Louise Breuil

Solène Marie

Sébastien Goutal

Sylvain Auvity

Charles Truillet

Wadad Saba

Oliver Langer

Fabien Caillé

Nicolas Tournier
email: n.tournier@universite-paris-saclay.fr

Comparative vulnerability of PET radioligands to partial inhibition of P-glycoprotein at the blood-brain barrier: a criterion of choice?

Keywords: ATP-binding cassette, drug-drug interaction, membrane transporter, neuropharmacology, pharmacokinetics

Only partial deficiency/inhibition of P-glycoprotein (P-gp, ABCB1) function at the blood-brain barrier (BBB) is likely to occur in pathophysiological situations or drug-drug interactions. This raises questions regarding the sensitivity of available PET imaging probes to detect moderate changes in P-gp function at the living BBB. In vitro, the halfmaximum inhibitory concentration (IC50) of the potent P-gp inhibitor tariquidar in P-gp-overexpressing cells was significantly different using either [ 11 C]verapamil (44 nM), [ 11 C]N-desmethyl-loperamide (19 nM) or [ 11 C]metoclopramide (4 nM) as substrate probes. In vivo PET imaging in rats showed that the half-maximum inhibition of P-gp-mediated efflux of [ 11 C]metoclopramide, achieved using 1 mg/kg tariquidar (in vivo IC50 = 82 nM in plasma), increased brain exposure by 2.1-fold for [ 11 C]metoclopramide (p<0.05, n=4) and 2.4-fold for [ 11 C]verapamil (p<0.05, n=4), whereby cerebral uptake of the "avid" substrate [ 11 C]N-desmethyl-loperamide was unaffected (p>0.05, n=4). This comparative study points to differences in the "vulnerability" to P-gp inhibition among radiolabeled substrates, which were apparently unrelated to their "avidity" (maximal response to P-gp inhibition). Herein, we advocate that partial inhibition of transporter function, in addition to complete inhibition, should be a primary criterion of evaluation regarding the sensitivity of radiolabeled substrates to detect moderate but physiologically-relevant changes in transporter function in vivo.

Introduction

P-glycoprotein (P-gp, ABCB1

) is an important efflux transporter of the ATP-binding cassette (ABC) transporter family, which is notably expressed at the vascular pole of endothelial cells forming the blood-brain barrier (BBB). P-gp selectively regulates the BBB transfer of many drugs, metabolites and endogenous compounds. [START_REF] Abbott | Structure and function of the bloodbrain barrier[END_REF] There is growing evidence that P-gp expression and function are impaired in various conditions, including normal ageing, Alzheimer's disease or neuroinflammatory states. [START_REF] Gil-Martins | Dysfunction of ABC transporters at the bloodbrain barrier: Role in neurological disorders[END_REF] Overexpression of P-gp has been reported in brains of rat epilepsy models [START_REF] Bartmann | Imaging of P-glycoproteinmediated pharmacoresistance in the hippocampus: proof-of-concept in a chronic rat model of temporal lobe epilepsy[END_REF] and in patients with epilepsy [START_REF] Löscher | Drug resistance in brain diseases and the role of drug efflux transporters[END_REF][START_REF] Liu | Neuropathology of the blood-brain barrier and pharmaco-resistance in human epilepsy[END_REF] , suggesting a relationship with resistance to anti-seizure medication. Active P-gp-mediated efflux is therefore increasingly regarded as a dynamic biomarker of BBB function, which may be linked to the onset of neurological disorders. [START_REF] Gil-Martins | Dysfunction of ABC transporters at the bloodbrain barrier: Role in neurological disorders[END_REF] Preclinical studies have shown that the brain distribution of many positron emission tomography (PET) imaging radioligands is restricted by P-gp function at the BBB [START_REF] García-Varela | Pharmacokinetic Modeling of [18F]MC225 for Quantification of the P-Glycoprotein Function at the Blood-Brain Barrier in Non-Human Primates with PET[END_REF][START_REF] Pike | PET radiotracers: crossing the blood-brain barrier and surviving metabolism[END_REF] . However, only few radiolabeled substrates have so far been used as dedicated probes to selectively assess P-gp function at the human BBB. These include [ 11 C]verapamil [START_REF] Muzi | Imaging of cyclosporine inhibition of P-glycoprotein activity using 11C-verapamil in the brain: studies of healthy humans[END_REF] (racemic or the (R)-enantiomer analogue of the calcium channel blocker verapamil), [ 11 C]Ndesmethyl-loperamide [START_REF] Kreisl | P-glycoprotein function at the blood-brain barrier in humans can be quantified with the substrate radiotracer 11C-N-desmethyl-loperamide[END_REF] (a radiolabeled metabolite of loperamide, an antidiarrheal opioid drug) and [ 11 C]metoclopramide 10 (analogue of metoclopramide, a dopamine receptor antagonist used as an antiemetic drug)

. The particular characteristics of P-gp-mediated transport at the BBB level as compared to conventional neuroimaging targets (e.g. receptor proteins) have prompted the need for new criteria to describe and select effective PET probes for P-gp activity [START_REF] Kannan | Imaging the function of Pglycoprotein with radiotracers: pharmacokinetics and in vivo applications[END_REF] . A first prerequisite is selectivity for P-gp over breast cancer resistance protein (BCRP, ABCG2), the other major ABC transporter at the BBB. Second, a limited contribution of radiometabolites to the brain PET signal is preferred for correct estimation of P-gp function at the BBB. Third, "avid" substrates were initially preferred as probes, with the aim to generate maximal contrast in the PET signal between the baseline condition, in which P-gp is fully functional, and a pharmacological inhibition challenge in which P-gp-mediated efflux is attenuated or abolished. Strains of P-gp deficient mice are useful tools to assess this last criterion in vivo [START_REF] Doran | The impact of P-glycoprotein on the disposition of drugs targeted for indications of the central nervous system: evaluation using the MDR1A/1B knockout mouse model[END_REF][START_REF] Dagenais | Development of an In Situ Mouse Brain Perfusion Model and its Application to mdr1a P-Glycoprotein-Deficient Mice[END_REF] . However, from a clinical perspective, complete deletion or genetic deficiency of P-gp expression is extremely rare in humans. [START_REF] Baudou | Serious Ivermectin Toxicity and Human ABCB1 Nonsense Mutations[END_REF] Indeed, only partial decreases in P-gp expression have been found in histology or proteomics studies of human specimens. [START_REF] Gil-Martins | Dysfunction of ABC transporters at the bloodbrain barrier: Role in neurological disorders[END_REF][START_REF] Sauna | Genomics and the mechanism of P-glycoprotein (ABCB1)[END_REF] This raises the important question of the sensitivity of available PET probes to detect the functional impact of moderate changes in P-gp expression at the BBB.

PET studies using either [ 11 C]verapamil or [ 11 C]N-desmethyl-loperamide have clearly shown the efficacy of pharmacological inhibitors to dose dependently decrease P-gp function at the human BBB. [START_REF] Kreisl | Increased permeabilityglycoprotein inhibition at the human blood-brain barrier can be safely achieved by performing PET during peak plasma concentrations of tariquidar[END_REF][START_REF] Bauer | Pgp-mediated interaction between (R)-[11C]verapamil and tariquidar at the human blood-brain barrier: a comparison with rat data[END_REF] Tariquidar is a highly potent and specific P-gp inhibitor that can be safely used to achieve an almost maximal decrease in P-gp function at the human BBB. [START_REF] Kreisl | Increased permeabilityglycoprotein inhibition at the human blood-brain barrier can be safely achieved by performing PET during peak plasma concentrations of tariquidar[END_REF][START_REF] Bauer | Pgp-mediated interaction between (R)-[11C]verapamil and tariquidar at the human blood-brain barrier: a comparison with rat data[END_REF][START_REF] Bauer | Approaching complete inhibition of P-glycoprotein at the human blood-brain barrier: an (R)-[11C]verapamil PET study[END_REF] Many marketed drugs are described as P-gp inhibitors in vitro, although with a lower potency or efficacy than tariquidar. [START_REF] Traxl | Inhibition of ABCB1 and ABCG2 at the Mouse Blood-Brain Barrier with Marketed Drugs To Improve Brain Delivery of the Model ABCB1/ABCG2 Substrate [11C]erlotinib[END_REF][START_REF] Matsson | Identification of Novel Specific and General Inhibitors of the Three Major Human ATP-Binding Cassette Transporters P-gp, BCRP and MRP2 Among Registered Drugs[END_REF] Clinical PET data using [ 11 C]verapamil or [ 11 C]N-desmethylloperamide as model P-gp substrates have been used as reference points to estimate the risk for P-gp-mediated drug-drug interactions at the human BBB. [START_REF] Kalvass | Why clinical modulation of efflux transport at the human blood-brain barrier is unlikely: the ITC evidence-based position[END_REF] These results led to the assumption that a high degree of P-gp inhibition by marketed drugs is very unlikely to occur, so that only partial inhibition can be achieved in clinical practice. [START_REF] Kalvass | Why clinical modulation of efflux transport at the human blood-brain barrier is unlikely: the ITC evidence-based position[END_REF][START_REF] Eyal | Drug interactions at the blood-brain barrier: fact or fantasy?[END_REF] [ 11 C]Metoclopramide has been designed as an alternative P-gp probe for PET imaging, based on the need for a PET probe to detect induction of P-gp function at the BBB, starting from substantial baseline brain uptake, when P-gp is normally expressed. [START_REF] Zoufal | Imaging P-Glycoprotein Induction at the Blood-Brain Barrier of a β-Amyloidosis Mouse Model with 11C-Metoclopramide PET[END_REF][START_REF] Pottier | Imaging the Impact of the P-Glycoprotein (ABCB1) Function on the Brain Kinetics of Metoclopramide[END_REF] [ 11 C]Metoclopramide corresponds to the definition of a "weak" P-gp substrate, thus mimicking the characteristics of many CNS active drugs including antiepileptic drugs, antidepressants, antipsychotics or opioids. [START_REF] Löscher | Drug resistance in brain diseases and the role of drug efflux transporters[END_REF] The overall increase in brain exposure to [ 11 C]metoclopramide after a high degree or nearly complete P-gp inhibition was consequently lower than that observed with the "avid" substrates [START_REF] Pottier | Imaging the Impact of the P-Glycoprotein (ABCB1) Function on the Brain Kinetics of Metoclopramide[END_REF][START_REF] Wanek | Radioligands targeting P-glycoprotein and other drug efflux proteins at the blood-brain barrier[END_REF][START_REF] Bauer | Imaging P-glycoprotein function at the blood-brain barrier as a determinant of the variability in response to CNS drugs[END_REF] However, it may be hypothesized that [ 11 C]metoclopramide with its lower transport rate and higher BBB permeability may show better sensitivity to detect partial P-gp inhibition compared with previous PET radioligands. [START_REF] Bauer | Imaging P-glycoprotein function at the blood-brain barrier as a determinant of the variability in response to CNS drugs[END_REF] This would imply a higher "vulnerability" to clinically relevant drug-drug interactions with P-gp inhibitors, resulting in increased brain exposure.

