Catherine M Gedik

Andrew Collins
email: a.r.collins@medisin.uio.no

Jacques Dubois

Pierre Duez

Léonard Kouegnigan

Jean-François Rees

Steffen Loft

Peter Möller

Annie Jensen

Henrik Poulsen

Peter Møller

Bente Riis

Allan Weimann

Jean Cadet

Thierry Douki

Jean-Luc Ravanat

Sylvie Sauvaigo

Andrea Hartwig

Piero Dolara

Lisa Giovannelli

Maura Lodovici

Francesco Guglielmi

Ryszard Olinski

Karol Bialkowski

Marek Foksinski

Daniel Gackowski

Zdena Duračková

Jana Muchová

Peter Korytar

Monika Sivonová

Mária Dušinská

Csilla Mislanová

Helena Petrovská

José Viña

Ana Lloret

Guillermo Sáez

Lennart Möller

Tim Hofer

Hanna Eriksson

Eric Gremaud

Frank Kelly

Christina Dunster

Sharon Wood

Nicholas Vaughan

Establishing the background level of base oxidation in human lymphocyte DNA: results of an interlaboratory validation study

Keywords: DNA oxidation, 8-oxodGuo, HPLC, comet assay, FPG

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

citation purposes)

xidative damage within the body arises through the attack of highly reactive forms of oxygen, released during normal respiration and also as part of the inflammatory response.

This kind of damage to various biomolecules, such as lipids, proteins, and DNA, has been implicated in many chronic and degenerative human diseases, as well as in aging. In particular, oxidative DNA damage may be one of the causes of cancer. It is therefore especially important that we should be able to measure this damage reliably. The most commonly used biomarker of DNA oxidation, in epidemiological as well as experimental studies, is 8-oxo-7,8-dihydro-2'deoxyguanosine . Yet, estimates of the concentration of 8-oxodGuo in DNA from normal human cells vary by as much as 1000-fold, depending largely on which procedure is used to measure it [START_REF] Collins | Problems in the measurement of 8oxoguanine in human DNA. Report of a workshop, DNA Oxidation[END_REF].

The European Standards Committee on Oxidative DNA Damage (ESCODD) was set up in 1997 to attempt to identify the problems; to devise standard, reliable techniques; and to reach a consensus on the true background level of damage in normal human cells. ESCODD gained funding from the European Commission in 1999 and has operated since February 2000 as a Concerted Action with 25 member laboratories in Europe and one in Japan. Previous reports [START_REF] Escodd | Inter-laboratory validation of procedures for measuring 8-oxo-7,8dihydroguanine/8-oxo-7,8-dihydro-2'-deoxyguanosine in DNA[END_REF][START_REF] Riis | Comparison of results from different laboratories in measuring 8oxo-2'-deoxyguanosine in synthetic oligonucleotides[END_REF][START_REF] Escodd | Comparative analysis of baseline 8-oxo-7,8-dihydroguanine in mammalian cell DNA, by different methods in different laboratories: an approach to consensus[END_REF][START_REF] Escodd | Measurement of DNA oxidation in human cells by chromatographic and enzymic methods[END_REF] have described the analysis in participating laboratories of solutions of 8-oxodGuo, oligonucleotides with defined amounts of 8oxoGua, calf thymus DNA, calf thymus DNA with experimentally induced 8-oxoGua, pig liver DNA, HeLa cells, and HeLa cells with experimentally induced 8-oxoGua. In this latest round, aimed at validating methods for use in human population studies, laboratories recruited volunteers in their respective countries to provide lymphocytes for analysis of DNA damage, and identical samples of HeLa cells were provided to each laboratory to act as a reference standard.

