Francis Berthias

Yali Wang

Eskander Alhajji

Bernard Rieul

Fathi Moussa

Jean-François Benoist

Philippe Maître

Identification and quantification of amino acids and related compounds based on Differential Mobility Spectrometry

des établissements d'enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

Introduction

Metabolomics-based strategies are routinely used for modern clinical research, allowing for more adequate diagnosis and prognosis. 1,2 Quantitative and robust methods 3 are required for the analysis of biomarkers of metabolic diseases (MDs) involved in a complex network of metabolic pathways. Amino acids (AAs) and related compounds constitute a class of biomarkers which is routinely analyzed for early diagnosis of MDs. 4 Separation of these metabolites within, for example, urine or blood samples, has been initially performed by high-performance liquid chromatography (HPLC), which was coupled to spectrophotometric detection. 5 For this detection purpose, postcolumn and precolumn derivatization had to be carried out, leading to time-consuming methods. Nowadays, metabolomics profiling is based on chromatographic separation coupled with tandem mass spectrometry (MS/MS). 6 Nuclear magnetic resonance (NMR) is also an important tool, but the main advantage of MS/MS is its sensitivity that requires a minimal amount of biological material. 7 Identification of the metabolites at their naturally occurring physiological concentrations is routinely achieved by comparing their fragmentation spectra with standard reference spectra. 7 Some major drawbacks of the current methods, such as time-consuming procedures, poor reproducibility of the derivatization, and problems of retention have been shown to be overcome using hydrophilic interaction liquid chromatography (HILIC) for the separation. Rapid quantification of underivatized AAs in plasma using a HILIC column, 8 and more recently mixed mode stationary phase and a binary gradient of elution, 9 coupled with tandem massspectrometry have recently been reported. Despite these improvements providing high degree of selectivity, separation and identification of isomeric metabolites remains a difficult task. 2 Additional or alternative dimensions of separation could thus be of interest for the separation of isobaric and isomeric metabolites. [10][11][12][13] Ion mobility spectrometry (IMS) is an interesting alternative for the separation of small molecular weight ions such as ionized metabolites. 14 Both time-and space-type IMS separation such as drift-time IMS (DT-IMS) and differential mobility spectrometry (DMS, also often referred to as Field Asymmetric Ion Mobility Spectrometry (FAIMS)), 15 respectively, have been successfully used for the separation of metabolites. DT-IMS a. Université Paris-Saclay, CNRS, Institut de Chimie Physique, 91405, Orsay, France. b. Université Paris-Saclay, Lipides, Systèmes Analytiques et Biologiques, 92296, Châtenay-Malabry, France. c. Service de biochimie hormonologie, Hôpital Universitaire Robert Debré, APHP, 48 Bd Serrurier, 75019 Paris. Electronic Supplementary Information (ESI) available: composition of standard mixture; concentrations of AAs and related compounds in plasmas; composition and concentrations of calibration solutions; compensation voltage and collision cross section values for a set of AAs and related compounds; Limit of detection values; Day-to-day confidence interval on CV values. See DOI: 10.1039/x0xx00000x

Please do not adjust margins Please do not adjust margins coupled to high resolution time of flight mass spectrometry has been employed for the profiling of human blood metabolome, and separation of 300 isomeric metabolites within various classes of metabolites could be achieved. 16 Time separation of IMS (millisecond) allows for the coupling of IMS-MS/MS to chromatographic separation. An LC-IMS-MS/MS protocol has been recently proposed for metabolomics and lipidomics. 17 Efforts toward separation of isobaric and isomeric species led to the development of high resolution instruments and/or methodologies. Within a decade, important progresses have been made with new types of instruments time- 18,19 and spacemobility 20,21 separation. Significant enhancement of the resolving power can also be achieved by introducing a fraction of He 22 or polar molecules 23 in the carrier gas, as found in particular for AAs. 24,25 Separation of diastereoisomeric chiral compounds such as AAs and peptides has been addressed recently. For the separation of D/L AAs, both derivatization 26 and formation of a proton bound dimers 27 with a chiral agent have been shown to be successful. Mobility separation of peptides with 4-29 residues including a single D-AA at different residues and locations has recently been reported. 28 High resolution DMS has also been recently successfully employed for the separation of isobaric peptides containing one or two methylated arginine residues. 29 Both space-and time-separation IMS approaches allow for a reduction of the signal-to-noise ratio. In addition, it has been shown that the number of false positive assignments can be reduced using DMS. 30 DMS has also been shown to be more orthogonal to MS/MS than DT-IMS or related time-separation methods, in the case of peptides, for example. 20,31 Coupled to MS/MS, it was shown that the coverage achieved with the DMS is comparable to that achieved with the chromatographic methods. 32 DMS hyphenated to MS/MS has thus been used for analysis of complex biological fluids such as urine and plasma, and especially for the separation of co-eluting isomers. 10,24 DMS-MS/MS workflows for lipid analysis has also been shown to overcome some limitation of LC-MS/MS such as low chromatographic resolution and the convolution of mass spectra from isomeric and isobaric species. 12 Nevertheless, "despite several success stories concerning DMS in metabolomics, the application range of this technique is not fully understood. Systematic studies investigating the metabolome selectivity of DMS are required". 32 Our aim is to evaluate the analytical performance of DMS hyphenated to MS/MS for the analysis of AAs and related compounds. More precisely, we will focus on the capacity of DMS for providing an additional metabolite identifier. As recently discussed by Barran and coworkers, 33 this idea has been pursued using DT-IMS approaches, and it had been shown that mobility drift-time, 34 or even collisional cross sections (CCS), 35,36 could be used for identification of pesticides. This idea of using CCS as identifier was further pursued for metabolites. [37][38][39][40] Using DMS, the additional identification parameter would be the so-called compensation voltage (CV) which has to be tuned for the transmission of a specific metabolite ion.

We first report on the analysis of the 20 common AAs, and the performance in terms of peak-to-peak resolution and peak capacity is compared to those using drift tube type IMS instruments, [13][14][15][16][17][18] in particular for the separation of isobars and isomers. Potential effects of matrix or concentration of the analytes on the CV values are also evaluated, and found to be negligible. An interesting probe of the robustness of the method was also provided by the stability of the CV position for series of DMS-MS experiments carried over 6 months, during which the DMS device was often removed and replaced. Isotopically labelled internal standards have been used for the quantification 41 of a selected sets of metabolites in human plasmas. The method linearity, intra-assay and inter-assay precision, detection limit, quantification limit and trueness analysis were found to be adequate in both physiological and pathological conditions. Concentration levels of metabolites derived with our DMS-MS protocols were found to be in good agreement with those obtained with routine LC-MS/MS protocols used for the diagnosis of MDs at the Hospital Robert Debré (Paris).

Experimental methods

Reagents. Methanol and water were used as solvent (50:50, vol:vol) and acidified with formic acid (0.5%). Methanol was purchased from Merck KGaA (Darmstadt, Germany) and formic acid from Fisher Scientific (Hampton, New Hampshire, USA), both are reagent grade. Deionized water is obtained using Millipore Direct-Q 3 UV (Lilipore corporation, Burlington, Massachusetts, USA). Certified graduated pipettes were used throughout for the sample preparation. Mixture of reference standards. A standard mixture made of 33 AAs and related compounds as listed in Table S1 (left column) was used for calibration. This mixture is based on a commercial AA standard solution (Type B, AN-II purchased from FUJIFILM Wako Pure Chemical Industries, Osaka, Japan). It was complemented by glutamine (Gln), asparagine (Asn), tryptophan (Trp), pipecolic acid, alloisoleucine (Ail), and sulfocysteine which were purchased from Sigma-Aldrich (St. Louis, Missouri, USA). Calibration curves were obtained using 6 calibration concentration levels in plasma. Mixture of Internal standards. A mixture of stable isotope internal standards (IS) was made using reference compounds from Eurisitop (Saint Aubin, France). The list of these IS as well as their final concentrations in the samples is given in Table S1 (right column). This IS mixture was used for the calibration and quantification protocols. Samples and subjects. Blood samples were obtained from patients suspected of MD based on routine diagnostic. Blood was collected by venous puncture into heparin containing tubes. In practice, blood was centrifuged rapidly after collection. Plasma was subsequently deproteinized by adding sulfosalicylic acid 30% (10:1 v:v) and then stored at -20°C. Just before analysis, samples were diluted in acidified water:methanol (50:50 v:v) solution, and 20-fold diluted plasma samples were used for electrospray injection.