[ 11 C]verapamil or [ 11 C]N-desmethyl-loperamide.
The present study aimed at comparing the "vulnerability" of [ 11 C]verapamil, [ 11 C]N-desmethyl-loperamide and [ 11 C]metoclopramide to partial P-gp inhibition. To this end, standardized in vitro uptake assays using cells expressing human P-gp and in vivo PET experiments in rats were performed with these 3 substrates, in the absence or presence of increasing doses of tariquidar. Based on these data, we propose new and important criteria to describe and compare the sensitivity of radiolabeled substrate probes to detect moderate changes in P-gp function at the BBB.

MATERIALS AND METHODS

Chemicals and radiotracers

Tariquidar used for P-gp inhibition was purchased from Eras Labo (France). Tariquidar solutions for intravenous injection were freshly prepared at the selected concentration on the day of the experiment by dissolving tariquidar dimesylate in dextrose solution (5%, w/v), followed by dilution with sterile water. Three radiolabeled P-gp probes suitable for human use were compared. Ready-to-inject [ 11 C]metoclopramide, racemic [ 11 C]verapamil and [ 11 C]N-desmethyl-loperamide were synthetized as previously described [START_REF] Caillé | Positron Emission Tomography Imaging Reveals an Importance of Saturable Liver Uptake Transport for the Pharmacokinetics of Metoclopramide[END_REF][START_REF] Damont | Imaging the impact of cyclosporin A and dipyridamole on P-glycoprotein (ABCB1) function at the blood-brain barrier: A [(11)C]-N-desmethylloperamide PET study in nonhuman primates[END_REF][START_REF] Luurtsema | Fully automated high yield synthesis of (R)-and (S)-[11C]verapamil for measuring P-glycoprotein function with positron emission tomography[END_REF] starting from cyclotron-produced [ 11 C]carbon dioxide (Cyclone-18/9 cyclotron; IBA, Belgium) by [ 11 C]methylation of normetoclopramide (Toronto Chemicals, Canada), racemic nor-verapamil (ABX advanced biomedical compounds, Germany) or in-house synthesized di-desmethyl-loperamide 28 using a TRACERLab FX CPro synthesizer (GE Healthcare, France). Quality control was performed by radio-high-performance liquid chromatography (HPLC) to assess the identity, radiochemical and chemical purities of the radiotracers and their molar activities.

Cells

Culture media and buffers were obtained from Fischer Scientific, France. Stably transfected MDCKII-MDR1 cells were obtained from Dr. Alfred Schinkel (National Cancer Institute, The Netherlands) and were grown under a controlled atmosphere at 37°C, 5% CO2. The culture medium was composed of DMEM Glutamax (Dulbecco's Modified Eagle Medium, 4.5 g/Ldextrose, 1 mM pyruvate) supplemented with 10% fetal bovine serum and 1% antibiotics (penicillin and streptomycin 5000 U/mL).

Uptake assay in P-gp overexpressing cells

P-gp-mediated transport of [ 11 C]verapamil and [ 11 C]N-desmethyl-loperamide and [ 11 C]metoclopramide was compared in cells expressing human P-gp. Cells were seeded in 24-well plates (30,000 cells per well) in 500 μL culture medium. Cells were grown to confluence (~2 days). On the day of the experiment, culture medium was removed and replaced by 200 μL of incubation buffer (10% HBSS (Hanks' Balanced Salt Solution) + 1.26 mM CaCl2 + 0.49 mM MgCl2) containing 1 mM pyruvate and 10 mM HEPES (4-(2-hydroxyethyl)-1piperazineethanesulfonic acid, 37°C). The incubation buffer contained the tested radiolabeled P-gp substrate (~37 MBq/40 mL corresponding to <1 µg/40 mL of the unlabeled compound) and tariquidar at the selected concentration. Tariquidar was dissolved in DMSO and tariquidar concentrations ranged from 0 to 200 nM (1% v/v final DMSO concentration). After 30 minutes of incubation at 37°C, buffer was removed and cell monolayers were rapidly washed with 300 μL of ice-cold buffer Dulbecco's phosphate buffer. Cells were then lysed with 500 μL of NaOH (10 mM, 10 min). Then, 400 μL of cell lysate was collected from each well (n = 4 wells per condition) and gamma-counted using a Cobra Quantum (Perkin-Elmer, France).

Animals

Thirty-six male Sprague Dawley rats (Janvier, France) were used for the study (mean weight: 354±76 g). Animals were housed and acclimatized for at least 3 days before the experiments. Rats had free access to chow and water. All animal experiments were in accordance with the recommendations of the European Community (2010/63/UE) and the French National Committees (law 2013-118) for the care and use of laboratory animals.

The experimental protocol was approved by a local ethics committee for animal use (CETEA) and by the French ministry of agriculture (APAFIS#7466-20 1611 04 1 7049220 v2). Samples size for each group was based on previous studies. [START_REF] Pottier | Imaging the Impact of the P-Glycoprotein (ABCB1) Function on the Brain Kinetics of Metoclopramide[END_REF][START_REF] Farwell | Imaging P-glycoprotein function in rats using [11C]-N-desmethyl-loperamide[END_REF] Reporting of animal data is in compliance with the ARRIVE (Animal Research: Reporting in Vivo Experiments) guidelines.

PET experiments

PET protocol

PET acquisitions were performed using an Inveon microPET scanner (Siemens, Knoxville, TN, US). Anesthesia was induced and then maintained using 3.5% followed by 1.5-2.5% isoflurane in pure oxygen. Thirty-minute dynamic scans were acquired, starting with intravenous bolus injection of

P-gp inhibition protocol

Tariquidar was used to modulate P-gp function in rats. Tariquidar doses up to 8 mg/kg were shown to be well tolerated with a negligible impact of tariquidar on the arterial input function and peripheral metabolism of

Determination of tariquidar in plasma

Blood samples (~2 mL) were collected under anesthesia at the end of the PET acquisition (i.e. at 65 ± 2 min after tariquidar injection) by intracardiac puncture. Afterwards animals were killed using 180 mg/kg i.v. pentobarbital (Dolethal ® , France). Samples were centrifuged and plasma was collected and stored at -80°C until analysis.

Determination of tariquidar concentration in plasma samples was performed using a validated HPLC method using elacridar as an internal standard. [START_REF] Auvity | P-Glycoprotein (ABCB1) Inhibits the Influx and Increases the Efflux of 11C-Metoclopramide Across the Blood-Brain Barrier: A PET Study on Nonhuman Primates[END_REF] Calibration curves for the analysis of plasma samples were generated by analyzing different dilutions of a tariquidar solution in drug-free human plasma (obtained from Etablissement Français du Sang, France).

Data analysis and statistics

In vitro data

Counting values obtained in each well (R) were normalized using either the concentration of radioactivity in the incubation buffer (=R/C0) or using the following equation:

where I% is the extent of inhibition (I%) between 0 and 100%, R is the radioactivity in the 4 tested wells, mR0 is the mean radioactivity of each radioligand in the 4 wells without tariquidar, and mRmax is the mean radioactivity of each radioligand in the 4 wells containing the highest concentration of tariquidar (200 nM).

𝐼% = 𝑅 -𝑚𝑅

! 𝑚𝑅 "#$ -𝑚𝑅 ! × 100
The in vitro half-maximum inhibitory concentration (IC50) was estimated by non-linear regression using the "Onesite binding Hill equation" function in Graphpad Prism software (V8.0, San Diego, CA, USA) with maximal uptake constrained to 100%. IC50 values obtained for each radiotracer were considered different when their 95%

confidence interval [CI95%] did not overlap. The uptake ratio was calculated as Rmax / mR0 (mean ± S.D) for each P-gp substrate. Normality of the data was checked using the Shapiro-Wilk's test and homoscedasticity using the Brown-Forsythe's test. Uptake ratios were then compared using one-way ANOVA and Tukey's post hoc test.

PET data analysis

Images were reconstructed with the Fourier rebinning algorithm and the 3-dimensional ordered-subset expectation maximization algorithm including normalization, attenuation, scatter, and random corrections. Image analysis and quantification of radioactivity uptake were performed using PMOD software (version 3.9; PMOD Technologies, Zürich, Switzerland). A region of interest was drawn over the whole brain to generate time-activity curves (TACs) with time frame durations of 0.25 min, 2 x 0.5 min, 0.75 min, 4 x 1 min, 1.5 min, 4 x 2 min, 3 x 2.5 min, 3 x 3 min and 3.5 min. Brain radioactivity was corrected for 11 C decay, injected dose and animal weight, and expressed as the standardized uptake value (SUV).

In rats, the determination of an arterial input function during the PET scan is very challenging. The impact of P-gp inhibition was therefore estimated from the area under the brain TAC, which was determined from 10 to 30 minutes after radiotracer injection (AUC10-30 min) to limit the pharmacokinetic variability associated with the bolus injection procedure. First, the doses that inhibited 50% (ID50%/Meto) and 100% (ID100%/Meto) of the P-gp-mediated transport of Graphical and statistical analysis were performed using Graphpad ® prism software. Normality of the data was checked using the Shapiro-Wilk's test and homoscedasticity using the Spearman's test. Brain AUC10-30 min values obtained under all tested conditions were compared using 2-way ANOVA analysis and compared with the baseline condition using the Dunett's post-hoc test.

RESULTS

In vitro experiments

The in vitro P-gp inhibitory potency of tariquidar was tested in P-gp overexpressing cells using either

In vivo experiments

Under the baseline condition, [ 11 C]metoclopramide PET signal in the rat brain was lower than that in tissues surrounding the skull (Fig. 2). The PET signal reached similar levels as peripheral tissues in rats injected with 1 mg/kg of tariquidar. Tariquidar doses > 1 mg/kg dose dependently increased the brain PET signal with a limited impact on radioactivity in peripheral tissues (Fig. 2). This is consistent with a localization of P-gp at the BBB and a negligible impact of tariquidar on the peripheral pharmacokinetics of [ 11 C]metoclopramide. [START_REF] Pottier | Imaging the Impact of the P-Glycoprotein (ABCB1) Function on the Brain Kinetics of Metoclopramide[END_REF] Dose-response and concentration-response curves of tariquidar for inhibition of P-gp-mediated transport of [ 11 C]metoclopramide at the rat BBB are shown in Figure 2. The estimated ID50%/Meto of tariquidar was 1 mg/kg [0.78-1.5 mg/kg] (IC50%/Meto ~82 nM). Maximal P-gp inhibition was observed for tariquidar doses ≥ 2 mg/kg (IC100%/Meto ~300 nM) using [ 11 C]metoclopramide as a substrate probe.

