the low endogenous concentrations of Aa-derived F 2 -isporostanes (approximately 10 2 pmol/L in healthy human plasma) from various biological samples [START_REF] Milne | F2-isoprostanes as markers of oxidative stress in vivo: an overview[END_REF], and now ensure their routine detection and quantification by LC-MS. Biological concentrations of isoprostanes are similar to those of NO 2 -FA in healthy human plasma; thus, further development of NO 2 -FA enrichment protocols could improve their detection in biological samples. In the present study, Dha-derived nitration products were detected for the first time. Although F 4 -neuroprostanes derived from Dha, like Aa derived F 2 -isoprostanes, have been related to various cardiovascular and neurodegenerative disorders [START_REF] Montuschi | Isoprostanes: markers and mediators of oxidative stress[END_REF][START_REF] Ricciotti | Prostaglandins and Inflammation[END_REF], here we can only speculate about the beneficial effects of Dha-derived nitration products, as was shown for AaNO 2 [START_REF] Gonzalez-Perilli | Nitroarachidonic acid prevents NADPH oxidase assembly and superoxide radical production in activated macrophages[END_REF][START_REF] Trostchansky | Synthesis, isomer characterization, and antiinflammatory properties of nitroarachidonate[END_REF][START_REF] Bonilla | Regulation of protein kinase C by nitroarachidonic acid: impact on human platelet activation[END_REF] and all other singly nitrated fatty acids. Lipid peroxidation products usually posses proinflammatory properties [START_REF] Lee | Role of lipid peroxidation-derived α, ßunsaturated aldehydes in vascular dysfunction[END_REF]; however current data on biological activities of NO 2 -FA support their anti-inflammatory effects via both cGMPdependent and cGMP-independent mechanisms. Although the overproduction of NO 2 -FA was observed during inflammation, a series of anti-inflammatory effects were reported for LaNO 2 [START_REF] Lim | Nitrolinoleate, a nitric oxide-derived mediator of cell function: synthesis, characterization, and vasomotor activity[END_REF][START_REF] Coles | Nitrolinoleate inhibits platelet activation by attenuating calcium mobilization and inducing phosphorylation of vasodilatorstimulated phosphoprotein through elevation of cAMP[END_REF][50] , OaNO 2 [START_REF] Baker | Fatty acid transduction of nitric oxide signaling: multiple nitrated unsaturated fatty acid derivatives exist in human blood and urine and serve as endogenous peroxisome proliferator-activated receptor ligands[END_REF][START_REF] Kansanen | Electrophilic nitrofatty acids activate NRF2 by a KEAP1 cysteine 151-independent mechanism[END_REF], and AaNO 2 [START_REF] Gonzalez-Perilli | Nitroarachidonic acid prevents NADPH oxidase assembly and superoxide radical production in activated macrophages[END_REF][START_REF] Trostchansky | Synthesis, isomer characterization, and antiinflammatory properties of nitroarachidonate[END_REF][START_REF] Bonilla | Regulation of protein kinase C by nitroarachidonic acid: impact on human platelet activation[END_REF]. The question about biological activities of multiply nitrated and nitro-oxidized PUFA remains open owing to the lack of the data on their biological availability.

Conclusion

A novel analytical approach to explore the diversity of NO 2 -FA was developed using in vitro nitration mixtures of Oa, La, Aa, and Dha for optimization of highly specific and sensitive targeted LC-ESI-MS/MS. Specific MRM transitions were optimized for 42 reaction products-that is, singly and multiply nitrated as well as nitro-oxidized fatty acids. The optimized methods were applied for the detection and relative quantification of NO 2 -FA in a cardiomyocyte model of nitrosative stress. Six fatty acid nitration products-namely, Oa(NO 2 ) 2 , AaNO 2 , DhaNO 2 , Dha(NO 2 ) 2 , DhaNO 2 OH, and Dpa(NO 2 ) 2 OH-were detected and quantified over a period of 70 min after stress induction, providing the kinetics of NO 2 -FA formed in cells during nitrosative stress.