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Please do not adjust margins Four plasma samples (P1, P2, P3, and P4) not specifically collected for validation of this study were analyzed anonymously. The quality control P1 plasma is a pool of plasmas taken from healthy volunteers. Plasma P2 and P3 were collected from patients with Phenylketonuria and Tyrosinemia, respectively. P4 is a random plasma. Plasma samples were collected for diagnosis purpose as part of the patient's routine care. Informed consent was obtained from patients for the use of leftover samples for analytical developments. The study was approved by the ethics committee of Robert Debré University Hospital (Paris, France), and in compliance with French Public health regulations (Code de la Santé Publique -Article L1121-3, amended by Law n°2011-2012, December 29, 2011 -Article 5). Concentration of the AAs of interest in these plasmas has been quantified by a liquid chromatography tandem mass spectrometry (LC-MS/MS) method validated according to the ISO 15189 standards. These quantification experiments were performed in an accredited university hospital laboratory where MD diagnoses are routinely performed. In brief, after butylation, compounds are separated by reverse phase liquid chromatography, then ionized by electrospray in the positive mode and identified by multiple reaction monitoring (MRM). Concentrations in samples and corresponding uncertainties for the molecules are given in Table S2. Differential Mobility Spectrometry (DMS) coupled to MS/MS instrument. DMS-MS experiments were carried out using a modified Esquire 3000+ quadrupole ion trap mass spectrometer (Bruker-Daltonics, Bremen, Germany) featuring an ElectroSpray Ionization (ESI) source. 42 The setup is based on the one developed by Glish and coworkers. 43 Our home-built DMS device has been successfully used for the separation and identification of isomers of alanine 44 and aminobutyric acids. 45 The DMS housing is mounted between the ESI emitter and the glass transfer capillary of the mass spectrometer. It is composed of two parallel electrodes (0.7 × 6 × 20 mm for gap height, width, and length, respectively) between which ions are transported from the atmospheric pressure to a ~10 -3 mbar source region pressure of the mass spectrometer through the glass transfer capillary. In order to enhance the transmission of ions, the standard glass transfer capillary has been replaced with a custom flared glass transfer capillary. 33,34 The DMS cell is designed such as to replace the spray shield. The N 2 desolvation gas is redirected through the outer housing of the DMS device in such a way that part of the N 2 flow (~0.9 l.min -1 ) is used to transport the ions between the electrodes towards the glass capillary. An NIGen LCMS 40-1 (Claind, Tremezzina, Italy) N 2 generator was used to produce the N 2 gas with a purity up to 99.99%. Since ion mobility is very sensitive to the composition of the carrier gas, a custom built chamber has been built to accommodate the DMS device, and allowing to adjust the position of the ESI nebulizer needle with respect to the entrance of the DMS device. Identical positive ESI conditions have been used throughout. The voltage of the ESI emitter was set to 4.5 kV and the entrance glass transfer capillary was grounded. The flow rate of the sample was 40 µL.h -1 .

DMS allows for space separation of the ions, and the working principles can be found, for example, in a textbook [START_REF] Shvartsburg | Differential Mobility Spectrometry: Nonlinear Ion Transport and Fundamentals of FAIMS[END_REF] and a review. 15 In practice, a bisinusoidal waveform is applied between the electrodes thus creating an alternating electrical field perpendicularly to the direction of the carrier gas flow. This asymmetric radiofrequency is generated by applying a sinusoidal waveform at 1.7 MHz to one of the DMS electrodes. A second sinusoidal waveform at the second harmonic, phase shifted by approximately 90°, and at 50% of the amplitude of the first sinusoidal waveform is applied to the second electrode. The zero-to-peak voltage (V 0p ) is referred to dispersion voltage (DV). Ions are transmitted through the DMS device only if the transverse displacements under low and high electric field offset one another. Transmission of ions of interest can be achieved by applying a DC offset to one DMS electrode, referred to as compensation voltage (CV). DMS-filtered ions are then transmitted to the ion trap where they are accumulated before being subjected to MS/MS sequences. Considering the gap between the electrodes (0.7 mm), and assuming that the carrier gas is near atmospheric pressure and a temperature around 70°C (measured temperature of the electrodes using a thermal camera), the reduced electric field value corresponding to a voltage of 1800 V can be estimated to ~120 Td. In the following, the DMS spectra correspond to extracted ion chromatogram plotted as a function of the CV value, where a 1 V CV value corresponds to an electric field of 1.43 kV.m -1 or ~0.07 Td. DMS Spectra: Data processing. DMS spectra are generated by plotting extracted ion chromatograms (XIC) as a function of CV value, which is scanned using a 0.1 V increment. DMS peak centroid (CV) and full width at half-maximum (FWHM) are extracted by fitting each DMS spectrum using Gaussian functions. The fitting procedure is performed using a homebuilt software based on SciPy libraries. An optimization algorithm based on a sequential least squares programming (SLSQP) method is used for the Gaussian fits. DMS spectra of the components of the standard mixture and dispersion plots of individual AA. Unless otherwise stated, DMS spectra presented here were recorded with a DV value set to 1.8 kV. All the CV values corresponding to optimal transmission of a particular metabolite given in the text refer to these conditions. Dispersion plots, i.e. evolution of CV value for the optimal transmission of each AA as a function of the DV value, were recorded. DMS spectra were recorded at different DV values ranging from 600 to 1800 kV. Solutions of AAs were prepared in methanol:water solvent mixture acidified with formic acid (0.5%) at a final concentration of 100 µM. Calibration curve and protocol for quantification of metabolites in Plasma. The set of standards of AAs and related compounds (Table S1) was spiked into the P1 plasma at six dilution factors spanning 2 orders of magnitude of concentration (Table S3). Except for pipecolic acid and alloisoleucine, which are found at very low concentrations in plasma, the dilution levels were chosen so that the pathophysiological AA concentrations in analyzed plasma lie in the range of the spiked P1 solutions.

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In practice, the calibration solutions spiked in the P1 plasma sample at a final 1/20 dilution factor were prepared as follow. Following the deproteinization with sulfosalicylic acid, 13.8 µL of the supernatant was mixed with 25 µL of the IS mixture, and diluted in a water:methanol (50:50) solution containing standards and 1.25 µL of formic acid to reach a total volume of 250 µL. Linear calibration curve parameters were obtained from the plot of the AA/related IS peak area ratio versus the nominal AA concentration. The CV, FWHM, and DMS peak area values were derived from the above mentioned fitting procedure. Quantification of AAs and related compounds in the four plasma samples (P1-P4) was performed. The sample preparation protocol is similar to that used for the calibration. 13.8 µL of deproteinized plasma was added to 25 µL of IS mixture and diluted in acidified water:methanol to a final volume of 250 µL.

In order to check for potential carry-over, a blank run was systematically performed between two plasma runs. The concentrations of unknown samples were calculated from their corresponding area ratios using the corresponding calibration curve.

Results and discussion

The aim of this work is twofold. First, it is to evaluate whether the CV parameter could provide a means for the identification of metabolites. Our second goal is the absolute quantification of AAs and related compounds for targeted metabolomics analysis of plasma samples from healthy or MD patients. In this perspective, the performance of DMS for the separation of AAs is first evaluated, with a special focus on peak capacity and isobars/isomers resolution. Then, the compensation voltage value of the AAs and related compounds within a mixture of standards are determined, in both water:methanol solution and plasma. Based on the good robustness and repeatability of the measurements, the calibration results are then presented. Finally, the quantification of a set of AAs and related compounds relevant for the diagnosis of MDs is presented for four sets of plasma samples. DMS separation of the 20 common AAs. In order to probe the effect of the DV value on peak capacity and resolution, DMS spectra were recorded at DV values ranging from 0.6 kV to 1.8 kV. The set values of flow rate of the N 2 desolvation gas and its temperature are 6.5 L.min -1 and 220 °C, respectively. This was found to provide a good balance between DMS resolution and ion transmission. Results of separation are reported in Figure 1. DMS dispersion plots, i.e. the evolution of CV as a function of DV for each individual AA, are reported in Figure 1a. Three classes of AAs can be distinguished based on the evolution of CV as a function of DV (Figure 1a). These three types of behaviors have been observed for other systems [START_REF] Buryakov | [END_REF] and interpretations have been proposed 15,[START_REF] Shvartsburg | Differential Mobility Spectrometry: Nonlinear Ion Transport and Fundamentals of FAIMS[END_REF] based on the interactions between ions with the carrier gas. These three types of ions, often referred to as A, B, or C ions, are observed for AAs under the present DMS conditions. [START_REF] Buryakov | [END_REF] Protonated asparagine, aspartic acid, threonine, cysteine, proline, serine, alanine and glycine are associated to type A ions, for which ionmolecule clustering and declustering are supposed to occur at low and high electric field, respectively. Type C refers to ions with a decreasing mobility coefficient K as a function of the electric field strength. These ions are supposed to experience hard sphere type collisions with the carrier gas. Protonated arginine and tryptophan were found to be the only two systems of type C under our experimental conditions. For type B, the ion mobility initially increases, reaches a maximum and decreases for larger DV values. As a result, the DMS dispersion curve presents a minimum. It has recently been proposed that the minimum CV values can be correlated to ion-solvent binding energy. 48 Protonated lysine, histidine, phenylalanine, tyrosine, methionine, glutamine, isoleucine, leucine, glutamic acid and valine are type B ions. Note that some of these ions have been already found to behave as B type ions, such as protonated leucine, 49 and protonated glutamic acid. 50 DMS spectra of each individual AA for three DV values, 1.4, 1.6, and 1.8 kV, are given in Figures 1b-d, respectively. The width (FWHM) of the DMS peaks does not vary with the DV value. A nearly linear increase (R 2 =0.88) of the FWHM value with m/z is observed, the values for the AAs ranging from 0.52 (Gly) to 0.88 V (Trp). A plot of the CV values of the set of AAs, and also some related compounds which can be found in plasmas, as a function of m/z is given in Figure 2. Overall, the observed trend is quite similar to that observed for the collision cross section (CCS) values (Å 2 ) derived using time-separation IMS instrument. 17,33,38,51 With increasing m/z, the CV value increases from A-, to B-, and C-type protonated AAs ions.

Separation of AAs: DMS versus IMS peak capacity. Separation of protonated AAs has been investigated using space 24,25 and time separation based ion mobility instrument such as drifttube 52,53 or radio frequency-confining drift cell, i.e. with a Traveling Wave Ion Mobility Spectrometer (TWIMS). 51 In the Please do not adjust margins Please do not adjust margins latter study, the effect of the nature (He, Ar, N 2 , CO 2 , and N 2 O) of the drift gas on both peak-to-peak resolution (R pp ) and peak capacity (P c ) has been evaluated. R pp and P c are defined in equations ( 1) and ( 2), where CV min and w min (CV max and w max ) are the CV peak position and full width at half maximum (FWHM) values, respectively, corresponding to the minimum (maximum) CV value.