The 1 mg/kg dose (ID50%/Meto) and the 8 mg/kg dose (ID100%/Meto) of tariquidar were then selected to test the impact of partial and maximal P-gp inhibition using [ 11 C]verapamil and [ 11 C]N-desmethyl-loperamide as probes. Figure 3 shows representative [ 11 C]verapamil, [ 11 C]N-desmethyl-loperamide and [ 11 C]metoclopramide PET images obtained at baseline and after injection of either 1 or 8 mg/kg of tariquidar. Maximal P-gp inhibition using 8 mg/kg tariquidar visually enhanced the brain PET signal (Fig. 3) and significantly increased AUC10-30 min compared to baseline (p<0.0001) for all three radiotracers (Fig. 4D). The rank order of increases was [

DISCUSSION

PET imaging using radiolabeled P-gp substrates has convincingly demonstrated the role of this efflux transporter in limiting the brain distribution of its substrates across the BBB in humans. [START_REF] Tournier | Impact of P-Glycoprotein Function on the Brain Kinetics of the Weak Substrate 11C-Metoclopramide Assessed with PET Imaging in Humans[END_REF][START_REF] Kreisl | Increased permeabilityglycoprotein inhibition at the human blood-brain barrier can be safely achieved by performing PET during peak plasma concentrations of tariquidar[END_REF][START_REF] Bauer | Approaching complete inhibition of P-glycoprotein at the human blood-brain barrier: an (R)-[11C]verapamil PET study[END_REF][START_REF] Sasongko | Imaging P-glycoprotein transport activity at the human blood-brain barrier with positron emission tomography[END_REF][START_REF] Bauer | Assessment of regional differences in tariquidar-induced P-glycoprotein modulation at the human blood-brain barrier[END_REF] Efforts have been made to develop optimized PET probes to quantitatively address the importance of P-gp function at the BBB in neurological disorders with good sensitivity. [START_REF] Kannan | Imaging the function of Pglycoprotein with radiotracers: pharmacokinetics and in vivo applications[END_REF][START_REF] Tournier | Imaging techniques to study drug transporter function in vivo[END_REF][START_REF] Luurtsema | PET Tracers for Imaging of ABC Transporters at the Blood-Brain Barrier: Principles and Strategies[END_REF] Previous studies have preferentially classified radiolabeled probes as "avid"

or "weak" P-gp substrates which refers to their maximal capacity of transport. However, a direct in vitro and in vivo comparison of all three PET probes currently available for human use, performed in parallel conditions, has not yet been performed. Moreover, the present work focused on the impact of partial rather than complete P-gp inhibition at the BBB on the brain distribution of radiolabeled P-gp substrates with the aim to reflect clinically relevant situations of a moderate decrease in P-gp function. Strikingly, our results point to significant differences in the "vulnerability" of radiolabeled substrates to P-gp inhibition at the BBB, which contributes to their "sensitivity"

to detect changes in P-gp function using PET imaging.

Differences in the substrate properties of [ C]N-desmethylloperamide (Fig. 1). The magnitude of the uptake ratios was consistent with the transport ratios of these compounds measured using conventional bidirectional transport assays, which were 2.1 for verapamil, 3.8 for loperamide and 1.4 for metoclopramide. [START_REF] Feng | In vitro P-glycoprotein assays to predict the in vivo interactions of P-glycoprotein with drugs in the central nervous system[END_REF] Of note, the uptake assay employed in our work is more suitable to test compounds radiolabeled with short-lived isotopes such as carbon-11 than the conventional bidirectional transport assay. In vitro, [ 11 C]metoclopramide was identified as one of the "weakest" P-gp substrates (lowest uptake ratio, similar to that of [ 11 C]verapamil) but showed the highest "vulnerability" to P-gp inhibition (lowest IC50 of tariquidar).

Interestingly, [ 11 C]verapamil appeared to be a "weaker" P-gp substrate than [ 11 C]N-desmethyl-loperamide (lower uptake ratio) but showed a lower "vulnerability" to P-gp inhibition (higher IC50). Importantly, these comparative in vitro data suggest that the "avidity" and the "vulnerability" of P-gp substrates are independent factors that are not directly or inversely correlated.

PET experiments were needed to compare the "vulnerability" of the three probes to P-gp inhibition in the living brain. Several factors other than P-gp function may additionally influence the PET signal obtained with the investigated radiotracers in the living brain. For instance, lysosomal trapping was shown to account for the irreversible uptake of [ 11 C]N-desmethyl-loperamide in the brain. [START_REF] Kannan | Lysosomal trapping of a radiolabeled substrate of P-glycoprotein as a mechanism for signal amplification in PET[END_REF] The [ 11 C]verapamil brain PET signal comprises a considerable proportion of radiometabolites. [START_REF] Luurtsema | Evaluation of (R)-[11C]verapamil as PET tracer of P-glycoprotein function in the blood-brain barrier: kinetics and metabolism in the rat[END_REF] [ 11 C]Metoclopramide shows a significant brain-to-blood efflux component that contributes to limit brain exposure. [START_REF] Pottier | Imaging the Impact of the P-Glycoprotein (ABCB1) Function on the Brain Kinetics of Metoclopramide[END_REF] These particular kinetic properties may also contribute to the in vivo sensitivity of radiolabeled P-gp substrates to detect moderate changes in P-gp function using PET [START_REF] Luurtsema | Evaluation of (R)-[11C]verapamil as PET tracer of P-glycoprotein function in the blood-brain barrier: kinetics and metabolism in the rat[END_REF] .

Dedicated kinetic models have been developed to estimate P-gp function, taking the individual properties of each radioligand into account. [START_REF] Pottier | Imaging the Impact of the P-Glycoprotein (ABCB1) Function on the Brain Kinetics of Metoclopramide[END_REF][START_REF] Lubberink | Evaluation of tracer kinetic models for quantification of P-glycoprotein function using (R)-[11C]verapamil and PET[END_REF][START_REF] Müllauer | Pharmacokinetic modeling of P-glycoprotein function at the rat and human blood-brain barriers studied with (R)-[11C]verapamil positron emission tomography[END_REF][START_REF] Farwell | Imaging P-glycoprotein function in rats using [(11)C]-N-desmethyl-loperamide[END_REF][START_REF] Lubberink | Kinetic Models for Measuring P-glycoprotein Function at the Blood-Brain Barrier with Positron Emission Tomography[END_REF] However, these models do not apply to all 3 radioligands. Tariquidar was selected as an inhibitor because doses up to 8 mg/kg were shown to not significantly impact the peripheral pharmacokinetics and metabolism of the tested radioligands in rats. [START_REF] Pottier | Imaging the Impact of the P-Glycoprotein (ABCB1) Function on the Brain Kinetics of Metoclopramide[END_REF][START_REF] Farwell | Imaging P-glycoprotein function in rats using [11C]-N-desmethyl-loperamide[END_REF][START_REF] Bankstahl | Tariquidar-induced Pglycoprotein inhibition at the rat blood-brain barrier studied with (R)-11C-verapamil and PET[END_REF] Using kinetic modeling and an arterial input function, brain AUC has been previously validated as a simplified parameter to describe the response to tariquidar for [ 11 C]metoclopramide 32 and [ 11 C]N-desmethyl-loperamide [START_REF] Kreisl | Increased permeabilityglycoprotein inhibition at the human blood-brain barrier can be safely achieved by performing PET during peak plasma concentrations of tariquidar[END_REF] and (R)-[ 11 C]verapamil (personal communication, Dr. Oliver Langer). We therefore used the brain AUC as a model-independent parameter to describe the increase in the brain exposure obtained after different degrees of P-gp inhibition, assuming negligible changes in the plasma kinetics of the investigated radioligands in the presence of increasing doses of tariquidar up to 8 mg/kg.

Mice that are heterozygous for the genes encoding P-gp (Abcb1a/1b (+/-) ) were shown to have approximately 50% lower P-gp expression levels at the BBB than wild-type mice. [START_REF] Umbenhauer | Identification of a Pglycoprotein-deficient subpopulation in the CF-1 mouse strain using a restriction fragment length polymorphism[END_REF] Interestingly, no significant increases in the brain uptake of (R)-[ 11 C]verapamil and [ 11 C]N-desmethyl-loperamide were observed in Abcb1a/1b (+/-) mice as compared with wild-type mice. [START_REF] Wanek | Factors Governing P-Glycoprotein-Mediated Drug-Drug Interactions at the Blood-Brain Barrier Measured with Positron Emission Tomography[END_REF] This suggested that (R)- ]verapamil, although the latter is a much more "avid" substrate of P-gp at the rat BBB (Fig. 4). A direct comparison remains however difficult because radiometabolites may contribute to the brain PET signal for [ 11 C]verapamil but not for [ 11 C]metoclopramide. [START_REF] Pottier | Imaging the Impact of the P-Glycoprotein (ABCB1) Function on the Brain Kinetics of Metoclopramide[END_REF][START_REF] Luurtsema | Evaluation of (R)-[11C]verapamil as PET tracer of P-glycoprotein function in the blood-brain barrier: kinetics and metabolism in the rat[END_REF] Nevertheless, this suggests that the "weak" P-gp substrate

[ 11 C]metoclopramide offers an adequate sensitivity to detect moderate decreases in P-gp function, which is probably linked to its higher "vulnerability" to P-gp inhibition. [START_REF] Pottier | Imaging the Impact of the P-Glycoprotein (ABCB1) Function on the Brain Kinetics of Metoclopramide[END_REF] In rats, [ 11 C]N-desmethyl-loperamide behaved as an "avid" and "refractory" substrate, with a limited response to partial P-gp inhibition. Given its excellent metabolic stability in humans and its favorable neuropharmacokinetic properties, [ 11 C]N-desmethyl-loperamide should be preferably employed to explore situations in which a great degree of inhibition/repression of P-gp function is expected, with an expected large increase in the PET signal. [START_REF] Kreisl | Increased permeabilityglycoprotein inhibition at the human blood-brain barrier can be safely achieved by performing PET during peak plasma concentrations of tariquidar[END_REF][START_REF] Farwell | Imaging P-glycoprotein function in rats using [11C]-N-desmethyl-loperamide[END_REF][START_REF] Lazarova | Synthesis and Evaluation of [Nmethyl-11C]N-Desmethyl-loperamide as a New and Improved PET Radiotracer for Imaging P-gp Function[END_REF][START_REF] Goutal | Physical blood-brain barrier disruption induced by focused ultrasound does not overcome the transporter-mediated efflux of erlotinib[END_REF] Altogether, our results suggest that imaging "sensitivity" of PET probes to detect partial inhibition of P-gp at the BBB in vivo does not only depend on the "avidity" (i.e. the ability to generate contrast between situations of full and abolished P-gp function, related to the uptake ratio), as it also depends on the "vulnerability" of the substrate (i.e. the ability to reach nearly maximal inhibition, related to the IC50). rodents. [START_REF] Uchida | Quantitative targeted absolute proteomics of human blood-brain barrier transporters and receptors[END_REF][START_REF] Shawahna | Transcriptomic and quantitative proteomic analysis of transporters and drug metabolizing enzymes in freshly isolated human brain microvessels[END_REF] Moreover, many compounds show disparate P-gp-mediated transport between human and rodent Pgp in vitro. [START_REF] Feng | In vitro P-glycoprotein assays to predict the in vivo interactions of P-glycoprotein with drugs in the central nervous system[END_REF][START_REF] Tournier | Discrepancies in the P-glycoprotein-mediated transport of (18)F-MPPF: a pharmacokinetic study in mice and non-human primates[END_REF][START_REF] Baltes | Differences in the transport of the antiepileptic drugs phenytoin, levetiracetam and carbamazepine by human and mouse P-glycoprotein[END_REF] However, our comparative in vitro experiments, performed using cells overexpressing human Pgp, pointed to a higher "vulnerability" of The neuropharmacokinetic consequences of partial P-gp inhibition on the brain exposure to "weak" but "vulnerable" P-gp substrates in humans remain to be further explored. Using [ 11 C]metoclopramide PET imaging in rats, we show here that partial P-gp inhibition may have a relatively higher impact for brain exposure compared with more "avid" but "inhibition-refractory" substrates like [ 11 C]N-desmethyl-loperamide. At the molecular level, the factors that account for the "avidity" and "vulnerability" of P-gp substrates remain poorly understood. This may imply the complex molecular interaction of substrates and inhibitors at multiple binding domains of P-gp. [START_REF] Prajapati | Translocation mechanism of P-glycoprotein and conformational changes occurring at drug-binding site: Insights from multi-targeted molecular dynamics[END_REF] From a PET imaging perspective, this suggests a higher sensitivity of [ 11 C]metoclopramide to explore situations of moderate changes in P-gp function at the BBB. From a neuropharmacological perspective, our results suggest that partial inhibition/deficit of P-gp at the BBB may contribute to clinical variability in efficacy of CNS-targeted "vulnerable" P-gp substrates, which may deserve further attention.