A variety of methods were used. The chromatographic approach was represented by HPLC with electrochemical detection (HPLC-ECD), both amperometric and coulometric, and HPLC with tandem mass spectrometry (HPLC-MS/MS). A set of methods described as "enzymic methods" make use of the bacterial DNA repair enzyme formamidopyrimidine DNA glycosylase (FPG) to convert 8-oxoGuas to apurinic sites and then, via its associated lyase activity, to DNA breaks; these are measured using the comet assay (single cell gel electrophoresis), alkaline unwinding, or alkaline elution. In recent rounds of ESCODD, the enzymic approach has generally given estimates of 8oxoGua that are several times lower (comparing median results from different laboratories) than those obtained using chromatography, and fall within a narrower range. There is convincing evidence that substantial oxidation of guanine occurs during preparation of samples for HPLC analysis, and in ESCODD, we have designed protocols for DNA extraction and hydrolysis specifically to minimize this effect. In contrast to the protracted and vigorous procedures that take place before chromatographic analysis, the enzymic methods involve minimal processing of the sample (e.g., in the comet assay, cells are simply centrifuged, embedded in agarose, incubated with FPG, then with alkali, and electrophoresed); samples are therefore less likely to suffer from spurious oxidation. The present trial was set up to test our success in controlling the artifact, to compare the merits of the two approaches, and to obtain a realistic estimate of background DNA oxidation in normal human cells. O Page 4 of 25(page number not for citation purposes)

MATERIALS AND METHODS

Human lymphocytes: selection of subjects and collection of samples

The aim of the study was to recruit groups of up to 20 healthy male nonsmokers, aged 20-30 years, in different countries. Eleven partner laboratories took part. Local ethical approval was obtained by each participant.

Five CPT vacutainers containing sodium heparin anticoagulant (Becton-Dickinson, Franklin Lakes, NJ) were used to collect blood (40 ml in total) from each volunteer by venepuncture on a single occasion. Blood samples were coded, and lymphocytes were obtained by centrifugation within 2 h. They were washed in Roswell Park Memorial Institute (RPMI) medium, centrifuged, resuspended in 1 ml of ice-cold freezing medium (90% heat-inactivated fetal calf serum, 10% DMSO), and diluted with freezing medium to a density of 6 × 10 6 cells/ml. Two 200 µl aliquots in cryovials were chilled slowly (in boxes of expanded polystyrene) to -80°C for the comet assay (if needed) and stored at this temperature. The remainder, for DNA extraction and HPLC analysis, was divided into ~3 ml aliquots in 15 ml tubes, packed upright, and slowly chilled to -80°C as above.

Preparation of HeLa cells as reference standard

HeLa (human transformed epithelial) cells were prepared by the coordinating laboratory. They were grown to confluence in 36 50-ml flasks (Nalge Nunclon International, Roskilde, Denmark), in Glasgow-modified Eagle's minimal essential medium (GMEM) with 5% fetal calf serum, 5% calf serum, supplemented with glutamine and nonessential amino acids. The cells were harvested using trypsin/EDTA, and the cells from each flask were transferred to a polystyrene roller bottle (850 cm 2 surface area; Corning Inc., Corning, NY) in 80-100 ml medium in an ~6% CO2 atmosphere. The bottles were placed on a roller at 37°C, and after 7 days, the medium was changed. After a further 4 days, 30 ml medium was added to each bottle, and the next day the cells were harvested from two bottles at a time using trypsin/EDTA. Medium with serum was added to inactivate the trypsin, and the cells were transferred to sterile glass bottles and stored in a 37°C incubator until all had been harvested. Cell suspensions were combined and gently inverted to mix, total volume was noted, and cell density was determined using a haemocytometer. Cells were then collected by centrifuging at 700 × g for 7 min at 20°C. The cell pellet was resuspended in 1 ml freezing medium (GMEM containing 20% fetal calf serum, with 10% DMSO). Additional freezing medium was added to bring the density to 5 × 10 6 cells/ml, and the cell suspension was divided into aliquots of the following: 1) 30 × 10 6 cells (6 ml) for chromatographic methods; 2) 1.5 × 10 6 cells (0.3 ml) for the comet assay; 3) 6 × 10 6 cells (1.2 ml) for alkaline unwinding; and 4) 10 × 10 6 cells (2 ml) cells for alkaline elution.

The cells were chilled slowly in expanded polystyrene boxes to -80°C. Aliquots were sent to participating laboratories on dry ice, and all arrived frozen. They were then stored at -80°C. Samples retained for use by the coordinating laboratory were kept on dry ice for 24 h to mimic conditions of postage and then stored at -80°C until use.

HeLa cells were sent to 17 laboratories for HPLC analysis with electrochemical detection (HPLCECD) and to two laboratories for HPLC-MS/MS analysis. Each participant received three aliquots for the ESCODD standard DNA extraction/hydrolysis method (see below) and a further three citation purposes)

for the participant's own method (if it differed from the standard method). Additional tubes for practising the standard method were provided on request. Participants were requested 1) to use as many cells as required for one DNA extraction from each tube, 2) to extract the DNA from each tube on different occasions if possible, and 3) to make triplicate injections of each sample.