𝑅 𝑝𝑝 = 1.178 x 𝐶𝑉 1 -𝐶𝑉 2 𝑤 1 +𝑤 2

(1)

𝑃 𝑐 = 2 x 𝐶𝑉 𝑚𝑎𝑥 -𝐶𝑉 𝑚𝑖𝑛 𝑤 1 +𝑤 2 (2)
As can be seen in Figure 1 CV as metabolite identifier: no matrix, nor concentration effects. DMS data recorded for a selected set of AAs and related metabolites in two matrices, plasma and water:methanol, are reported in Figure 3. 2D plot of m/z as a function of CV values for a mixture of standards diluted in water:methanol is given in Figure 3a. The corresponding 2D plot for the pool of plasma sample (P1) containing IS is given in panel b. In each case, the mass spectrum integrated over the whole CV range is given on the right hand side in panels d) and e), respectively. XIC for a selected set of m/z values specific of molecules of interest for plasma study are given in Figure 3c. The blue and red lines correspond to standards in water:methanol solution and plasma sample, respectively. All the curves are normalized to maximum and smoothed using a rolling average over two points.

As expected, a higher density of contributions both in CV and m/z dimensions is observed in plasma. Nethertheless, the 2D plots of standards and plasma show common features. As can be seen in Figure 3c, no significant matrix effect on CV position nor in the DMS peak width are observed. This is confirmed through the comparison of the CV values determined for a selected set of metabolites in a plasma or diluted in water:methanol solution (Table 1). The standard deviation (SD) derived using a set of 36 plasma measurements is smaller or equal to 0.1 V, except for glycine (0.2 V), alanine (0.3 V), and serine (0.2 V). Considering the SD values, there is an excellent agreement between the two sets of values, and thus no evidence of matrix effects. Concentration effects on DMS peak width and position could also be expected. DMS peaks were characterized for a set of AAs and related compounds when spiked in plasma P1 using six dilution factors spanning two orders of magnitude of concentration (1-100 µmol). Overall, both FWHM and position of DMS peaks were found constant over these experiments. For the molecules of interest here, the mean value of CV standard deviation was 0.1 V. The maximum CV standard deviation (0.18 V) was found for protonated glycine. Similarly, the peak width was also found not to be correlated with the concentration value. The mean FWHM value was 0.72 V, and the standard deviation was 0.1 V in the concentration range examined.

A full assignment of the XIC is out of the scope of this study, and unassigned peaks are labeled with a "*" symbol in Figure 3c. DMS peaks associated to positive CV values, however, have to be analyzed carefully. It is indeed well-known that protonated homo-and hetero-dimers are typically transmitted at positive CV values. 54 And these dimers are partially fragmented in the source region, at the exit of the glass capillary, just downstream of the DMS device. 54 Due to this so-called "post-DMS fragmentation", their corresponding protonated monomers are observed at the corresponding transmission CV of the dimers. 55 Hence, in addition to the DMS peak of a given protonated monomer at m/z, peaks are also observed at positive CV values corresponding to maximum transmission of their proton bound dimers. This is evidenced in Figures 3a and3b where the 2D Fig. 2 CV and CCS as a function of m/z for a selected set of AAs and related species (See list in Table S4). CV values (in V), 1 V corresponding to an electric field of 1.

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Please do not adjust margins plots in the positive CV range do not significantly depend on the m/z value.

In addition, a slight fragmentation of some protonated monomer of standards may also occur. This is the case for phenylalanine (m/z 166→120), and methionine (m/z 150→104) as shown in Figure 3c with "#" marks. These fragments can be easily assigned because they are generated through post-DMS dissociation and are thus found at the CV position of the corresponding precursor ion. Finally, one must also pay attention to isotopic peaks (see "&" in Figure 3c). In the case of arginine (MH + m/z 175), which is abundant in the plasma and observed at 0.95 V, a significant contribution associated with the first isotopic peak is also observed in the m/z 176 XIC.

Similarly, contributions of the first and second isotopic peaks of leucine/isoleucine are found in the m/z 133 and 134 XIC, respectively.

Separation of isobaric and isomeric AAs and related species.

Considering the limited mass-resolution of our quadrupole ion trap instrument, the separation of isobaric species is an important issue. The performance of DMS relative to drift-tube instruments can be assessed by inspecting Figure 2 where CV and CCS are plotted as a function of m/z. The corresponding values for a selected set of isobaric and/or isomeric species discussed below are given in Table S4. Apparent disagreements in terms of DMS peak position are in fact the result of the different relative concentrations of (partially) unresolved isomeric compounds in the two matrices (water:methanol and plasma). This the case for example for the broad DMS peak observed near -7.5 V for m/z 90 ions (Figure 3c). It results from contribution of α-and β-alanine, 44 the latter being observed at slightly more negative (-0.3V) CV value than the former. The difference between the two m/z 90 XIC near -7.5 V (Figure 3c) is thus simply due to the fact that β-alanine is only found as trace amounts in the plasma while it is a component of our standard mixture. More important is the issue of the separation of α-alanine and sarcosine which cannot be distinguished based on their fragmentation mass spectra. These two ions were found to be baseline separated, the sarcosine peak being observed at -10.6 V. This DMS peak in observed for the standard mixture, but not for the plasma, where Sarcosine is expected at a very low concentration.

A similar issue occurs in the XIC at m/z 132 (Figure 3c). Indeed, leucine/isoleucine have similar concentrations in the standard mixture, but not in the plasma. As a result, the shape of the unresolved DMS peak in the XIC at m/z 132 (Figure 3c) changes accordingly. In addition, note that Allo-isoleucine (Ail) is also included in the standard mixture, while it is not expected in plasma P1 which is from healthy volunteers. Measurements on each individual Leu isomer were performed, and the CV values of Leu, Ile, and Ail were found to be -3.0, -2.8, and -2.9 V, respectively. Better peak-to-peak resolution is obtained with isobaric cis-and trans-hydroxyproline, which were found to be transmitted at -3.6 and -4.0 V, respectively. For m/z 133, three baseline separated peaks were observed at -4.0, -2.6, and -1.4 V. Based on measurements on individual metabolites, the peaks at -4.1 and -1.3 V can be assigned to Asn and Orn, respectively. As can be seen in Figure 2, the DMS resolution is better than that obtained with a drift-tube instrument. For example, Orn and Asn measured CCS were found to be 130 and 132 Å 2 , respectively. 38 Similarly, protonated glutamine and lysine were found to be baseline resolved and found at -2.7 and -0.3 V, respectively. Their measured CCS, however, were found very close (133 and 132 Å 2 , respectively). 38 DMS performance in terms of separation of isomeric species can be illustrated with 1-and 3- Please do not adjust margins Please do not adjust margins methylhistidine. DMS measurement on individual isomers show that they are found at -0.2 and 0.4 V.

CV as metabolite identifier: reproducibility and robustness.

The robustness of the method in terms of accuracy on the peak position over time is also an important issue in the perspective of using CV as a metabolite identifier. Figure 4 illustrates the stability of DMS peak position for a selected set of six AAs over a period of 6 months. These DMS spectra include experiments performed with plasma (filled circles) and standards in water:methanol solutions (open circles). The mean CV and associated standard deviation values of the DMS peaks given in Figure 4 can be found in Table 1. The maximum standard deviations were found for serine (0.15 V) and proline (0.12 V) other standard deviation values are smaller than the CV scan step (0.1 V). DMS specificity. The filtering effect of DMS over a short range of m/z values containing protonated proline (m/z 116) and valine (m/z 118) is illustrated in Figure 5. The mass spectrum of a pool of plasmas (P1 sample) recorded with RF off and CV=0 (DMS Off or transparent mode) is given in black in Figure 5a. A significant reduction of background signal is observed when DMS is turned on with the CV value set to -3.7 and -5.3 V for optimal transmission of protonated valine (blue) and proline (red) as illustrated in Figure 5. The mass spectra in panel b) are those recorded at the optimal CV value, while mass spectra integrated over the targeted DMS peak are given in panel c). The DMS specificity is also illustrated in Figure 5. When the DMS filtering is turned on, only the targeted ions are transmitted, as well as their corresponding IS, protonated Val-d 8 and Pro-d 7 .

Note that the maximum transmission of these IS is achieved at a CV value slightly different from that of the corresponding standard. This explains the fact that the signal at m/z 126 and 123 is lower when the DMS filtering mode is turned on than under transparent mode. The MS instrument parameters were Please do not adjust margins Please do not adjust margins tuned so as to avoid saturation of the trap. This is the reason why the peak intensity of protonated valine (m/z 118) and proline (m/z 116) are the same using either DMS-off or DMS-on mode. If the mass spectra integrated over the DMS peaks are considered (Figure 5c), a 10-fold intensity increase is observed.