CONCLUSION

The sensitivity of PET probes to detect moderate changes in P-gp function at the BBB appeared different among clinically validated radiolabeled substrates. Our results suggest that the "weak" but "vulnerable" P-gp substrate [ 11 C]metoclopramide offers good sensitivity as a PET probe to detect moderate changes in P-gp function in vivo.

We advocate that the future development of radiolabeled substrates for imaging the function of P-gp or other ABC efflux transporters at the BBB should also include an evaluation under conditions of partial transporter inhibition.

This may improve our understanding of the impact of clinically relevant situations of a moderate decline in transporter function at the BBB.

  [ 11 C]verapamil (40 ± 8 MBq; 2.0 ± 1.5 µg), [ 11 C]N-desmethyl-loperamide (35 ± 5 MBq, 1.7 ± 1.3 µg) or [ 11 C]metoclopramide (35 ± 5 MBq, 3.4 ± 1.3 µg), via a catheter inserted in the caudal lateral vein.

[ 11 C

 11 ]verapamil 31 , [ 11 C]N-desmethyl-loperamide 30 or [ 11 C]metoclopramide 24 . Tariquidar was injected in a volume of 200-300 µL into the caudal vein at 15 minutes before radiotracer injection. For [ 11 C]metoclopramide, a range of different tariquidar doses was tested (vehicle-0, 1, 2, 3, 4 or 8 mg/kg). PET acquisitions using [ 11 C]verapamil or [ 11 C]N-desmethyl-loperamide were performed after injection of vehicle and 1 or 8 mg/kg of tariquidar (n=4 per condition).

[ 11 C

 11 ]metoclopramide were estimated by fitting plots of brain AUC10-30 min versus the injected dose of tariquidar (n=1-4 animals per dose) with the non-linear one-site binding Hill equation. Brain AUC10-30 min was also plotted versus the plasma concentration of tariquidar to estimate IC50%/Meto, corresponding to the plasma concentration of tariquidar that inhibited 50% of the P-gp-mediated efflux of [ 11 C]metoclopramide at the BBB. Then, PET experiments were performed using [ 11 C]verapamil and [ 11 C]N-desmethyl-loperamide at baseline and after administration of tariquidar at ID50%/Meto or ID100%/Meto (n=4 per condition).

[ 11 C

 11 ]verapamil, [ 11 C]N-desmethyl-loperamide or [ 11 C]metoclopramide as substrate probes. The fitted concentration-inhibition curves for each radiotracer are shown in Fig. 1. Cellular uptake of the tested substrates was not significantly higher in the presence of 200 nM vs 100 nM tariquidar suggesting that maximal inhibition was achieved using 200 nM tariquidar. The estimated IC50 of tariquidar was 44 nM [33-53 nM] for [ 11 C]verapamil, 19 nM [13-25 nM] for [ 11 C]N-desmethyl-loperamide and 4 nM [2-8 nM] for [ 11 C]metoclopramide. The CI95% of theIC50 values of the different tested substrates did not overlap, suggesting that each substrate's "vulnerability" to partial P-gp inhibition was significantly different (Fig.1). Uptake ratios were 1.4 ± 0.09 for [ 11 C]verapamil, 3.9 ± 0.64 for [ 11 C]N-desmethyl-loperamide and 1.4 ± 0.07 for [ 11 C]metoclopramide. For [ 11 C]metoclopramide, maximal inhibition was achieved with ~50 nM of tariquidar. At this concentration, only 58% of the P-gp-mediated transport of [ 11 C]verapamil and 70% of the transport [ 11 C]N-desmethyl-loperamide was inhibited (Fig.1).

  [START_REF] Kannan | Imaging the function of Pglycoprotein with radiotracers: pharmacokinetics and in vivo applications[END_REF] C]verapamil(7.4-fold) > [ 11 C]N-desmethyl-loperamide (6.8-fold) > [ 11 C]metoclopramide (3.3-fold). The intermediate dose of tariquidar (1 mg/kg) visually increased the brain PET signal for [ 11 C]verapamil and [ 11 C]metoclopramide with a limited impact on brain uptake of [ 11 C]N-desmethyl-loperamide (Fig. 3). The corresponding increase in AUC10-30 min relative to baseline was significant for [ 11 C]verapamil (2.4-fold, p<0.05) and [ 11 C]metoclopramide (2.1-fold p<0.05), but not for [ 11 C]N-desmethyl-loperamide (1.8-fold, p>0.05, Fig. 4D).

  [ 11 C]verapamil and [ 11 C]N-desmethyl-loperamide lack the sensitivity to detect the consequences of a moderate decline (< 50%) in P-gp function at the BBB in vivo. In our study, the dose of tariquidar that inhibited 50% of the P-gp-mediated transport of [ 11 C]metoclopramide at the rat BBB increased the brain uptake of [ 11 C]verapamil, but not that of [ 11 C]N-desmethyl-loperamide. From a quantitative PET imaging perspective, the in vivo response to partial inhibition was similar for both [ 11 C]metoclopramide and [ 11 C

[ 11 C

 11 ]Metoclopramide and [ 11 C]verapamil PET imaging in rats showed similar sensitivity (i.e. enhancement of the brain PET signal) to partial P-gp inhibition, although [ 11 C]metoclopramide is a "weaker" substrate than [ 11 C]verapamil in vivo, as revealed by the lower response to maximal P-gp inhibition (Fig.4). The in vivo IC50 of tariquidar using [ 11 C]verapamil as probe has not been determined in this work to enable a direct comparison with the IC50 obtained with [ 11 C]metoclopramide. It may nonetheless be hypothesized that the higher in vivo "vulnerability" of [ 11 C]metoclopramide may compensate its lower "avidity" compared with [ 11 C]verapamil to offer similar sensitivity to partial inhibition. Our preclinical data should be interpreted with caution with respect to their clinical translatability. Differences in P-gp expression at the BBB have been reported between humans and

  [ 11 C]metoclopramide to P-gp inhibition compared with [ 11 C]verapamil and [ 11 C]N-desmethyl-loperamide.

Fig. 1

 1 Fig.1 Uptake of [ 11 C]verapamil, [ 11 C]N-desmethyl-loperamide and [ 11 C]metoclopramide in MDCKII-MDR1 cells in presence of increasing doses of tariquidar (TQD). Intracellular uptake is expressed as the percentage of radioactivity in the cell lysate relative to the incubation buffer (%C0, in A) and as the percentage of maximal inhibition (I%, assuming maximal inhibition was achieved using tariquidar 200 nM, in B). Lines in B represent fits of Hill model employed to estimate the in vitro half-maximum inhibitory concentration (IC50) with corresponding 95% confidence interval (CI95%, n=4 per condition, reported in C). In D, uptake ratios are reported as mean ± S.D (n=4 per condition). ****p<0.0001, ns = not significant (one-way ANOVA with Tukey's post-hoc test for multiple comparisons).

Fig. 2 .

 2 Fig.2. Dose-and concentration-response of tariquidar (TQD) for inhibition of P-gp-mediated transport of [ 11 C]metoclopramide at the rat BBB. In A, representative summed brain PET images are shown for each condition. Brain exposure (AUC10-30 min) of [ 11 C]metoclopramide as a function of administered tariquidar dose (B) or tariquidar plasma concentration measured at the end of the PET scan (C). Dotted lines in B and C represent fits of the employed Hill model.

Fig. 3 .

 3 Fig. 3. PET images obtained with [ 11 C]verapamil, [ 11 C]N-desmethyl-loperamide and [ 11 C]metoclopramide after selected doses of the P-gp inhibitor tariquidar. Representative PET images are summed from 0 to 30 min and were obtained at baseline, after partial P-gp inhibition (1 mg/kg tariquidar) and complete P-gp inhibition (8 mg/kg tariquidar).

Figure 4 .

 4 Figure 4. Impact of P-gp inhibition of the brain kinetics of radiolabeled substrates. Time-activity curves obtained with [ 11 C]verapamil (A), [ 11 C]N-desmethyl-loperamide (B) and [ 11 C]metoclopramide (C) in presence of selected states of P-gp inhibition. In D, the corresponding brain exposures (AUC10-30 min) are shown. Data are reported as mean ± S.D with n=4 per condition. *p<0.05, ****p<0.0001, ns = not significant (2-way ANOVA with Dunett's post-hoc test for comparison with baseline).

  [START_REF] Kannan | Imaging the function of Pglycoprotein with radiotracers: pharmacokinetics and in vivo applications[END_REF] C]verapamil, [ 11 C]N-desmethyl-loperamide and [ 11 C]metoclopramide were first compared in vitro, using a standardized uptake assay in cells overexpressing human P-gp. Maximal Pgp inhibition, using 200 nM tariquidar, allowed determining the uptake ratios of the tested substrates. The results confirmed that [ 11 C]metoclopramide and [ 11 C]verapamil are "weaker" P-gp substrates than[START_REF] Kannan | Imaging the function of Pglycoprotein with radiotracers: pharmacokinetics and in vivo applications[END_REF]

  [START_REF] Löscher | Drug resistance in brain diseases and the role of drug efflux transporters[END_REF][START_REF] Bauer | Imaging P-glycoprotein function at the blood-brain barrier as a determinant of the variability in response to CNS drugs[END_REF] Bauer et al. have recently shown a significant decline in P-gp function at the BBB of elderly subjects using [ 11 C]metoclopramide PET imaging.[START_REF] Bauer | Impaired clearance from the brain increases the brain exposure to metoclopramide in elderly subjects[END_REF] Interestingly, differences in brain distribution between elderly and young subjects were greater for [ 11 C]metoclopramide than for (R)-[ 11 C]verapamil.[START_REF] Bauer | Age dependency of cerebral Pgp function measured with (R)-[11C]verapamil and PET[END_REF] This is consistent with a higher "vulnerability" of [ 11 C]metoclopramide to moderate and physiological changes in P-gp function at the human BBB.[START_REF] Bauer | Imaging P-glycoprotein function at the blood-brain barrier as a determinant of the variability in response to CNS drugs[END_REF] Interestingly, differences between young and elderly subjects could be observed using (R)-[ 11 C]verapamil in the presence of partial P-gp inhibition (3 mg/kg tariquidar).[START_REF] Bauer | Effect of P-glycoprotein inhibition at the blood-brain barrier on brain distribution of (R)-[11 C]verapamil in elderly vs. young subjects[END_REF] This suggested that partial P-gp inhibition enhanced the "vulnerability" of (R)-[ 11 C]verapamil, thus improving the sensitivity of the technique. Such a partial inhibition protocol is difficult to implement because tariquidar is no longer available for human use and may cause drug-drug interactions with concomitant treatments in patients. This supports the use of [ 11 C]metoclopramide as a "change-sensitive" PET probe to safely explore P-gp function at the human BBB.