Eight laboratories carrying out the comet assay were sent three aliquots of HeLa cells for analysis by the participant's own method on different days; each sample was to be analyzed in triplicate.

Participants whose comet assay method differed significantly from the ESCODD protocol were sent an extra set of cells so that both methods could be used.

Three samples of HeLa cells were sent to two laboratories using alkaline unwinding, and to one laboratory using alkaline elution, to be analyzed on three separate occasions in triplicate.

DNA extraction and hydrolysis for HPLC analysis

A recommended procedure for extraction and hydrolysis of DNA was devised on the basis of comparative tests of different methods carried out by members of ESCODD [START_REF] Ravanat | Cellular background level of 8-oxo-7,8-dihydro-2'deoxyguanosine: an isotope based method to evaluate artefactual oxidation of DNA during its extraction and subsequent work-up[END_REF]. Frozen aliquots of HeLa and lymphocyte suspensions were thawed in a 37°C water bath with gentle agitation. As soon as they were thawed (lymphocyte aliquots from one volunteer being combined in a single 50 ml centrifuge tube), the cell suspensions were made up to 45 ml with ice-cold PBS (phosphate-buffered saline; 0.14 M NaCl, 3 mM KCl, 8 mM Na2HPO4, 1 mM KH2PO4, pH 7.4) and centrifuged at 700 × g for 7 min at 4°C. The supernatant was discarded and the cell pellet dispersed in 1 ml of PBS, diluted to 45 ml with PBS, and recentrifuged. The pellet was dispersed in 1.5 ml of ice-cold buffer A (10 mM Tris, 0.32 M sucrose, 5 mM MgCl2, 0.1 mM deferoxamine mesylate; adjusted to pH 7.5; then 1% Triton X-100 added, and buffer made up to final volume) and transferred to a 2 ml Eppendorf tube. The nuclei from the lysed cells were pelleted by centrifugation at 1500 × g for 5 min at 4°C. The supernatant was removed and 1.5 ml buffer A added to the pellet, which was vortex-mixed for 10 s and then centrifuged again at 1500 × g for 5 min at 4°C.

The supernatant was discarded, and 0.6 ml of buffer B (10 mM Tris, 5 mM Na2EDTA, 0.15 mM deferoxamine mesylate, pH 8.0) was added to the pellet. The pellet was dispersed by vortexing, 35 µl of 10% SDS added, and vortexing repeated. Three µl of RNase IIIA (Sigma, Poole, UK; 100 mg/ml) and 8 µl of RNase T1 (Sigma; 1 U/µl), both in 10 mM Tris, pH 8.0, were added, and the mixture gently vortexed for 10 s. After incubation for 15 min at 50°C, and cooling to 37°C, 30 µl of protease (Qiagen; dissolved at 20 mg/ml in water) was added, and the mixture was gently vortexed for 10 s and incubated at 37°C for 1 h.

The mixture was cooled to 4°C and transferred to a 15 ml centrifuge tube with 1.2 ml of cold NaI solution (40 mM Tris, 20 mM Na2EDTA, 7.6 M NaI puriss. grade [Fluka, St Gallen, Switzerland], 0.3 mM deferoxamine mesylate, pH 8.0), and vortexed vigorously for 30 s. Two ml of 2-propanol was added, and the tube was gently inverted several times to mix before centrifuging at 5000 × g for 5 min at 20°C. The supernatant was aspirated off, the pellet was washed with 40% 2-propanol and centrifuged at 5000 × g for 5 min at 20°C, and the pellet was then washed with 70% ethanol (-20°C) and centrifugation was repeated at 5000 × g for 5 min but at 4°C. The ethanol was removed with a citation purposes) pipette, and the DNA was left to dry in the tube for 5 min. The DNA was suspended in 0.1 mM deferoxamine mesylate, immediately frozen, and stored at -80°C.