A similar discussion the DMS specificity has been reported by Chen and coworkers in a study of two radiation biomarkers in human and non-human plasmas using a triple-quadrupole instrument. 56 Validation of the calibration method. Calibration curves were constructed by plotting the ratio of the peak area of each reference compounds and its corresponding IS versus the corresponding reference compound concentration. Six concentration levels of the reference compounds spiked in P1 plasma were used. The results given in Table S5 were derived from 4 independent experiments performed over 6 months. In most cases, linear response profiles with R 2 >0.99 were obtained in the concentration ranges of 1-300 µM after sample dilution. This concentration range covers the expected pathophysiological range for this class of molecules. Limit of detection and limit of quantification. Limits of detection (LOD) and of quantification (LOQ) were derived from the linear regressions discussed above, where LOD can be defined as 3*SD b /S, where S is the slope and SD b is the standard deviation of the intercepts of the regression lines. This method is an alternative to the well-known signal-to-noise method when the MS analysis method does not involve background noise. 57 LOD values for targeted AAs and related compounds are given in Table S5. Quantification of AA levels in plasmas. Plasma samples were analyzed, and concentration levels of metabolites of interest are presented in Figure 6. Results are given for 13 AAs and related compounds. Quantification results obtained with LC-MS/MS for a purpose of diagnosis of MD at the hospital are given in blue bars. Results obtained with the present method are given in red bars. Inter-day quantification experiments were repeated at least three times. A calibration experiment was performed before each quantification experiment. Concentration levels determined for plasma P1 from healthy patients are given in panel a. Overall, our method performs well compared to LC-MS/MS except in the case of Ala. This issue has been discussed above, and is probably related to the presence of β-alanine in our standard mixture used for the calibration procedure. One can also expect similar issue for the quantification of leucine, isoleucine, and allo-isoleucine (not presented). While the two formers can be separated using helium-rich carrier gas, 22 these three isomers are poorly separated using N 2 as a carrier gas. Error bars for the LC-MS/MS protocol were derived for each compound from the corresponding inter-day standard deviation (SD) of an equivalent concentration calculated from plasma controls daily analyzed over a period of 5 months. For the present DMS-MS method, the error bars correspond to the Please do not adjust margins Please do not adjust margins standard deviation over the series of inter-day experiments. Taking into account the SD values, whose mean value is 18%, there is a good quantitative agreement between the two approaches in 77% of the cases. As already discussed, concentration of alanine cannot be accurately determined. In the cases of lysine and methionine, concentrations are slightly overestimated using our DMS-MS approach as compared to LC-MS/MS. Inter-day quantification experiments were also carried out on two plasmas from MD patients in order to analyze AA profiles for diagnosis of known MD. Results are given for plasmas P2 and P3 (see panels b and c, respectively) from patients with Phenylketonuria and Tyrosinemia diseases, respectively. Overall, similar qualitative observations as for plasma P1 can be made. More importantly, the increase of the biomarker for these two MD diseases determined by the DMS-MS method is in excellent agreement with those obtained upon diagnosis using LC-MS/MS. In the case of Phenylketonuria P2 plasma, phenylalanine concentration was calculated at 449±8 µM with our method, compared to 423±24 µM with an LC-MS/MS protocol used in routine for MD diagnosis. In the case of the plasma P3, tyrosine concentration was found to be 448±25 µM with DMS-MS, compared to 436±24 µM with LC-MS/MS. Intra-day quantification experiments were performed on random plasma (P4). The concentration levels determined by our method and LC-MS/MS are given in Figure 7. AAs and related metabolites profiles were analyzed 6 times, and a blank was recorded between two plasma analyses. Overall, there is a good qualitative agreement between the two sets of concentration derived by the two methods. Compared to the pool of plasmas (P1), a concentration increase of 4 out of the 13 selected metabolites (Lysine, threonine, tyrosine, methionine) is found with DMS-MS, in good agreement with LC-MS/MS. In the case of Met, however, the increase is slightly overestimated using the DMS-MS protocol. For the three other biomarkers (Lysine, threonine, tyrosine), there is a good quantitative agreement between the DMS-MS results (363±91, 385±73, 205±45, respectively) and the LC-MS/MS ones (397±24, 412±26, 184±10, respectively). The CV values of the different metabolites were found stable. The mean value of the CV standard deviations was 0.05 V, lower than the overall value (0.1 V) derived over 6 month experiments. A similar value (0.05 V) was found for the mean standard deviation value of the IS. It is worth to note that the CV positions of standard metabolites are found to be slightly different from that of their corresponding IS. The latter were (almost) systematically found to be more positive by 0.09 V (mean value). Such isotopic shifts have been observed by Guevremont and coworkers for small ions such as 37 Cl -and 35 Cl -. 58 In recent studies performed on high-definition DMS by Shvartsburg and coworkers, these isotopic CV shifts have been found to allow for identification of isomers of dichloroanilines, 59 and to be structurally informative for dibromoanilines and tribromoanilines. 60

Conclusions and perspectives

We herein provide evidence that DMS hyphenated to an MS/MS instrument is a valuable alternative approach for the separation, identification, and quantification of AAs and related compounds. As retention time in LC-MS/MS, compensation voltage (CV) value allowing for the maximum transmission of a specific metabolite could allow for its identification. It is thus important to stress that the position of the DMS peak position values of metabolites of interest do not depend on concentration and/or matrix effects. Variations of CV positions were found to be stable within ±0.1 V, as compared to peak width (FWHM) of 0.72 V. For a set of AAs and related compounds, we show here that CV values are reproducible over a large period of time (6 months). The proposed DMS method itself is robust since these experiments were carried out on an open MS/MS platform, and the DMS device thus had to be mounted and unmounted several times. DMS could thus be an alternative separation method for MS/MS based targeted metabolomics for the quantification of biomarkers. In particular, two applications dedicated to MD diagnosis could be performed in order to address the topics of emergency in metabolic diagnosis and newborn rapid screening. Biomarker analysis time could be significantly reduced using DMS-MS compared to LC-MS/MS approaches. Even using an ion-trap as in the present study, a full CV scan can be performed within ~5 minutes. Furthermore, a full CV scan is no longer needed once the CV is known for the targeted metabolites/biomarkers. Quantification of these biomarkers could be achieved by recording mass spectra at selected identified CV values. Finally, as discussed in the recent literature DMS can also be hyphenated to LC-MS/MS. 11,[61][62][63][64] Two dimensional separation LC-IMS-MS/MS method allows for resolving chromatographic peaks, and an increase of signal-tonoise ratio. 11 In the perspective of using DMS for identification purpose, the addition of selected organic molecules to the carrier gas is undoubtedly the approach of choice for improving DMS specificity. Drastic improvement of ion, and especially isomeric ions, separation has been observed by adding alcohol or other polar and/or proton-acceptor molecules in the buffer gas, 23, 65- 67 and in the case of AAs in particular. 24,25 A systematic study of the effect of different modifiers on the separation of AAs and related compounds is under study in our group. Using methanol as an added modifier to N 2 as carrier gas, we already showed baseline separation of isomeric ions could be obtained in the case of sarcosine, α-and β-alanine, 44 as well as for the α-, β-, and -aminobutyric acid isomers. 45 As reported in the literature, adding of a fraction of He to the carrier gas 20,22,29 or increasing the waveform amplitude 21 should also be considered for high resolution DMS. Quantification of AAs and related compounds were achieved using our 3D quadrupole ion trap despite the intrinsic limitations of these instruments in terms of sensitivity and linear dynamic range. However, FT-ICR or QIT instruments are interesting for the coupling with IR lasers for DMS-peak assignment using IRMPD spectroscopy. 42,44,45 Ultimately, however, faster quantification will be performed with DMS hyphenated to a triple quadrupole instrument.

Fig. 1

 1 Fig. 1 Differential ion mobility data for the 20 protonated AAs recorded at different dispersion voltage (DV) using N2 as carrier gas. (a) DMS dispersion plots, or evolution of CV as a function of DV; DMS spectra for each individual AA at DV = 1.4, 1.6, 1.8 kV are given in panels (b), (c), and (d), respectively. A CV value of 1 V corresponds to an electric field of 1.43 kV.m -1 or ~0.07 Td. To facilitate visualization, each DMS peak is individually normalized.

  , P c increases with DV, from P c = 18.1 at 1.4 kV to P c = 23.8 at 1.8 kV. For more details, the evolution of P c (black curves) and R pp (blue curve) as a function of the DV value is shown in FigureS1. The best separation performance is observed for DV value of 1.8 kV except for few AA pairs. For example, protonated lysine (B type) and protonated tryptophan (C type) are poorly separated at 1.8 kV (R pp = 0.1), while a better separation is observed at lower DV (R pp = 0.48 at DV = 1.2 kV). In a consistent manner, the fraction of the pairwise combinations of protonated AAs which are baseline resolved (R pp =1.5) increases from 29 % at DV=1 kV to 81 % at DV = 1.8 V. It can thus be concluded that these R pp and P c values are very close to those estimated (P c ~15 and ~80% resolved AAs (R pp >1.0)) for a high performance drift tube instrument using N 2 as a carrier gas.51

  Fig.2CV and CCS as a function of m/z for a selected set of AAs and related species (See list in TableS4). CV values (in V), 1 V corresponding to an electric field of 1.43 kV.m -1 or ~0.07 Td. CCS (in Å 2 ) are taken from (a) ref. 36, (b) ref. 31, (c) ref. 17, and (d) ref. 49.

Fig. 3

 3 Fig.3DMS data recorded for a set of standards (AAs and related compounds) in water:methanol and for a plasma sample (P1) containing IS. The corresponding 2D plots, with m/z as a function of CV, are given in panels a) and b), respectively. Extracted Ion chromatograms (XIC) of a selected set of metabolites of interest are given in panel c) where data for the plasma (red) and standards in water:methanol solution (blue) are superimposed. Mass spectra integrated over the CV range corresponding to 2D plots of panels a) and b) are given in panels d) and e), respectively. Peak assignment in the XIC panel c) is provided. 3-letter nomenclature symbols are used for AAs, as well as for Aminoadipic acid (AAA), Aminobutyric acid (ABA), Citrulline (Cit), Ornithine (Orn), and Sarcosine (Sar). Peaks corresponding to post-DMS (in-source) fragments or also first and second isotopic peaks are labelled with an "#" and an "&" symbol, respectively. Unassigned DMS peaks are symbolized by a "*" mark. A CV value of 1 V corresponds to an electric field of 1.43 kV.m -1 or ~0.07 Td.

Table 1 .