Acknowledgements

We thank Maud Goislard, Thierry Lekieffre, Christine Coulon and Kevin Pansavath for helpful technical assistance. Louise Breuil received funding from the joined AP-HP/CEA grant. This work was performed on a platform partially funded by the France Life Imaging network (grant ANR-11-INBS-0006).

Author contribution statement

Nicolas Tournier, Louise Breuil and Oliver Langer designed the research; Solène Marie, Sébastien Goutal, Charles Truillet, Wadad Saba and Fabien Caillé performed the research; Louise Breuil, Sylvain Auvity and Sébastien Goutal analyzed the data; Louise Breuil and Nicolas Tournier drafted the paper. All authors revised the manuscript critically for important intellectual content and approved the final version.

Conflict of interest

The authors declare that there is no conflict of interest.


Ivo Juki¢

Bernarda Lovrin£evi¢
email: bernarda@pmfst.hr

Martina Poºar

Aurélien Perera

Universal features in lifetime distribution of clusters in hydrogen bonding liquids

Hydrogen bonding liquids, typically water and alcohols, are known to form labile structures (network, chains, etc...), hence the lifetime of such structures is an important microscopic parameter, which can be calculated in computer simulations. Since these cluster entities are mostly statistical in nature, one would expect that, in the short time regime, their lifetime distribution would be a broad Gaussian-like function of time, with a single maximum representing their mean lifetime, and weakly dependent on criteria such as the bonding distance and angle, much similarly to nonhydrogen bonding simple liquids, while the long time part is known to have some power law dependence. Unexpectedly, all the hydrogen bonding liquids studied herein, namely water and alcohols, display highly hierarchic three types of specic lifetimes, in the sub-picosecond range 0-0.5ps The dominant lifetime very strongly depends on the bonding distance criterion and is related to hydrogen bonded pairs. This mode is absent in non-Hbonding simple liquids. The secondary and tertiary mean lifetimes are related to clusters, and are nearly independent of the bonding criterion. Of these two lifetimes, only the rst one can be related to that of simple liquids, which poses the question of the nature of the third lifetime. The study of alcohols reveals that this 3rd lifetime is related to the topology of H-bonded clusters, and that its distribution may be also aected by the alkyl tail surrounding ?bath?. This study shows that hydrogen bonding liquids have a universal hierarchy of hydrogen bonding lifetimes with a timescale regularity across very dierent types, and which depend on the topology of the cluster structures

Introduction

Labile structures in associated liquids and mixtures pose the problem of the role of kinetics of said structures, and its inuence on the thermophysical and dynamical properties of these systems [START_REF] Goshe | Supramolecular recognition: On the kinetic lability of thermodynamically stable hostguest association complexes[END_REF][START_REF] Paul | Proteinpeptide association kinetics beyond the seconds timescale from atomistic simulations[END_REF]. Such structures play an important role in soft matter, as for example with micelles and lamellae [START_REF] Glatter | Nonionic micelles near the critical point:â micellar growth and attractive interaction[END_REF][START_REF] De | Novel gemini micelles from dimeric surfactants with oxyethylene spacer chain. small angle neutron scattering and uorescence studies[END_REF][START_REF] Berghausen | Inuence of watersoluble polymers on the shear-induced structure formation in lyotropic lamellar phases[END_REF], and more particularly in biology, wherein the labile character of various functional molecular entities, such as enzymes for example, wears a fundamental operational nature [START_REF] Augenstein | Optimization in the recovery of a labile intracellular enzyme[END_REF][START_REF] Spahn-Langguth | Microsomal acyl glucuronidation: Enzyme-kinetic studies with labile glucuronides[END_REF][START_REF] Kim | Dynamic kinetic resolution of amines and amino acids by enzymeâmetal cocatalysis[END_REF]. One might even postulate that biological systems have been built by the increasing role played by such labile structures in the early evolution of primitive biochemical systems [START_REF] Trevors | From self-assembly of life to presentday bacteria: a possible role for nanocells[END_REF][START_REF] Deamer | Self-assembly processes in the prebiotic environment. Philosophical transactions of the Royal Society of London[END_REF]. Based on these premises, it is important to better understand the interplay between labile nature of molecular assemblies and the role of kinetics in their dynamics [START_REF] Goshe | Supramolecular recognition: On the kinetic lability of thermodynamically stable hostguest association complexes[END_REF][START_REF] Augenstein | Optimization in the recovery of a labile intracellular enzyme[END_REF][START_REF] Spahn-Langguth | Microsomal acyl glucuronidation: Enzyme-kinetic studies with labile glucuronides[END_REF][START_REF] Kim | Dynamic kinetic resolution of amines and amino acids by enzymeâmetal cocatalysis[END_REF].

A rst step in that direction would be to analyze the lifetime distribution of the hydrogen bonding process which is at the root of molecular association [START_REF] Almásy | Microscopic origin of the scattering pre-peak in aqueous propylamine mixtures: X-ray and neutron experiments versus simulations[END_REF][START_REF] Juki¢ | Comparative analysis of ethanol dynamics in aqueous and non-aqueous solutions[END_REF]. Since H-bonding is essentially a quantum mechanical process, it is aected by various intramolecular motions, and in turn it aects intermolecular motions. It is not clear if the various experimental techniques, which allow to probe the frequencies associated with these motions, can unambiguously answer the question posed above [START_REF] Lake | Far infrared studies of hydrogen bonding in alcohols[END_REF][START_REF] Murthy | Spectroscopic studies of the hydrogen bond[END_REF][START_REF] Craven | Far infrared group frequencies. ii. primary amines[END_REF][START_REF] Craven | Far-infrared group frequencies. i. aliphatic alcohols[END_REF]. Since this is a many body quantum mechanical phenomenon, it is not even clear if classical approximate theories can provide an alternative approach to this question. On the other hand, computer simulation provide a direct access to the statistics of molecular motions, and are able to answer this question. One could even answer this question at the level of classical physics, where the hydrogen bond is modeled by the Coulomb pairing of opposite charges [START_REF] Jorgensen | Quantum and statistical mechanical studies of liquids. 11. transferable intermolecular potential functions. application to liquid methanol including internal rotation[END_REF][START_REF] Starr | Hydrogen-bond dynamics for the extended simple point-charge model of water[END_REF], and whose pertinence has been amply proven by more than fty years of computer simulations and force eld development.

In fact, the question posed above has been already partly answered by Luzar and Chandler in their 1996 Nature paper [START_REF] Luzar | Hydrogen-bond kinetics in liquid water[END_REF]. In this paper, the authors focus essentially in the long time kinetics of H-bonding in water, beyond the initial 0.5 ps which they mention as the transient regime. This choice is amenable to a theoretical approach of the H-bonding kinetics, which is shown to be nonexponential, and further supported by classical computer simulations, but it does not explain the origin of the transient behaviour observed at short times.

The present work aims at revealing surprising repeatability of this transient part across several H-bonding liquids. It is initially motivated by the fact that H-bonding liquids other than water, such as alcohols and amines, are known to form short chain-like clusters, both from scattering experiments [START_REF] Pierce | X-ray studies on liquids: the inner peak for alcohols and acids[END_REF][START_REF] Warren | X-ray diraction in long chain liquids[END_REF][START_REF] Magini | On the structure of methyl alcohol at room temperature[END_REF][START_REF] Narten | Hydrogen bonding in liquid methanol and ethanol determined by x-ray diraction[END_REF][START_REF] Vahvaselkä | Determination of liquid structures of the primary alcohols methanol, ethanol, 1-propanol, 1-butanol and 1-octanol by x-ray scattering[END_REF][START_REF] Karmakar | On the structure function of liquid alcohols at small wave numbers and signature of hydrogenbonded clusters in the liquid state[END_REF][START_REF] Yamaguchi | The structure of liquid methanol revisited: a neutron diraction experiment at â80 °c and +25 °c[END_REF][START_REF] Benmore | The structure of liquid ethanol: A neutron diraction and molecular dynamics study[END_REF][START_REF] Tom²i£ | Structural properties of pure simple alcohols from ethanol, propanol, butanol, pentanol, to hexanol: Comparing monte carlo simulations with experimental saxs data[END_REF][START_REF] Sahoo | The probable molecular association in liquid d-1-propanol through neutron diraction[END_REF][START_REF] Vrhov²ek | Hydrogen bonding and molecular aggregates in liquid methanol, ethanol, and 1-propanol[END_REF][START_REF] Cerar | Performance of various models in structural characterization of n-butanol: Molecular dynamics and x-ray scattering studies[END_REF][START_REF] Almásy | Microscopic origin of the scattering pre-peak in aqueous propylamine mixtures: X-ray and neutron experiments versus simulations[END_REF][START_REF] Poºar | On the x-ray scattering pre-peak of linear mono-ols and the related microstructure from computer simulations[END_REF], spectroscopy investigations [START_REF] Hagemeister | Density functional theory calculations of the structures, binding energies, and infrared spectra of methanol clusters[END_REF][START_REF] Murdoch | Infrared spectroscopy of ethanol clusters in ethanolâhexane binary solutions[END_REF][START_REF] Wrzeszcz | Microheterogeneity in binary mixtures of methanol with aliphatic alcohols: Atr-ir/nir spectroscopic, chemometrics and dft studies[END_REF][START_REF] Pogorelov | Structural transformations in bulk and matrix-isolated methanol from measured and computed infrared spectroscopy[END_REF][START_REF] Balanay | Revisiting the formation of cyclic clusters in liquid ethanol[END_REF] and computer simulations [START_REF] Kosztolányi | Hydrogen bonding in liquid methanol, methylamine, and methanethiol studied by molecular-dynamics simulations[END_REF][START_REF] Ludwig | The structure of liquid methanol[END_REF][START_REF] Zorani¢ | Microstructure of neat alcohols: A molecular dynamics study[END_REF][START_REF] Perera | Microstructure of neat alcohols[END_REF][START_REF] Lehtola | Structure of liquid linear alcohols[END_REF][START_REF] Poºar | A re-appraisal of the concept of ideal mixtures through a computer simulation study of the methanol-ethanol mixtures[END_REF][START_REF] Poºar | On the micro-heterogeneous structure of neat and aqueous propylamine mixtures: A computer simulation study[END_REF]. Exploring this transient regime, essentially in the sub picosecond region, we have uncovered a universal dynamical behaviour common to all H-bonding liquids, which is unexpected in this time domain, where the dierences in the molecular interactions play an important role. Indeed, such universality would be more expected in the long time kinetics regime, where common features of the association process are likely to settle [START_REF] Paul | Proteinpeptide association kinetics beyond the seconds timescale from atomistic simulations[END_REF][START_REF] Schreiber | Fundamental aspects of proteinprotein association kinetics[END_REF].