  24) and nitrosodinitro-Oa [Oa(NO 2 ) 2 NO, m/z 398.19]. The signal at m/z 342.23 may indicate hydroxynitro-Oa (OaNO 2 OH), an oxidized nitro species. Peaks corresponding to nitrooctadecadienoic acid [La(NO 2 ) 2 , m/z 369.20] and nitrosonitrooctadecadienoic acid (LaNO 2 NO; m/z 353.21) showed that Oa nitration can yield products containing two double bonds, such as octadecadienoic acid.

Fig. 1

 1 Fig.1Electrospray ionization (ESI) Orbitrap mass spectrometry spectra recorded after 1 h of oleic acid (Oa; a), linoleic acid (La; b), arachidonic acid (Aa; c) and docosahexaenoic acid (Dha; d) nitration with nitronium tetrafluoroborate

Fig. 3

 3 Fig. 3 (a) ESI Orbitrap mass spectrum of conjugated La (cLa) incubated with the peroxynitrite donor 2,3,5,6-tetramethyl-4-(methylnitrosoamino)phenol for 15 min. (b) Kinetics of cLa nitration monitored by multipole reaction monitoring (MRM) optimized for 12 cLa derivatives for 2 h

Fig. 4

 4 Fig. 4 Extracted ion chromatograms for the quadrupole 1/quadrupole 3 transition pairs corresponding to a Oa(NO 2 ) 2 , b AaNO 2 , c DhaNO 2 , d DhaNO 2 OH, e Dha(NO 2 ) 2 , and f Dpa(NO 2 ) 2 OH in cardiomyocyte (CM) lipid extract (upper panels) and corresponding polyunsaturated fatty acids in in vitro nitration mixtures (lower panels). Identification of modified fatty acids was based on the compound-specific MRM transitions and retention times. Measured values were normalized to the values for the

Table 1

 1 Summary of NO 2 BF 4 -and 2,3,5,6-tetramethyl-4-(methylnitrosoamino)phenol-mediated nitration products of oleic acid (Oa), linoleic acid (La), arachidonic acid (Aa), docosahexaenoic acid (Dha), and conjugated La (cLa). The elemental composition and the m/z values for deprotonated ions are shown Fatty acid Oxidation derivativesLa was observed at m/z 279.23, indicating that La was not fully consumed within 1 h in the reaction with NO 2 + . Signals corresponding to LaNO 2 (m/z 324.22), dinitro-La [La(NO 2 ) 2 ; m/z 369.20], trinitro-La [La(NO 2 ) 3 ; m/z 414.19], and tetranitro-La [La(NO 2 ) 4 ; m/z 459.17] suggested that the two double bonds allowed incorporation of up to four nitro groups. Unlike Oa nitration, La nitration yielded only one nitroso product, observed at m/z 308.22 (LaNO). Similarly to Oa nitration, hydroxynitro-La (LaNO 2 OH; m/z 340.21) and hydroxynitrooctadecenoic acid (OaNO 2 OH; m/z 342.23) were also detected.

	Nitration derivatives

x

(NO 2 ) x (NO) x (NO 2 ) y (NO 2 )

x (OH) y Oa (18:1) C

  and m/z 327.23, respectively), and the relative signal intensities of all nitration products were below 10 %. This observation indicated significantly lower susceptibilities of Aa and Dha to NO 2 + -mediated nitration compared with Oa and La. Both Aa and Dha were oxidized to keto derivatives (m/z 317.23, AaO; m/z 341.21, DhaO) and hydroxy derivatives (m/z 319.23, AaOH; m/z 343.23, DhaOH). Mass spectra of both Aa and Dha nitration mixtures revealed the presence of nitro derivatives (m/z 348.22, AaNO 2 ; m/z 372.22, DhaNO 2 ), dinitro derivatives [m/z 393.20, Aa(NO 2 ) 2 ; m/z 417.20, Dha(NO 2 ) 2 ], and trinitro derivatives [m/z 438.19, Aa(NO 2 ) 3 ; m/z 462.19, Dha(NO 2 ) 3 ]. Additionally, nitroso-Aa (AaNO; m/z 332.22), dinitro-Aa [Aa(NO) 2 ; m/z 361.22], and nitrosonitro-Aa (AaNO 2 NO, m/z 377.21) as well as nitrosonitro-Dha (DhaNO 2 NO, m/z 401.21) and nitrosodinitro-Dha [Dha(NO 2 ) 2 NO, m/z 446.19] were detected. Weak signals were observed for nitro-oxidized Aa and Dha-namely, hydroxynitro-Aa (AaNO 2 OH; m/z 364.21), hydroxynitro-Dha (DhaNO 2 OH; m/z 388.21), and hydroxydinitrodocosapentaenoic acid [Dpa(NO 2 ) 2 OH; m/z 435.18,].