 1 Compensation voltage (CV) values recorded for a selected set of metabolites in a plasma or diluted in water:methanol solution. Mean CV and the corresponding standard deviation (SD) are given in Volt unit. Day-to-day reproducibility of the DMS peaks and correspondence between CV and electric field are given in

Fig. 4

 4 Fig.4 Selected DMS spectra recorded for serine (Ser), proline (Pro), glutamine (Gln), phenylalanine (Phe), lysine (Lys) and arginine (Arg) over a period of 6 months. Experiments were performed with plasma samples (filled circles) and standards of AAs and related compounds in water:methanol solutions (open circles). The mean CV and associated standard deviation values are given in Table1. A CV value of 1 V corresponds to an electric field of 1.43 kV.m -1 or ~0.07 Td.

Fig. 5

 5 Fig. 5 Mass spectra recorded with DMS-off (black) and DMS-on (color) for a pool of plasmas sample P1. (a) DMS off and CV set at 0 V. (b) CV set at -5.3 V (red) and -3.7 V (blue) for optimal transmission of protonated proline (m/z 116) and valine (m/z 118), respectively. (c) Mass spectra integrated over the DMS peaks of protonated valine (dark blue) and proline (dark red). A CV value of 1 V corresponds to an electric field of 1.43 kV.m -1 or ~0.07 Td.

Fig. 6

 6 Fig. 6 Concentration (µM) of 13 selected metabolites in three plasmas (P1-P3) samples. DMS-MS experiments (red bars) were performed on different days, and standard deviations are indicated as error bars. LC-MRM data are shown as blue bars, and error bars correspond to RSD calculated over 5 months.

Fig. 7

 7 Fig. 7 Concentration (µM) of selected metabolites in random plasma (P4). DMS-MS data (red bars) correspond to the mean value derived from 6 intraday experiments. Standard deviations are indicated as error bars. LC-MRM data are shown as blue bars, and error bars correspond to RSD calculated over 5 months.

Table S6 .

 S6

		Plasma		Water:MeOH	alanine	-7.6	0.3	-7.9	0.3
		CV	SD	CV	SD	serine	-6.6	0.2	-6.8	0.2
	glycine	-10.7	0.2			proline	-5.3	0.1	-5.4	0.2
						valine	-3.7	0.1	-3.9	0.2
						threonine	-4.8	0.1	-4.9	0.2
						asparagine	-4.2	0.1	-4.3	0.1
						glutamine	-2.7	0.1	-2.8	0.1
						lysine	-0.3	<0.1	-0.3	0.2
						methionine	-2.3	0.1	-2.4	0.2
						histidine	-1.5	0.1	-1.5	0.2
						phenylalanine	-1.3	0.1	-1.4	0.2
						arginine	0.9	<0.1	1	0.2
						citrulline	-0.7	<0.1	-0.7	0.2
						tyrosine	-1.4	0.1	-1.4	<0.1
						tryptophan	0.2	0.1	0.2	0.1

This journal is © The Royal Society of Chemistry 2020Analyst, 2020, 00, 1-3 | 8

Acknowledgements

Financial support from the Agence Nationale de la Recherche (ANR-19-CE29-0016-01, ANR-10-LABX-0039) is gratefully acknowledged. YW would also like to acknowledge the (CSC) for China Scholarship Council (CSC) program for a full scholarship. EA is grateful for the support of the "Programme national d'aide à l'Accueil en Urgence des Scientifiques en Exil" (PAUSE) from the Collège de France.

Conflicts of interest

There are no conflicts to declare.


Maxence Lepers

Véronique Zehnlé

Jean Claude Garreau

Tracking quasi-classical chaos in ultracold boson gases

Keywords: numbers: 03.75.Nt, 05.45.Ac, 37.10.Jk

We study the dynamics of a ultra-cold boson gas in a lattice submitted to a constant force. We track the route of the system towards chaos created by the many-body-induced nonlinearity and show that relevant information can be extracted from an experimentally accessible quantity, the gas mean position. The threshold nonlinearity for the appearance of chaotic behavior is deduced from KAM arguments and agrees with the value obtained by calculating the associated Lyapunov exponent.

Recent advances in the physics of cold atoms paved the way for the investigation of fundamental quantum problems with unprecedented cleanness. "Quantum chaos" is one of the most fascinating among these problems, because in such case the correspondence between the dynamics of the quantum system and its classical counterpart is nontrivial. The reasons for that are twofold: First, quantum particles obey Heisenberg inequalities, hence their dynamics cannot be described in terms of phase-space trajectories. Second, sensitivity to initial conditions is not observed, as the Schrödinger equation is linear. For these reasons, quantum chaos has often been defined as "the behavior of a quantum system whose classical limit is chaotic". Whereas this is a reasonable definition, it is clear that the actual quantum dynamics has no direct relation to classical chaos. Much work thus concentrates in finding "reminiscences" of classical chaos that might survive in the quantum system, the so-called "signatures" of quantum chaos [START_REF] Haake | Quantum Signatures of Chaos[END_REF].

The realization in 1995 of the first Bose-Einstein condensate with laser-cooled atoms [START_REF] Anderson | [END_REF]3,4] opened a new way for investigating (truly) nonlinear dynamics in quantum systems. In an "ideal" (i.e. in the zero temperature limit) Bose-Einstein condensate (BEC), the atoms are indistinguishable and form a mesoscopic object which can be described by a "collective" single wavefunction. The BEC's dynamical behavior is then described by the Gross-Pitaevskii equation which includes a nonlinear term due to atomic interactions [5]. The solutions of such equation can -and do -present sensitivity to initial conditions, leading to "classical-like" instabilities, a possibility that attracted much attention from both theoreticians [6,7,8,9,10,11,12,13,14] and experimentalists [15,16].

In order to make these ideas clear, let us first consider a simpler system, the kicked rotor, whose experimental realization with laser cooled atoms [17] lead to an impressive burst of experimental work [18,19,20,21,22,23,24,25]. This system is formed by a particle periodically "kicked" by a sinusoidal force. The classical dynamics is characterized by a single parameter K, which is the normalized amplitude of the potential [17], whereas the "quanticity" of the quantum system is described by a "normalized Planck constant" k (the dynamics become classical as k → 0).

In order to characterize the dynamics, one may chose, for instance, the particle's square momentum as the relevant quantity (p 2 (t) in the classical case, p 2 (t) in the quantum case). By taking the Fourier transform of the above quantity and pinpointing the frequencies appearing in the evolution of the system for different values of K and fixed initial conditions, we can characterize in a common language the classical and the quantum versions. The result is plotted in Fig. 1.

The classical dynamics, Fig. 1a, shows resonances (or frequency-lock events) each time a frequency become very close to another one. For K 0.8, the frequencies in the spectrum "coalesce" to form of a dense spectrum, which is a signature of a chaotic dynamics. In the quantum case, Fig. 1b, the spectrum is richer and present numerous frequency crossings, but no dense spectrum is observed.

Let us now consider a one-dimensional BEC of particles of mass M placed in a sinusoidal lattice formed by counterpropagating laser beams of wavelength λ L , and subjected to a linear force. The system is described by the Gross-Pitaevskii equation (GPE)

i ∂ψ(x, t) ∂t = P 2 2m + V 0 cos(2πx) + F x + g |ψ| 2 ψ(x, t), (1)
with lengths measured in units of the lattice step d = λ L /2, energy measured in units of the recoil energy

E R = 2 k 2 L /(2M ) (with k L = 2π/λ L ), time in units of /E R , the force F in units of E r /d, P = -i∂/∂x, the reduced mass is m = π 2 /
2 and g is the (1D) nonlinear parameter. The eigenstates of the linear part of the Hamiltonian (1), obtained by setting g = 0, are the so-called Wannier-Stark (WS) states [6,26,27]. In order to simplify the discussion, let us we suppose that the BEC energy is low enough that its wavefunction ψ can be expanded only on the lowest-energy WS state for each potential well, hereafter noted as ϕ n (x) (corresponding to the n th well). These states are centered at each potential well, obey the symmetry relation ϕ n+m (x)= ϕ n (x -m), and form a For each value of K, we numerically calculate p 2 (t) (classical case) and p 2 (t) (quantum case) up to 2000 kicks, perform the Fourier transform and pinpoint the frequencies whose Fourier amplitude is above a threshold (here 1/500 of the maximum amplitude). (a) Frequencies in the classical dynamics: A dense spectrum appears for K 0.82 (initial conditions x = 0 and p = 0.56). (b) Frequencies in the quantum dynamics: The spectrum is complex, but always discrete (k = 2.89, the initial state is a Gaussian in momentum space of FWHM 5k centered at p = 0.56).

ladder of eigenenergies E n = nω B , where ω B = F d/ (F in our normalized units) is the Bloch frequency. Thus, putting ϕ(x) ≡ ϕ 0 (x)

ψ(x, t) = n I n e -iθn ϕ(x -n), (2)
where the eigenstates population I n and phase θ n are real functions of the time. Following the approach of [6], we insert Eq. ( 2) in Eq. ( 1), and obtain Hamilton equations for the populations and phases. The Hamiltonian then appears as a sum of an integrable part H 0 and a non-integrable perturbation H 1 :

H = H 0 (I) + ǫH 1 (I, θ), (3)
with ǫ ∝ gχ 01 , where χ 0i ≡ dxϕ 3 (x)ϕ(x -i) is a measure of the superposition of eigenfunctions correspond- ing to different wells. For small ǫ, the system is quasiintegrable and fits into the general frame of the KAM theorem [START_REF] Lichtenberg | Regular and Chaotic Dynamics[END_REF]. Therefore, the quantum-coherent evolution of a BEC displays classical-like KAM-structured chaos. We call this, "quasi-classical" chaos.

If we temporarily neglect the nonlinear term in Eq. ( 1), we see that the phases evolve as θ n (t) = θ n0 + nω B t.