In order to better appreciate the results presented herein, it is useful to picture the H-bonding process as essentially a random process at short times and distances, governed by the random molecular encounters. Intuitively, one would expect a broad distribution of H-bonding life times, as function of both time and H-bonding distance, centered around some mean representative lifetime, which could be subsequently searched in the various experimental techniques investigating the relaxation processes. What we uncover here, is that while the distribution of H-bonding distances are indeed as hypothesized, the lifetime distributions exhibit several specic times, which vary from 1 at very short distances to 3 at larger ones, where the distances are picked around the main peak of the oxygen-oxygen distribution function, typically in the range 2.5 Å to 3.5 Å, this latter value being often used in the literature [START_REF] Haughney | Molecular-dynamics simulation of liquid methanol[END_REF][START_REF] Guardia | Dynamics in hydrogen bonded liquids: water and alcohols[END_REF][START_REF] Skarmoutsos | Local structural eects and related dynamics in supercritical ethanol. 2. hydrogen-bonding network and its eect on single reorientational dynamics[END_REF][START_REF] Cerar | Structural, rheological and dynamic aspects of hydrogen-bonding molecular liquids: Aqueous solutions of hydrotropic tert-butyl alcohol[END_REF], as in the Luzar paper mentioned above [START_REF] Luzar | Hydrogen-bond kinetics in liquid water[END_REF]. These specic times, and related distribution, appear as common to all three associated liquids we have investigated herein, namely water, alcohols and amine, and across dierent force eld representation. In contrast, the long time kinetics show specicities related to each type of liquid. Although we have used classical force eld, this universality strongly suggests that these times should be a real feature of the associated liquids. We suggest that these specic times correspond to three types of molecular associations: dimer, linear chain-like clusters and other types of clusters. This nding suggests that self-assembled labile structures have typical lifetimes related to their dierences in microscopic topology. In that, it helps understand how such structures, when complexied by appropriate molecular entities, could acquire an important role in the pre-biotic phenomena. More importantly, it shifts the interest to such structures, from the usual long time kinetics approach, where kinetic constants play an important role, to short times and distance, in the range of which new self assembled objects appear, and could possibly play the role of new molecular species in a very restricted spatio-temporal region.

2

Theoretical, models and simulation details

We would like to rst stress that the present manuscript deals with the H-bond lifetime distribution itself, and not the time auto-correlation of it, which has been the subject of previous works by other authors [START_REF] Luzar | Hydrogen-bond kinetics in liquid water[END_REF][START_REF] Luzar | Resolving the hydrogen bond dynamics conundrum[END_REF][START_REF] Martiniano | Insights on hydrogen-bond lifetimes in liquid and supercooled water[END_REF][START_REF] Voloshin | Hydrogen bond lifetime distributions in computer-simulated water[END_REF][START_REF] Geiger | Structure and dynamics of the hydrogen bond network in water by computer simulations[END_REF]. It is therefore important to clarify the dierences in the theoretical backgrounds.

The hydrogen bond is a real property of many associated liquids in many contexts. However, for calculation purposes, it is necessary do dene the two following parameters, the bonding distance r c between the two donor/acceptor atoms, and the corresponding bonding angle θ c . In computer simulations, two molecules i and j are considered as H-bonded, when the distance r ij between the corresponding donor/acceptor atoms A i and B j obeys r ij ≤ r c , and the angle θ ij = A i HB j obeys |θ ij | < θ c . The A i and B j atoms are typically oxygen atoms, such as in water, but they can also refer to nitrogen atoms, such as for 1-propylamine considered in this work. Below, we will refer as C the ensemble of atoms which verify both criteria. For each pair of molecules in this time inter-val, there are 2 characteristic times: that t ij when they rst bond and τ ij when they break apart for the rst time. For this pair, we dene a time dependent random variable H ij (t), such that

H ij (t) = H(t -t ij )H(τ ij -t) (1)
where H(t) is the Heaviside function, and H ij (t) = 1 for t ij < t < τ ij and zero elsewhere.

We believe that this is the rst proper denition of the function introduced in the previous literature [START_REF] Luzar | Hydrogen-bond kinetics in liquid water[END_REF][START_REF] Luzar | Resolving the hydrogen bond dynamics conundrum[END_REF][START_REF] Martiniano | Insights on hydrogen-bond lifetimes in liquid and supercooled water[END_REF][START_REF] Voloshin | Hydrogen bond lifetime distributions in computer-simulated water[END_REF][START_REF] Geiger | Structure and dynamics of the hydrogen bond network in water by computer simulations[END_REF]

h ij (t) = ˆdtδ(t -τ ij ) (2)
which is 1 when the H-bond breaks at time τ ij and zero elsewhere.

This variable is therefore adapted to built the lifetime histogram, and the associated lifetime distribution dened as

L(t) = 1 T 0 L 0 ij∈C h ij (t) (3)
where the normalisation factor L 0 is dened as

L 0 = 1 T 0 ˆT0 0 dt   ij∈C h ij (t)   (4)
It is easily veried that L(t) is a probability distribution which veries ˆT0

0 dtL(t) = 1 (5)
Eqs. [START_REF] Glatter | Nonionic micelles near the critical point:â micellar growth and attractive interaction[END_REF][START_REF] De | Novel gemini micelles from dimeric surfactants with oxyethylene spacer chain. small angle neutron scattering and uorescence studies[END_REF] provide a direct computational indication as how to evaluate L(t) in a given computer simulation. An average over all possible time origin is implict.

The auxiliary Gromacs program gmx hbon d module with the -life option, allows to compute L(t). We have also checked through our own code that it was consistent with the denitions given above. It should be noted that one can end the H-bonding as soon as two bonded atoms part away according to the chosen criterion. This is the strict denition, which has been adopted in this work, but also in Gromacs. However, in reality, a broken bond could be reformed quickly, and perhaps some margin should be allowed for rebinding, which could be added to the ensemble of criterion in C. This could be conveniently introduced by replacing the Dirac delta in Eq.( 2) by a Gaussian function, which we will consider in another context.

This study focuses on the typical hydrogen bonding liquids, which are water and alcohols, which are both based on the OH group.

Mono-ols such as methanol, ethanol and propanol have been studied. The SPC/E [START_REF] Berendsen | The missing term in eective pair potentials[END_REF] and TIP4P_2005 [START_REF] Abascal | A general purpose model for the condensed phases of water: Tip4p/2005[END_REF] models were used to simulate water. Alcohols were modeled with the OPLS-UA [START_REF] Jorgensen | Optimized intermolecular potential functions for liquid alcohols[END_REF] and TRaPPE-UA [START_REF] Chen | Monte carlo calculations for alcohols and their mixtures with alkanes. transferable potentials for phase equilibria. 5. united-atom description of primary, secondary, and tertiary alcohols[END_REF] forceelds.

The program package Gromacs, version 2018.1 [START_REF] Pronk | Gromacs 4.5: a high-throughput and highly parallel open source molecular simulation toolkit[END_REF][START_REF] Abraham | Gromacs: High performance molecular simulations through multi-level parallelism from laptops to supercomputers[END_REF] was used for all molecular dynamics simulations. The simulation protocol has been the same for all neat systems. The initial congurations of N molecules were generated by random molecular positioning with the program Packmol [START_REF] Martínez | Packing optimization for automated generation of complex system's initial congurations for molecular dynamics and docking[END_REF], which were then energy minimized. The values for N considered here are typically N=1000 or N=2048. The systems were then equilibrated in the isobaric constant NpT ensemble for at least 5 ns, followed by a production run of at least 5 ns. Finally, an additional production run of 300 ps was performed to gather sucient data for the analysis of dynamic quantities (every conguration was sampled).

The integration algorithm of choice was the leap-frog [START_REF] Hockney | The potential calculation and some applications[END_REF] and the time step was 2 fs. The electrostatics were handled with the PME method [START_REF] Darden | Particle mesh ewald: An n*log(n) method for ewald sums in large systems[END_REF] and the constraints with the LINCS algorithm [START_REF] Hess | Lincs: A linear constraint solver for molecular simulations[END_REF]. The short-range interactions were calculated within the 1.5 nm cut-o radius. Each neat liquid was simulated at ambient conditions. Temperature was maintained at T = 300 K using the NoseHoover [START_REF] Nose | A molecular dynamics method for simulations in the canonical ensemble[END_REF][START_REF] Hoover | Canonical dynamics: Equilibrium phase-space distributions[END_REF] thermostat, while pressure was kept at p = 1 bar with the ParrinelloRahman barostat [START_REF] Parrinello | Crystal structure and pair potentials: A molecular-dynamics study[END_REF][START_REF] Parrinello | Polymorphic transitions in single crystals: A new molecular dynamics method[END_REF]. The temperature algorithms had a time constant of 0.2 ps, while the pressure algorithm was set at 2 ps.

The lifetime calculation is part of the post simulation analysis Gromacs this study, while we vary the bonding distance, the angle is maintained to the accepted value of 180 • ±30. However, the angular dependence is briey discussed in section 3.5, and shown to be very similar to the distance dependence.

Static structural properties of H-bonding

The universality in kinetics claimed in this work is supported by the static properties, in particular by the structural properties. We turn to the H-bond donor-acceptor distance distributions, which is shown in Fig. 2, for the same models as in Fig. 1. D HB (r) is dened as the histogram of the number of bonded pairs for a given distance r, and calculated here for systems of N=2048 molecules. We immediately note the expected feature discussed in the Introduction, namely that the distribution is indeed a broad one, centered around a distance which corresponds to the central peak in Fig. 1, around 2.7 Å. It conrms that the underlying bonding is essentially randomly distributed. This nding is somewhat in contrast with what snapshots of the cluster analysis reveal, where such clusters in water appear in various patterns, while for alcohols it has been proven that chains and loops are predominant.

The curves in Fig. 2 tend to indicate that donor-acceptor distribution is es- The analysis above clearly indicates that the clustering dierences between water and alcohols cannot be clearly dierentiated through the spatial H-bond distribution analysis, which in a way is disappointing. But, we show now that this is not the case for the temporal distributions.