Table 2

 2 Relative quantification of unmodified, oxidized, nitrated, and nitro-oxidized fatty acids detected after 15, 30, and 70 min of 3morpholinosydnonimine treatment in rat primary cardiomyocytes shown as a percentage relative to the control (control=100 %)

	Profiling and relative quantification of multiply nitrated
	Class	Analyte	15 min		30 min		70 min
			Percentage relative	p	Percentage relative	p	Percentage relative	p
			to control±SD		to control±SD		to control±SD
	Fatty acid	Oa	193±16	0.014	198±85	0.184	276±60	0.049
		La	161±22	0.012	184±18	0.0007	208±33	0.005
		Aa	183±30	0.012	171±22	0.006	171±27	0.015
		Dha	155±17	0.006	161±18	0.005	174±20	0.003
	Oxidized fatty acid	OaOH	200±46	0.047	234±61	0.046	226±35	0.011
		LaOH	147±15	0.009	168±31	0.033	1,470±53	0.205
		LaOOH	117±3	0.011	150±14	0.036	164±64	0.221
		AaOH	163±12	0.0005	169±12	0.0002	157±21	0.014
	Nitrated fatty acid	Oa(NO 2 ) 2	173±41	0.075	199±23	0.006	228±78	0.095
		AaNO 2	141±43	0.218	148±6	0.007	115±28	0.323
		DhaNO 2	174±45	0.121	230±32	0.028	190±55	0.122
		Dha(NO 2 ) 2	332±5	0.0002	371±40	0.011	226±22	0.014
	Nitro-oxidized fatty acid	DhaNO 2 OH	131±24	0.118	134±16	0.030	131±26	0.133
		Dpa(NO 2 ) 2 OH	345±33	0.009	210±28	0.031	369±54	0.019
	SD standard deviation

-(m/

z385.1975)

Acknowledgments The authors are grateful to Ralf Hoffmann (Institute of Bioanalytical Chemistry, University of Leipzig) for providing access to his laboratories and instruments. Financial support from the European Regional Development Fund (European Union and the Free State of Saxony; 100146238 and 100121468 to M.F.) and a stipend to I.M. provided by Universität Leipzig are gratefully acknowledged.

Authors' contributions I.M. performed all experimental work, corresponding data evaluation, and contributed to the writing of the manuscript. E.G. and V.V. designed the cell model of nitrosative stress and performed cell culture experiments. N.I., H.N., and N.M. provided the peroxynitrite donor 2,3,5,6-tetramethyl-4-(methylnitrosoamino)phenol. J.M.G., C.O., and T.D. provided internal standards used for liquid chromatography-tandem mass pectrometry. M.F. conceived and designed all experiments and contributed to the writing of the manuscript. Thierry Durand is a group leader at the Institute of Biomolécules Max Mousseron. His research interests include the total synthesis of oxygenated cyclic and noncyclic metabolites of polyunsaturated fatty acids, mainly leukotrienes, isoprostanes, phytoprostanes, and neuroprostanes, as well as dihydroxylated polyunsaturated fatty acids, and the understanding of the role of such bioactive lipids by developing collaborations with chemists, biochemists, biologists, and clinicians all over the word.

Coles B, Bloodsworth

Maria Fedorova is a group leader in the Institute of Bioanalytical Chemistry, Faculty of Chemistry and Mineralogy, at the University of Leipzig. Her research is focused on analysis of oxidized lipids and proteins as well as lipid-protein adducts in variety of in vitro and in vivo systems.