To the first order in ǫ, the effect of the nonlinearity is to introduce a population-dependent correction to this phase, producing a harmonic evolution with frequencies ω n = nω B + gχ 00 I n . The intensities are constants of motion, I n (t) = I n (t = 0) and the "self-coupling" parameter χ 00 depends only V 0 and F . The BEC dynamics is thus governed by the Bohr frequencies

ω nm ≡ ω n -ω m = (n -m)ω B + gχ 00 (I n -I m ). (4)
In order to simplify further our description, let us restrict the dynamics to three adjacent potential wells, i.e. we set I n ≡ 0 if n = -1, 0, 1 in Eq. (2). The system is then four-dimensional, the dynamical variables being two populations (since I -1 + I 0 + I 1 = 1) and two relative phases.

The onset of chaos in this system is shown in Fig. 2, which is the equivalent of Fig. 1 for the BEC dynamics, with the difference that we used the average position x(t) instead of p 2 (t) . Strikingly, the plot resembles more closely to the classical (Fig. 1a) than to the quantum (Fig. 1b) dynamics of the kicked rotor.

We can interpret the main features of Fig. 2 from simple arguments. At g = 0 there are only three Bohr frequencies in the model: Two of them are degenerated, ω 0-1 = ω 10 = ω B , and correspond to energy difference between neighbor wells, whereas ω -11 = 2ω B corresponds to next-to-neighbor energy difference. The low-g structure can be understood using Eq. ( 4). The frequencies present around g = 0.05, for instance, in Fig. 2 are of the form Ω pqr = pω 0-1 + qω 10 + rω 1-1 (5) with p, q, r integer. Namely, from top to bottom, we found ω 1-1 , 2ω 10 , ω 0-1 , ω 10 and ω 0-1 -ω 10 . These frequencies are only weakly perturbed by the nonlinearity, which manifests itself by the slight curvature of the branches. A KAM-type expansion in powers of ǫ shows that the weight of a frequency Ω pqr in the spectrum is proportional to g/Ω pqr [START_REF] Lichtenberg | Regular and Chaotic Dynamics[END_REF], thus, the higher the value of g, the larger the number of frequencies that will be present (remember that Fig. 2 displays the frequencies whose amplitude is above a threshold), and smaller frequencies will be favored. The intersection of two branches correspond to a resonance, that is, the condition Ω pqr = 0 is fulfilled for some p, q, r. Close to a resonance the KAM perturbation theory breaks down, and non-integrable behaviors appear that may lead to chaos. In Fig. 2 the arrows indicate the resonances Ω 1q0 = 0 (q = -2) for g ≈ 0.093, and q = -3, -4, -5 for, respectively, g ≈ 0.14, g = 0.17, g = 0.196 . For higher g values, resonances with higher and higher q values become significative, as seen on the left of the chaotic region: Each frequency crossing produces a multi-frequency structure (indicated by the arrows). Finally, for g 0.2 one observes at least four successive windows of dense spectrum, corresponding to a chaotic behavior. Fig. 2 represents a "local route" (in the sense that its detailed geometry depends on the initial conditions) to quasi-classical chaos. The above argument suggests however that this route is characteristic of KAM systems, and globally independent of initial conditions. Its universal character is confirmed by comparing it to Fig. 1a, where an analogous structure is observed in a completely different KAM system, the classical kicked rotor. We have thus put into evidence a route to KAM chaos in a nonlinear quantum system, which is potentially observable with state-of-art experiments.

In order to confirm our conclusions, we have also calculated the maximum Lyapunov exponent (MLE) associated to the dynamics, which is a direct signature of the sensitivity to the initial conditions, and thus of chaos. In order to calculate MLEs in our quantum system, we adapted the classical Jacobian method [START_REF] Eckmann | [END_REF]. We represent the system evolution by a trajectory in a sixdimensional "generalized quantum phase-space", formed by the real and imaginary parts of each WS state coefficient, i.e. √ I n cos θ n and √ I n sin θ n , for n = -1, 0, 1 (this is numerically more stable than using I n and θ n ), and calculate the divergence of neighbor WS states, from which we can extract the MLE. The result is presented in B = ωB/2π) as a function of g. One observes a good correspondence between the existence of a non-zero Lyapunov exponent and the presence of dense-spectrum regions in Fig. 2. The observed dense-spectrum is thus a reliable signature of the chaotic behavior. Same parameters and initial conditions as in Fig. 2. Fig. 3. One sees a clear transition to chaos for g ≈ 0.194, which is in good agreement with the value that can be deduced from Fig. 2. We can also identify the four successive chaotic windows.

For given initial conditions, we can obtain an estimate for the critical value of g at which chaos appears. KAM theory shows that chaos generally appears along a separatrix. In our system there are two main kinds of trajectories [6]: "Passing" trajectories correspond to Bloch oscillations slightly perturbed by the nonlinearity (for the low values of g we are considering); they appear when the three populations are comparable. Bound, periodic orbits correspond to a motion essentially confined at a potential well, and appear when one population dominate the others. Between these two kinds of trajectories, there is a separatrix. For a given trajectory (i.e. for fixed initial conditions), chaos appears when the changing in the value of g brings it close to the separatrix. The condition for that can be simply estimated by confining the BEC to only two wells (I -1 = 0), in which case the system is integrable. This "reduced" system does not exhibit chaos, but it has a phase-space structure analogous to that of the 3-wells, displaying bound and passing trajectories, and, between them, a separatrix. In the reduced system we can calculate both the energy E 0 corresponding to the trajectory (with I -1 = 0) and the energy E s of the unstable point to which the separatrix is connected. We can then numerically determine the value of g for which E 0 = E s , which gives the critical value. Fig. 4 shows the result for various initial conditions (solid line). The dotted lines indicate the limits of the chaotic region as inferred from the Lyapunov exponent calculation in the 3-well system. The agreement is very good, even when the chaotic zone is narrow.

In conclusion, we have characterized the phenomenon of quasi-classical chaos using an experimentally accessible signal, the mean position of the boson gas. We have studied a "local route" to quasi-classical chaos and shown that the information obtained from this approach agrees well with that provided by a hallmark signature of chaos, the sensitivity to initial conditions, quantified by a positive Lyapunov exponent. The understanding of the structure of such a route allowed us to determine the critical value of the nonlinearity parameter, which is in good agreement with the one deduced from the calculation of the Lyapunov exponent. We think that the present work might stimulate an experimental observation of quasi-classical chaos, which would, in turn, stimulate new investigations on the nature of quantum chaos.

The authors are happy to thank M. Lefranc for fruitful discussions and for his help with the calculation of Lyapunov exponents.

Figure 1 :

 1 Figure1: Classical and quantum dynamics of the kicked rotor. For each value of K, we numerically calculate p 2 (t) (classical case) and p 2 (t) (quantum case) up to 2000 kicks, perform the Fourier transform and pinpoint the frequencies whose Fourier amplitude is above a threshold (here 1/500 of the maximum amplitude). (a) Frequencies in the classical dynamics: A dense spectrum appears for K 0.82 (initial conditions x = 0 and p = 0.56). (b) Frequencies in the quantum dynamics: The spectrum is complex, but always discrete (k = 2.89, the initial state is a Gaussian in momentum space of FWHM 5k centered at p = 0.56).

Figure 2 :

 2 Figure 2: Frequencies present in the spectrum of x(t) vs g (the threshold is 1/100 of the maximum amplitude). For small g, the Bohr frequencies [cf. Eq. (4)] are ω10 = ωB -0.4gχ00, ω0-1 = ωB + 0.55gχ00 and ω1-1 = 2ωB + 0.15gχ00, corresponding to branches a, b, and c, respectively. Harmonics of ω10 are also seen (branches d and e) and Ω1,-1,0 (branch f). For 0.22 g 0.26 one observes dense-spectrum windows corresponding to quasi-classical chaos. Parameters are ωB = 0.25, V = 5 (χ00 = 2); initial conditions are I0 = 0.65, I1 = 0.25, θ0 = θ-1 = 0, θ1 = π.

Figure 3 :

 3 Figure 3: Maximum Lyapunov exponent (in units of T -1B = ωB/2π) as a function of g. One observes a good correspondence between the existence of a non-zero Lyapunov exponent and the presence of dense-spectrum regions in Fig.2. The observed dense-spectrum is thus a reliable signature of the chaotic behavior. Same parameters and initial conditions as in Fig.2.

1 Figure 4 :

 14 Figure 4: Determination of the critical value of g. The solid lines with triangles are the estimated critical value of g, given by the condition that the trajectory energy equals the separatrix energy (cf. text). The dotted lines with crosses are the limits of the chaotic region, where the Lyapunov exponent is non-zero. The parameters are the same as in Fig 2, with θ0 -θ1 = π 2 .


Jean-François Lutz
email: jflutz@unistra.fr

Toward artificial Life-supporting Macromolecules

Abstract. Terrestrial Life is based on polymers. In all known living organisms, DNA stores genetic information, mutates, self-replicates and catalyzes the synthesis of messenger molecules. Although the function of nucleic acids is well-understood, the development of artificial macromolecular mimics remains very limited. Laboratory-synthesized nucleic acids still support Life and some nucleic acids analogues exhibit biological functions. Yet, after hundred years of polymer science, no other type of Lifesupporting macromolecule (i.e. non-nucleic acids) has ever been reported. In this context, the aim of the present viewpoint is to discuss important challenges that shall be addressed by polymer chemists to achieve artificial Life. Similarly to DNA, an artificial Life-supporting macromolecule shall store information, transfer information and mutate. Many tools, such as sequence-defined polymer synthesis, polymer modification, supramolecular polymer chemistry and dynamic chemistry, are already available to chemists to attain these properties. However, the design of artificial Life-supporting macromolecules is hindered by two main factors. First, the chemical search space is enormous and it is hard to predict promising structures, even with the help of combinatorial and chemoinformatic tools. Secondly, rational design is probably a limited approach to achieve macromolecules that shall be involved in non-equilibrium metabolic systems. Hence, a synergic combination of classical polymer chemistry with the more recent field of systems chemistry is probably the key toward the emergence of artificial Life-supporting macromolecules.