H-bond life times

We would like to stress that it is the lifetime of pair of associated atoms which is measured, and not that of clusters. One expects that the pair lifetime would inuence the cluster lifetime by some broadening or narrowing of the distribution.

But what we observe below is unexpected.

Water

Fig. 3 shows the H-bond lifetimes distributions L HB (t) for the SPC/E water model, in log-scale, for dierent distances encoded in the color codes of the dierent curves, and ranging from r c = 2.5 Å to r c = 3.5 Å , which cover the distances from under the rst peak in Fig. 1 until the rst minimum. In terms of the corresponding oxygen-oxygen potential of mean force, such as -ln(g OO (r)),

these correspond to the rst neighbour ranges, as shown in Fig. 1, for example. However, as the contact distance is increased, we witness the appearance of 2 secondary peaks, which grow at shorter times than the corresponding rst peak lifetime. For example, for the mean H-bonding distance of r c = 2.7 Å (green curve), the main peak is at 40 fs, while a broad shoulder at 15 fs witness the growth of the two secondary lifetimes. For the largest H-bonding distance of r c = 3.5 Å (in cyan), the secondary peaks are respectively at 20fs and 45fs, and their amplitude is far superior to that of the rst peak/shoulder at 130fs, since they are respectively of 10 and 7, almost one order of magnitude larger that the for the rst peak. In addition, we see that these two secondary peaks are nearly always at the same respective times of 20fs and 45fs. Anticipating the corresponding analysis for the alcohols, we infer that these 2 secondary peaks must correspond to lifetimes of bonded pairs within larger clusters, hence are signatures of such clusters. The overall picture which emerges from these ndings is that, tightly H-bound molecules (small r c ) have high probability but short lifetimes, but less tightly H-bound molecules (larger r c ) are less probable, but live longer since they belong to a cluster. But a new problem appears: if the secondary peaks corresponds to H-bonding within a cluster, why two such peaks? Following the same inferring approach, we deduce that two types of clusters are present. From the analysis of the cluster shapes in water, and also in alcohols, we infer that these peaks must correspond respectively to linear and non-linear (globular, or branched) clusters.

Figs.SI-1 and SI-2 of the SI material show that the shape of these lifetimes is very similar across water models. It must therefore be a genuine physical property of real water.

Finally, we note that the conclusions inferred from the analysis above are in contrast to a generic intuitive idea: that clusters could be made of highly and tightly H-bound molecules. The picture which emerges from the present analysis is that larger clusters are less probable than dimers, but they are also live longer. At the opposite range, dimer clusters are highly likely, but they also break faster than larger clusters. Somewhere in between these two extremes, one must have clusters which witness the existence of labile entities, and the 3 peaks of the present nding correspond to typical such clusters.

We now in position to confront our ndings with respect to that of Luzar and Chandler [START_REF] Luzar | Hydrogen-bond kinetics in liquid water[END_REF]. Fig. 4 shows the long time behaviour of L(t). It is readily seen that, as the H-bonding distance is increased, the fast exponential decay corresponding to smaller H-bonding distances converges towards a slow, and possibly non-exponential behaviour, as witnessed by the merging of the 3 curves corresponding to the range r c = 3.1 -3.5 Å. Luzar and Chandler have selected r c = 3.5 Å in their entire analysis, and this criterion seems to have been retained in all subsequent literature [START_REF] Guardia | Dynamics in hydrogen bonded liquids: water and alcohols[END_REF][START_REF] Skarmoutsos | Local structural eects and related dynamics in supercritical ethanol. 2. hydrogen-bonding network and its eect on single reorientational dynamics[END_REF][START_REF] Cerar | Structural, rheological and dynamic aspects of hydrogen-bonding molecular liquids: Aqueous solutions of hydrotropic tert-butyl alcohol[END_REF]. In view of the present analysis, this approach would correspond to a global, almost macroscopical view of the kinetics of association in H-bonded liquids. While this may seem reasonable to make contact with macroscopic physics, we show here that the microscopic physics does not lead to a picture dominated by random distributions, but on the contrary to a selective trinitary view of clustering.

What about other H-bonded liquids? These are usually not studied under the same perspective as water. For example, H-bonding in water is often discussed in terms of ickering clusters [START_REF] Frank | Ion-solvent interaction. structural aspects of ion-solvent interaction in aqueous solutions: a suggested picture of water structure[END_REF] or network [START_REF] Perram | Cooperative hydrogen bonding and the question of ickering clusters in water[END_REF]. These specicities do not apply to other H-bonding liquids throughout the literature. Is there a common clustering specicity to all H-bonding liquids?

Alcohols

Fig. 5 shows the H-bond lifetimes distributions L(t) for the OPLS methanol model. The comparative analysis for other models is shown in the SI document.

A comparison of Fig. 5 with water in Fig. 3 shows immediately that the same global features are present, namely a high rst peak which decreases to larger lifetimes for larger r c distances, while intermediate 2 lifetime peaks emerge at distances around the attractive minimum of the mean force potential.

The principal dierences are seen in the values of the maximum amplitudes and positions, which are model dependent to a large extent as one should expect for underlying dierences in interactions, but also in the secondary peaks. These latter peaks appear as more marked for methanol than for water, and this is more particularly true for the second peak, which is a true peak for the alcohol, while it was more of a shoulder in the case of water. Another similarity with water is the relative insensitivity of the positions of the second and third peaks to the H-bonding r c distance. The fact that the secondary peaks are more marked is in line with the known fact that alcohols form linearly shaped clusters [START_REF] Magini | On the structure of methyl alcohol at room temperature[END_REF][START_REF] Narten | Hydrogen bonding in liquid methanol and ethanol determined by x-ray diraction[END_REF][START_REF] Vahvaselkä | Determination of liquid structures of the primary alcohols methanol, ethanol, 1-propanol, 1-butanol and 1-octanol by x-ray scattering[END_REF][START_REF] Tom²i£ | Structural properties of pure simple alcohols from ethanol, propanol, butanol, pentanol, to hexanol: Comparing monte carlo simulations with experimental saxs data[END_REF] of dierent topology: chains, loops, lassos [START_REF] Sarkar | Molecular clusters and correlations in liquid methanol at room temperature[END_REF][START_REF] Sarkar | Molecular clusters in liquid ethanol at room temperature[END_REF][START_REF] Karmakar | On the structure function of liquid alcohols at small wave numbers and signature of hydrogenbonded clusters in the liquid state[END_REF][START_REF] Vrhov²ek | Hydrogen bonding and molecular aggregates in liquid methanol, ethanol, and 1-propanol[END_REF][START_REF] Benmore | The structure of liquid ethanol: A neutron diraction and molecular dynamics study[END_REF].

These shapes are not found in water, at least in computer simulation of popular water models. Therefore, the similarity of the peak cannot refer to topological specicity, but to characteristic H-bonding patterns. The cluster analysis of water models shows that water clusters are either globular like or chain like. A with dierent H-bonding distance r c . Symbols and line colors are same as in Fig. 3 The case of 1-propylamine is illustrated in Fig. 7 of the SI, and again shows very similar features for L(t), although this is a very dierent H-bonding liquid ,4). This particular liquid conrms that the short time universality for L(t) uncovered herein is a real feature of H-bonding liquids in general.

Similarly to Fig. 4 for water, we compare in Fig. 7 the long time decay of L(t) for several H-bonding liquids, and this for two dierent values of r c . Interestingly, we nd that for a given r c , all curves tend to lie quite close to one another, and that there is also a species dependence. This is illustrated for r c = 3 Å for water and alcohols. We also note that, since 1-propylamine is a nitrogen based H-bonding liquid, its long time behaviour cannot be compared with that of oxygen atom based ones. It is interesting to compare the long time analysis of L(t) with the approach of Luzar-Chandler which is based on the analysis of c(t) and the related H-bond kinetics. While the long time behaviour points to dierences in H-bonding liquids, the short time transient part, which we study here demonstrates, that the underlying transient dynamics are based on universal elementary cluster structures. This is perhaps the main message of the present ndings.

A test with the weak-water model

In order to test the present conclusions, and in particular the inference methodology, we have studied previously introduced models of weak-water [START_REF] Keºi¢ | A model for molecular emulsions: Water and âweak waterâ mixtures[END_REF]. This model is based on the SPC/E water for water, where the partial charges on the oxygen and hydrogen atoms are scaled by a parameter λ (0 < λ < 1), allowing to tune the hydrogen bonding from the original model (with λ = 1) to a simple Lennard-Jonesium (with λ = 0). It was found that from λ ≤ 0.6 the inuence of partial charges and hydrogen bonding were not relevant and the model was structurally similar to a simple Lennard-Jones liquid. This model appears here as a useful way to measure the cluster hypothesis for the complex time dependence of L(t). As λ is made smaller, the hydrogen bonding abilities decrease, and it is possible to test directly the inuence of hydrogen bonding clusters on the shape of L(t).

In the order to preserve the liquid state for small λ values and under ambient conditions, it was found necessary to increase the Lennard-Jones energy parameter = (λ) according to the decrease of λ. In the present test, we have bypassed this procedure by doing the test simulations in the NVT Canonical ensemble, hence keeping the volume xed at that of the real liquid water.

Since the structure of the weak-water liquid is strongly be aected by the decrease of the partial charges, the H-bonding distances must be adjusted appropriately. Fig. 8a shows the various g OO (r) for dierent λ values we have used here, namely λ = 0.8, 0.5 and 0.2. The selected bonding distances depend on the position of the maximum, and dier quite a bit from that of the initial SPC/E water, ranging now from 2.7 Å to 4.5 Å. It is important to note that, while we vary the bonding distance criteria, we keep the angular criterion the same as that for pure water, which is that the angle O-H-O is 180 ± 30. This means that, even though weaker water model may have dierent bonding angles, we will still select only a sub class of those obeying tetrahedral bonding directions. This is justied by the fact that these models do not have as strong H-bonding tendencies as SPC/E water, hence the angular bias is less signicant than the distance of bonding. But we will need to take into this bias in order to interpret the data.

Fig. 8b shows the H-bond lifetimes for λ = 0.2, which is very close to a Lennard-Jones system. For very small bonding distances r c ≤ 3.0 Å , we note that the L(t) do not exhibit a clear maximum, and sometimes even show a plateau-like behaviour, such as for R = 3.0 Å(blue curve). This is a direct consequence of the angular bias described above. However, from bonding distance r c ≥ 3.1 Å, we observe a clear maximum, and this maximum is nearly the same for all subsequent r c values. This is what we would expect in a quasi-LJ system, where pairing is nearly isotropic. We also note that this lifetime distribution is not necessarily about isolated pairs, and could involve those in larger clusters.

We note the presence of intriguing small shoulder-like features at large times, such as for the green, blue and yellow curves, but these features cannot be interpreted from Fig. 8b alone, but will become clearer from the analysis of the next cases. Fig. 8c shows the H-bond lifetimes for λ = 0.5, which has the same pair distribution as the λ = 0.2 case, as seen in Fig. 8a. We note that most of the features observed previously equally appear here -supporting the structural analogy, but the previously noted intriguing shoulder structures has now grown into peak structures and are very apparent. However, it still remain dicult to interpret them by simple inference. 3. But now, we can nally understand the smaller features in Fig. 8c, namely the origin of the secondary peaks, precisely because these were the dominant features in the previous Figs. 8b-c. Indeed, we have interpreted them as peaks related to bonded pairs within clusters. This is precisely the conclusion we have reached when discussing Figs. 3456, but by inference. To summarize, the study of the weak-water models allow us to conrm that the secondary peak features are indeed related to clusters.

Inuence of the alkyl tails

Although the present study reveals similarities in the H-bonding life times, one would expect that the presence of non-bonding alkyl tails would aect the lifetime. This is illustrated in Fig. 9 for all the liquids studied in this work, water, methanol to 1-octanol and 1-propylamine. Since the alkyl tail inuence is best seen at largest distance cuto we have considered r c = 3.5 Å for all systems except for propylamine, for which we have used appropriately r c = 4.0 Å.

For clarity, each curve has been shifted by log [START_REF] Deamer | Self-assembly processes in the prebiotic environment. Philosophical transactions of the Royal Society of London[END_REF] from the previous one.

The bottom black curve is for water, and serves only as a reference, since water has no alkyl tail, and consequently does not show anything particular other Figure 9: Inuence of alkyl tails on lifetime distribution for methanol (red), ethanol (green), propanol(blue), octanol (gold) and propylamine (purple). The black curve is for water and serves as a reference. The gray lines serve to delimitate the 2nd and 3rd peaks regions mentioned in the text. than the features discussed above. However, all the alcohol curves show tiny oscillations past the 3rd peak. Higher alcohols such as 1-propanol and 1-octanol have even their 3rd peak weakly modulated. We attribute these oscillations in lifetime to the presence of the alkyl tail "bath", which surrounds the OH clusters, and aects the decay of their lifetime. This interesting feature further supports the interpretation that the 3rd peak is associated with the evolution of the cluster topology in time. This feature is naturally absent from water because there are no alkyl groups. However, the case of the propylamine (the upper most curve in purple) is very interesting. In previous studies [START_REF] Poºar | On the micro-heterogeneous structure of neat and aqueous propylamine mixtures: A computer simulation study[END_REF], we have compared the clustering of neat 1-propanol to that of 1-propylamine, and shown that the clustering of the amine group was not so important, both in size and shape, as the chain patterns observed for 1-propanol. The alkyl chain, being the same between the 2 species, plays an indirect role in this incomplete clustering of the NH2 amine groups. What we observe in Fig. 9 is that the role of the alkyl tail is so important that it "bites" into the 3rd peak which is about cluster topology, and makes even this peak look less characteristic compared to those of the alcohols. These oscillations also last longer than in the case of alcohols. This gure reveals the dynamical role played by the seemingly neutral alkyl tail background. The role of these tails was emphasized in a previous study [START_REF] Poºar | On the x-ray scattering pre-peak of linear mono-ols and the related microstructure from computer simulations[END_REF] of the shape of the Xray pre-peak feature of alcohols. This role is conrmed through the present study.

Angle dependence of L(t)

We briey discuss here the H-bond donor-acceptor θ angle dependence of the lifetime distribution, which are seen to be quite similar to the r c dependent ones. This is illustrated in Fig. 10 While the overall behaviour looks the same, we observe a notable dierence in the dependence on θ of the two peaks, whereas in the case of the r c dependence the peaks position were nearly invariant. We have attributed this invariance to that of the lifetimes of the clusters on r c as soon as that value was large enough to account for cluster sizes. In the case of the angular dependence, we observe that clusters with larger allowed θ angles tend to live longer than those with more restricted ones. This appears to be a natural consequence of the fact that strong angle constraints lead to cluster breaking earlier. We conclude that the angle dependence is less a robust criteria than the distance dependence, in what concerns the universal behaviour. Finally, we note the alkyl tail dependence in the case of ethanol, with the presence of long time oscillations, very similar to those observed for the r c dependence.

Discussion and Conclusion

While chemical matter, and more importantly bio-chemical matter, is known to be made of labile objects, in addition to well dened atoms and molecules.

It raises the question whether such eeting objects could play a role as important as the permanent ones. The pertinence of this question is enforced by the recognition that liquid assemblies of atoms and molecules are subject to uctuations which depend on the nature of the interactions, and in particular highly directional ones such as the H-bond interaction. Self assembly is the driving mechanism of many complex systems, such as in soft-matter and biological systems. The microscopic interaction at the root of self-assembly is the H-bonding process. This mechanism can be relatively well captured by classical force eld simulation using Coulomb charge pairing.

The calculations reported herein show that, for a given H-bonding distance r c , the H-bond lifetime is essentially dominated by one mean lifetime for distances r c smaller than the rst peak of g(r), but that two secondary peaks appear for larger distances until the rst minimum of g(r), and that these become dominant as r c is increased. We have attributed these peaks to linear and non-linear cluster geometries. In addition, the present study reveals a previously unexpected similarity in H-bonding lifetime distribution in the small distance We have provided convincing empirical arguments for the existence of three type of H-bond based clustering specicities in three dierent associated liquids, water, mono-ols and amines. These arguments suggest that there exist two distinct families of clusters, inside which the dimer lifetime plays a very dierent role. The rst family concerns linearly shaped clusters, whether these are linelike or circular. The second family, we have referred as non-linear clusters, which covers all other forms, such as globular -as those found in water, or branched, such as lasso-shaped found in mono-ols [START_REF] Guo | Molecular structure of alcohol-water mixtures[END_REF][START_REF] Poºar | On the x-ray scattering pre-peak of linear mono-ols and the related microstructure from computer simulations[END_REF].

The calculations concern primarily lifetime of pair of H-bonded particles.

But such pairs often reside inside specic clusters, hence their lifetime is aected by the life of the entire cluster. Because of the cooperative motions of particles linked within a cluster, we expect that the these motions aect the lifetime statistics in particular ways, as to emerge the 3 peaks observed in the data reported herein.

  package (gmx H-bond module) . It follows the usual process of monitoring the lifetime of pairs of H-bonded O atoms across dierent water molecules. The O-O H-bonding distance r c and the O-H-O angle are input parameters. In

Fig. 1

 1 shows the various oxygen-oxygen pair correlation functions g OO (r) obtained from computer simulations of the SPC/E water model, and rst two OPLS alcohol models. The rst peaks, shown in the main panel, demonstrate that the contact pairing is dominated by the oxygen atom size as well as the strong H-bonding pairing induced by the O-H-O Coulomb association. It is important to note that these rst peaks are relatively robust across models, such that the distributions of bonding distances do not vary much across models. This is further enforced by the striking resemblance of distribution times across models, as shown below and in the SI material.

Figure 1 :

 1 Figure 1: Oxygen-oxygen distribution functions g OO (r) for water and rst alcohols. The vertical blue lines represent a sample of the H-bonding distances used in this work. The inset represent a wider range of these functions.

Figure 2 :

 2 Figure 2: H-bond spatial distribution function D HB (r) for the dierent models studied herein.

Fig. 3

 3 Fig.3shows two remarkable features. First, for small distance r c < 3.0 Å, one sees a series of peaks, centered around a single maximum which represents the mean bonding lifetime, and these peaks are seen to shift to larger times as r c is increased, as well as decreasing in magnitude. The shift to larger times can be explained as the bonding distance criteria is brought closer to the rst maximum of g(r) and beyond, one increases the probability of pair association stability, hence the mean lifetime.

Figure 3 :

 3 Figure 3: H-bond lifetime distribution L(t) for SPC/E water, with dierent Hbonding distances r c . The symbols on each curve signal the corresponding peak position (see text): dot for rst peak, square for second and triangle for third.

Figure 4 :

 4 Figure 4: Long time behaviour of the H-bond lifetime distribution L(t) for the SPC/E model.

Figure 5 :

 5 Figure 5: H-bond lifetime distribution for OPLS methanol, with dierent Hbonding distance r c . Symbols and line colors are same as in Fig.3

Figure 6 :

 6 Figure 6: H-bond lifetime distribution for OPLS ethanol (a) and propanol (b),

Figure 7 :

 7 Figure 7: Long time behaviour of the H-bond lifetime distribution for all the models studied herein

Figure 8 :

 8 Figure 8: Weak water models data. (a) Comparison of the RDF of various models; (b) H-bond lifetimes for λ = 0.2; (c); (b) H-bond lifetimes for λ = 0.5; (d) H-bond lifetimes for λ = 0.8.

Fig.8d is

  Fig.8d is for the case λ = 0.8, which is the closest to SPC/E water, and should be compared with Fig.3. The previous intriguing peaks have now grown to invade the entire gure, and, by comparison with Fig.3, represent the dominant contribution to the H-bond lifetimes for each R r c value, as we have interpreted them when discussing Fig.3. But now, we can nally understand the

  for the SPC/E water model and the OPLS ethanol model. Since the overall behaviour of the primary dimer peak is very similar, only the secondary and tertiary peaks are shown for xed H-bonding distance of r c = 3.5 Å corresponding to the minimum of g OO (r).

Figure 10 :

 10 Figure 10: Angle dependence of H-bond lifetimes for SPC/E water (left panel) and OPLS ethanol (right panel). Only the secondary and tertiary peaks are shown for r c = 3.5 Å

  and short time regime corresponding to distances under 3.5 Å and under 150 fs, and this independently of the nature of the H-bonding group. This interpretation is summarized in Fig.11 below, where we have shown sample L(t) curves for the SPC/E water (see Fig.3) for typical r c distances picked along the rst peak of the g OO (r) curve (see right panel), covering the contact, the peak height and 2 points at the minimum. The clusters shown above the right panel illustrate the rst peak/dimer base (in red) within larger clusters (with additional water molecules shown in gray), helping to visualize why long lived dimers at larger r c would give raise to shorter lived clusters.

Figure 11 :

 11 Figure 11: Illustration summarizing the correspondence between the hydrogen lifetime L(t) curves for dierent r c values taken along the g OO (r) curve. The water molecules dimer based clusters illustrate the correspondence between rst neighbours lifetimes and larger clusters lifetimes as r c is varied.

  In the present work, we focus on the lifetime distribution itself, and the appropriate random variable is h ij (t) can dened from H ij (t). We rst take the derivative dH ij (t)/dt = δ(t -t ij ) -δ(t -τ ij ), and remove the origin part. Then we dene h ij (t) as gauge variable (because of the derivative/integration operations) related to H ij (t) through

	under the denomination h(t). From
	this random variable one can measure several statistical averages and correla-
	tion, and in particular the auto-correlation function c(t) =< h(0)h(t) > which
	has been studied in the past.

ResultsUsually, H-bonding is discussed in terms of the spatial distribution, through pair correlation functions and their spatial representations, as proven by their overwhelming representation in the literature, as compared with their temporal one. Therefore, the nding of a corresponding temporal universality can be considered as an important feature, which should help better understand the role of labile structures.

Acknowledgments

This work has been supported in part by the Croatian Science Foundation under the project UIP-2017-05-1863 Dynamics in micro-segregated systems.