This viewpoint is written in the context of the centennial anniversary of the first article written by Hermann Staudinger about polymerization. [START_REF] Staudinger | Über Polymerisation[END_REF] All along his career, Staudinger has pointed out the similarities between man-made and biological macromolecules. In particular, in the last pages of his 1953 Nobel lecture, 2 he discussed the transition from inanimate to living matter and the fact that living organisms and heredity are operated by macromolecules. Earlier that year, [START_REF] Watson | Molecular Structure of Nucleic Acids: A Structure for Deoxyribose Nucleic Acid[END_REF] the structure of DNA was first revealed by Watson and Crick. More than sixty years later, the fact that Life is operated by polymers and the so-called central dogma of molecular biology [START_REF] Crick | Central Dogma of Molecular Biology[END_REF] involving DNA, RNA and proteins have become textbook knowledge. In particular, it is well understood that DNA is an informational polymer, which stores genetic information using controlled monomer sequences, as well as a molecular replicator, which has the ability to transfer its sequence information to macromolecules of similar (replication) or different (transcription) nature. It also undergoes mutations (i.e. random monomer alterations), which enable Darwinian evolution to proceed.

Yet, despite significant progress in polymer chemistry and molecular biology during the past decades, it is still not possible to create synthetic polymers that can support Life in the same way DNA does. Still, very important milestones have been reached with chemically-synthesized nucleic acids. For instance, it has been evidenced that fragments or even whole genomes can be prepared in a laboratory and still operate Life when included in a host organism. [START_REF] Gibson | Creation of a Bacterial Cell Controlled by a Chemically Synthesized Genome[END_REF][START_REF] Fredens | Total synthesis of Escherichia coli with a recoded genome[END_REF] It has also been reported that nucleic acids containing nonnatural backbones [START_REF] Pinheiro | Synthetic Genetic Polymers Capable of Heredity and Evolution[END_REF][START_REF] Taylor | Catalysts from synthetic genetic polymers[END_REF] (so-called xeno nucleic acids) or expanded genetic alphabets [START_REF] Malyshev | A semi-synthetic organism with an expanded genetic alphabet[END_REF][START_REF] Hoshika | Hachimoji DNA and RNA: A genetic system with eight building blocks[END_REF] still exhibit operational biological functions. Nevertheless, in those examples, natural molecular design (i.e. nucleic acids) is used, either as such or in a slightly-modified form. In this context, a crucial question shall be asked to polymer chemists: are nucleic acids the only type of macromolecule that can efficiently support Life in the Universe? In other words, is it possible to support Life using chemically-different (i.e. non-nucleic acids) informational polymers? In recent years, significant progress has been made on the synthesis of sequence-controlled polymers; [START_REF] Lutz | Sequence-Controlled Polymers[END_REF] including some advances for the design of abiotic informational polymers. [START_REF] Colquhoun | Information-containing macromolecules[END_REF][START_REF] Lutz | Coding Macromolecules: Inputting Information in Polymers Using Monomer-Based Alphabets[END_REF][START_REF] Rutten | Encoding information into polymers[END_REF] Yet, as mentioned above, artificial Life was never achieved using man-made polymers. In a recent Essay, [START_REF] Lutz | Can Life Emerge from Synthetic Polymers?[END_REF] I have discussed the historical, philosophical and scientific aspects of polymer-based artificial Life, and the reader is referred to this earlier text for broader information on the topic. The present viewpoint is specifically addressed to synthetic polymer chemists and discusses specific challenges that shall be overcome to achieve artificial Life-supporting macromolecules.

Figure 1.

Required properties and available chemical tools for achieving artificial Life-supporting macromolecules. As shown in the middle panel, in order to support Life, a synthetic macromolecule shall at least combine the three following properties: (i) information storage, (ii) information transfer and (iii) mutability. Although some of these properties can be attained by rational design using the polymer chemistry tools listed in the left panel, Life can only occur in non-equilibrium thermodynamic conditions as schematized on the right panel. Thus, artificial Life-supporting macromolecules will probably result from a synergic combination of polymer chemistry and systems chemistry.

What has already been achieved?

Although there is no universally-admitted definition of Life, [START_REF] Schrödinger | What is Life?[END_REF][START_REF] Cleland | Defining 'Life'. Origins Life Evol[END_REF][START_REF] Ruiz-Mirazo | A Universal Definition of Life: Autonomy and Open-Ended Evolution[END_REF][START_REF] Benner | Defining Life[END_REF] living matter can be described -at least from a chemist's point of view -as an individual entity that can exchange matter and energy with its environment, grow, reproduce, react to environmental variations, adapt and evolve. Hence, in order to support Life, a synthetic macromolecule shall exhibit at least the following properties: information storage, information transfer and mutability (Figure 1). [START_REF] Lutz | Can Life Emerge from Synthetic Polymers?[END_REF] Macromolecular storage, which is the possibility to store information using controlled monomer sequences, has now been widely demonstrated in synthetic copolymers. [START_REF] Lutz | Coding Macromolecules: Inputting Information in Polymers Using Monomer-Based Alphabets[END_REF][START_REF] Lutz | Aperiodic Copolymers[END_REF] A few years back, [START_REF] Lutz | Sequence-Controlled Polymers[END_REF] that property was restricted to nucleic acids. However, during the last five years, significant research has been dedicated to the synthesis of non-natural information-containing macromolecules; [START_REF] Colquhoun | Information-containing macromolecules[END_REF][START_REF] Rutten | Encoding information into polymers[END_REF][START_REF] Trinh | Synthesis of Molecularly Encoded Oligomers Using a Chemoselective "AB + CD" Iterative Approach[END_REF] including so-called digital polymers that show promise in data storage, [START_REF] Roy | Design and synthesis of digitally encoded polymers that can be decoded and erased[END_REF][START_REF] Martens | Multifunctional sequence-defined macromolecules for chemical data storage[END_REF][START_REF] Huang | Binary tree-inspired digital dendrimer[END_REF] cryptography, [START_REF] Boukis | Multicomponent reactions provide key molecules for secret communication[END_REF] traceability and anti-counterfeiting [START_REF] Karamessini | Abiotic Sequence-Coded Oligomers as Efficient In Vivo Taggants for the Identification of Implanted Materials[END_REF] applications. Such macromolecules are, in most cases, prepared by multistep growth approaches, [START_REF] Lutz | From Precision Polymers to Complex Materials and Systems[END_REF] in which coded monomer sequences are constructed step-by-step. Among recent progress, it has been shown that these syntheses can be greatly simplified by the use of protecting-group-free strategies, [START_REF] Trinh | Synthesis of Monodisperse Sequence-Defined Polymers Using Protecting-Group-Free Iterative Strategies[END_REF] the selection of fast coupling reactions, [START_REF] Roy | Design and synthesis of digitally encoded polymers that can be decoded and erased[END_REF] as well as the development of engineering protocols, which facilitate purification. [START_REF] Dong | Sequence-defined multifunctional polyethers via liquid-phase synthesis with molecular sieving[END_REF] In addition, macromolecular storage density can be increased by the use of expanded monomer alphabets, [START_REF] Lutz | Coding Macromolecules: Inputting Information in Polymers Using Monomer-Based Alphabets[END_REF][START_REF] Martens | Multifunctional sequence-defined macromolecules for chemical data storage[END_REF][START_REF] Boukis | Data storage in sequence-defined macromolecules via multicomponent reactions[END_REF] which exceed the number of monomers found in nucleic acids. Hence, the synthesis of small-and mediumsize abiotic informational polymers has almost become routine laboratory practice. The automated synthesis of relatively long chains is also possible, [START_REF] Ouahabi | Synthesis of Monodisperse Sequence-Coded Polymers with Chain Lengths above DP100[END_REF] even if the maximum chain-length attainable by chemical synthesis is still rather small in comparison with biological sizes. [START_REF] Lutz | Can Life Emerge from Synthetic Polymers?[END_REF] Beyond macromolecular information storage, examples of intermolecular information transfer in abiotic molecules are much more scarce. Although several examples of non-natural self-replicators have been reported, [START_REF] Vidonne | Making Molecules Make Themselvesthe Chemistry of Artificial Replicators[END_REF] most of them deal with non-informational low molecular weight molecules. [START_REF] Orgel | Molecular replication[END_REF] Very recently, Hunter and colleagues have reported the transfer of a trimer information sequence into a complementary copy using a covalent template. [START_REF] Núñez-Villanueva | Sequence information transfer using covalent template-directed synthesis[END_REF] A related strategy using dynamic diimine duplexes was also described by Prywes and coworkers; however only for the transfer of a single bit of information. [START_REF] Strom | Transfer of Sequence Information and Replication of Diimine Duplexes[END_REF] The translation of a nucleic acids sequence into an abiotic informational strand was also described by Liu and coworkers. [START_REF] Niu | Enzyme-free translation of DNA into sequence-defined synthetic polymers structurally unrelated to nucleic acids[END_REF] There is also almost no example of targeted mutations in synthetic copolymers, despite an abundant literature on polymer modification. [START_REF] Günay | Standing on the shoulders of Hermann Staudinger: Postpolymerization modification from past to present[END_REF] Yet, our group has reported very recently the sequence-directed photo-mutation of synthetic informational polymers. [START_REF] König | Photo-editable macromolecular information[END_REF] Open challenges for polymer chemists. To attain polymer-based artificial Life (if attainable at all), major challenges lie ahead for polymer chemists. First, research on sequence-controlled polymers is far from being finished. For instance, methods allowing preparation of long non-natural informational chains (i.e. above 100 monomers) would be an important breakthrough in the field. In addition, the reduction of synthesis times and the simplification (eventually the avoidance) of complex multi-step procedures remain important problems to solve. Still, the main challenges are elsewhere. For instance, intermolecular information transfer has never been demonstrated with "real" polymers, i.e. with synthetic structures containing at least 10-20 monomer units. In fact, the transition from low molecular weight models containing a few bits of information to real informational macromolecules is far from being trivial. Important progress still has to be done regarding the supramolecular association of complementary polymer strands as well as on templated polymerizations. For instance, there are still very few examples of sequence-specific hybridization of non-natural strands, [START_REF] Gong | Molecular Duplexes with Encoded Sequences and Stabilities[END_REF][START_REF] Stross | Sequence-Selective Formation of Synthetic H-Bonded Duplexes[END_REF][START_REF] Liu | Sequence-Controlled Stimuli-Responsive Single-Double Helix Conversion between 1:1 and 2:2 Chloride-Foldamer Complexes[END_REF][START_REF] Szczypiński | Building blocks for recognition-encoded oligoesters that form H-bonded duplexes[END_REF] and most of them deal with relatively short oligomers. Similarly, non-enzymatic self-replication has only been reported with natural oligomers [START_REF] Sievers | Self-replication of complementary nucleotide-based oligomers[END_REF][START_REF] Lee | A self-replicating peptide[END_REF] or with very small nonnatural models (vide supra). [START_REF] Vidonne | Making Molecules Make Themselvesthe Chemistry of Artificial Replicators[END_REF] Overall, the supramolecular association and polymerization of functional monomers on polymeric templates remain poorly-controlled, despite numerous attempts and studies. [START_REF] Tan | The synthesis of polymers by template polymerization[END_REF][START_REF] Połowiński | Template polymerisation and co-polymerisation[END_REF] Although template polymerizations may be useful to replicate some molecular parameters such as average chain-length or narrow polydispersity, [START_REF] Lo | Nucleobase-Templated Polymerization: Copying the Chain Length and Polydispersity of Living Polymers into Conjugated Polymers[END_REF][START_REF] Mchale | Biomimetic radical polymerization via cooperative assembly of segregating templates[END_REF] it is not possible at present to transfer accurately (i.e. without defects) a non-natural information sequence from parent to daughter-strand. Hence, sequence-specific hybridization and intermolecular information transfer remain important challenges in synthetic macromolecular chemistry. A crucial aspect of this problem is chirality and stereoregularity. Homochirality is essential in terrestrial molecular biology and is known to influence molecular informational content, [START_REF] Carroll | A new definition of life[END_REF] macromolecular folding [START_REF] Nanda | The role of protein homochirality in shaping the energy landscape of folding[END_REF] and self-replication. [START_REF] Frank | On spontaneous asymmetric synthesis[END_REF] While the preparation of stereoregular polymers remain a highly stimulating area of polymer science, [START_REF] Badi | Microstructure Control: An Underestimated Parameter in Recent Polymer Design[END_REF] it is probably easier to first study information transfer using polymers with achiral backbones, in which informational side-groups are attached to symmetric carbons [START_REF] Ouahabi | Synthesis of Non-Natural Sequence-Encoded Polymers Using Phosphoramidite Chemistry[END_REF] or nitrogen atoms (e.g. as in peptide nucleic acids, [START_REF] Egholm | Peptide nucleic acids (PNA). Oligonucleotide analogs with an achiral peptide backbone[END_REF] peptoids [START_REF] Sun | Peptoid Polymers: A Highly Designable Bioinspired Material[END_REF] and other unnatural polymers [START_REF] Núñez-Villanueva | Sequence information transfer using covalent template-directed synthesis[END_REF][START_REF] Mondal | Efficient Protocol for the Synthesis of "N-Coded" Oligoand Poly(N-Substituted Urethanes)[END_REF] ). Such studies could help to assess if homochirality is also a mandatory requirement for artificial Life. If the implementation of mutations in synthetic informational polymers seems easier from a polymer chemistry point of view, this aspect has barely been studied, as underlined in the previous section. Thus, research shall be conducted in order to design functional repeat units that can be deleted or altered by simple chemical or physical triggers.

The limits of rational design. As discussed in the previous section, synthetic polymer design is mandatory to achieve artificial Life-supporting macromolecules. Yet, it is highly unlikely that artificial Life will be reached by pure rational design. There are two main for that. The first one is that the chemical search space (i.e. number of possible reactions, building blocks and macromolecular parameters) is extremely wide, not to say infinite. Taking into account that the number of possible sequences in a 100mer protein (which is only a single type of polymer) exceeds the number of atoms in the observable Universe, 57 the situation will be even worse considering more types of monomers and polymers. Even with the help of chemoinformatics, including recent tools such as machine learning and deep learning, it is unlikely that an optimal Life-supporting macromolecule can be entirely predicted. The second limiting aspect is the fact that the largest majority of polymer structures that are synthesized by traditional polymer chemistry are stable molecules than can be isolated, stored and characterized. In contrast to that, living systems are not at thermodynamic equilibrium. [START_REF] Schneider | Life as a manifestation of the second law of thermodynamics[END_REF] In order to support Life, a macromolecular replicator has to be involved in an out-of-equilibrium metabolic network. Similarly, artificial Life shall rely on dissipative systems, in which a Life-supporting macromolecule shall be in constant interaction with nutrients, waste and eventually other components. Many of these aspects are covered by the emerging domain of systems chemistry, [START_REF] Ludlow | Systems chemistry[END_REF] even though very few works have been done so far in this area with synthetic mac-romolecules. Thus, the study of non-equilibrium chemical networks involving abiotic informational polymers could be a first step in the direction of polymer-based artificial Life. Furthermore, if a mutable selfreplicating artificial macromolecule could be achieved, such non-equilibrium systems would become adaptive. [START_REF] Lehn | Perspectives in Chemistry-Aspects of Adaptive Chemistry and Materials[END_REF] In other words, they would evolve through generations.

But a crucial question is where to start? Starting from a random mixture of low molecular weight components may lead to the emergence of Life, but, as learned from the development of our planet, it is an extremely long process (on the scale of hundreds of millions to billions of years). Therefore, the process shall be somehow accelerated by Human selection. For example, starting (macro)molecular components could be engineered by polymer chemists and afterwards involved in non-equilibrium systems. Consequently, the combination of conventional polymer design and systems chemistry could be the key towards the development of polymer-based artificial Life. The first fundamental aspect that could be optimized by chemists if the environment, in which such macromolecular systems shall proceed. One feature that is shared by all known living organisms is compartmentalization, [START_REF] Yewdall | The hallmarks of living systems: towards creating artificial cells[END_REF] which is the fact that a self-replicating chemical machinery is confined in a compartment (e.g. a spherical lipid membrane) that keeps it apart from the outside environment; while still allowing selective exchanges between internal and external milieu. Although isolation can be achieved in a variety of colloidal systems, [START_REF] Lutz | Can Life Emerge from Synthetic Polymers?[END_REF] model chemical networks shall probably be first studied in flow reactors allowing chemical input and output. Yet, another important parameter is the choice of the solvent. Even though water is an obvious choice, it would be highly important to assess if artificial Life could be obtained in non-aqueous solvents. For instance, the possibility to find Life in solvents, such as liquid ammonia, formamide and liquid alkanes, has already been discussed. [START_REF] Benner | Is there a common chemical model for life in the universe?[END_REF] Last but not least, monomer design could be optimized by Human selection. For instance, the formation of stable backbones (e.g. C-C) shall probably be avoided. As learned from Nature -for example, from the natural selection of phosphate linkages in water [START_REF] Westheimer | Why nature chose phosphates[END_REF][START_REF] Benner | DNA, and the Search for Nonterrean Life: A Second Generation Model for Genetic Molecules[END_REF] -heteroatom-containing backbones (e.g. including phosphodiesters, esters, amides, imines, carbonates or carbamates linkages) could be more useful for attaining Life-supporting properties since they may enable dynamic rearrangement, degradation and repair.

In addition, the chemical composition and stereochemistry of the building-blocks could be predefined by Human reasoning, combinatorial experiments and predictive chemo-informatics modeling. For instance, the development of pseudo-genetic monomer alphabets (e.g. type and number of symbols) and monomermonomer communication (e.g. use of specific side-chain associating motifs) could be pre-optimized by in silico design and information theory. [START_REF] Lutz | Coding Macromolecules: Inputting Information in Polymers Using Monomer-Based Alphabets[END_REF] Outlook. Nucleic acids, such as DNA and RNA, are probably not the only family of macromolecules that can support Life. This is one possible option that has emerged on Earth through very long period of times under environmental selective pressure. In other words, the formation of these macromolecules has been guided by the history of terrestrial chemical variations (assuming that the primordial soup theory is correct and that Earth was not seeded by extra-terrestrial sources). [START_REF] Lutz | Can Life Emerge from Synthetic Polymers?[END_REF] It could be that this option is, in fact, the only viable one to support Life in the whole Universe. Nevertheless, it is not unreasonable to assume that alternatives could exist. [START_REF] Benner | Is there a common chemical model for life in the universe?[END_REF] Other types of Life-supporting macromolecules may already have emerged somewhere else in the Universe; i.e. on planets having chemical compositions that differ from the one of Earth. Alternatively, they may be created by intelligent species like our one. Yet, as discussed in this viewpoint, there are still many obstacles to overcome, in terms of polymer chemistry and systems chemistry, before polymer-based artificial Life could be attained. Nonetheless, this is not science fiction. On the contrary, the quest for polymer-based artificial Life appears as an exciting challenge to start the second century of Human polymer science.

Acknowledgment

The excellence network Chemistry of Complex Systems (LabEx CSC) and the CNRS are acknowledged for recurrent financial support.