For hydrolysis, the DNA in deferoxamine mesylate was thawed, and to it was added 10 µl of nuclease P1 (Sigma) dissolved in 0.3 M sodium acetate, 1 mM ZnSO4, pH 5.3, at a concentration of 1 U/µl. The mixture (with still undissolved DNA) was vortex-mixed and incubated at 37°C for 2 h, vortexing again after 1 h of incubation. Then 0.2 µl of alkaline phosphatase at 20 U/µl (Roche Diagnostics Ltd., Lewes, UK) and 10 µl of 0.5 M Tris, 1 mM Na2EDTA, pH 8.0, were added, the mixture was vortexmixed and incubated for 1 h at 37°C, and vortexed again after 30 min. The mixture was centrifuged for 5 min at 5000 × g, and aliquots of the supernatant were taken for analysis.

Most partners used their own procedures alongside the standard ESCODD procedure. Details of variations from the standard protocol are given in Table 1 and Table 2.

Analysis of 8-oxodGuo by chromatography

HPLC-ECD: Separation of DNA hydrolysate, 8-oxodGuo, and dGuo samples on a C18 column (varying in dimensions and manufacturer) was followed by electrochemical detection (coulometric or, in two laboratories, amperometric) of 8-oxodGuo, and UV detection of dGuo. Results that were below the limit of detection are recorded as zero. (It is not possible to give such results an assumed value of, say, half the limit of detection; the limit of detection is an absolute amount or concentration, but our results are expressed relative to the amount of dGuo in the sample, which can vary.) HPLC-MS/MS: Two laboratories used liquid chromatography with online detection by MS. (Only one was able to return results.) Nucleosides were separated on a C18 column and injected into a triple quadrupole mass spectrometer for identification and quantitation of products using heavy isotopelabeled internal standards (necessary to control for variation in ionization before MS).

Analysis of FPG-sentitive sites

Comet assay: Cells were thawed by gentle agitation in a 37°C water bath and immediately centrifuged at 200 × g for 3 min at 4°C. Pelleted cells were dispersed in 0.4 ml of cold RPMI medium with 10% heat-inactivated fetal calf serum, and 30 µl aliquots were added to 1 ml of cold PBS. The cells were centrifuged again, dispersed in 85 µl of low-melting-point agarose at 37°C, and placed on a microscope slide. Lysis with Triton X-100 and 2.5 M NaCl leaves "nucleoids" (containing supercoiled DNA) embedded in the agarose. Alkaline incubation and electrophoresis follows, as described in detail previously [START_REF] Collins | Oxidation of cellular DNA measured with the comet assay[END_REF]. Essentially, the comet assay measures DNA breaks by their ability to relax DNA supercoiling; the relaxed DNA loops extend under electrophoresis to form a comet-like image when viewed under fluorescence microscopy. The percentage of DNA in the comet tail indicates the frequency of breaks. Calibration is indirect, based on a standard curve constructed using ionizing radiation to induce DNA breaks at known frequency (0.31 breaks per 10 9 daltons per Gray).

Laboratory 4 performed the comet assay by the standard method, in which alkaline denaturation and electrophoresis are carried out at an ambient temperature of 4°C, and also by their own method in which denaturation and electrophoresis are at 18°C. For both methods, they used their own washing procedure after lysis, which involved three washes with 2 ml 0.4 M Tris, pH 7.5, followed by three citation purposes)

washes with 2 ml of enzyme buffer, instead of the more usual method of immersing the slides in 100 ml enzyme buffer for three washes of 5 min. Laboratory 3 used the standard method but with a 45 min incubation (instead of 30 min) with FPG.

Alkaline elution: Cells were thawed and dispersed in 5 ml PBS at 0°C. Aliquots containing 10 6 cells were loaded on polycarbonate membrane filters (25 mm diameter, 2 µm pore size) and analyzed for single-strand breaks and FPG-sensitive modifications as described previously [START_REF] Epe | Oxidative DNA damage: endonuclease fingerprinting[END_REF][START_REF] Pflaum | Determination of steady-state levels of oxidative DNA base modifications in mammalian cells by means of repair endonucleases[END_REF]. Briefly, the rate at which single-stranded DNA moves through the filter depends on the size of the fragments. Again, calibration is by comparison with irradiated DNA.

Alkaline unwinding: Frozen aliquots of cell suspensions were thawed by gently shaking in a 37°C water bath. Immediately after thawing, they were transferred into a centrifuge tube and cell suspensions were diluted to [START_REF] Pflaum | Determination of steady-state levels of oxidative DNA base modifications in mammalian cells by means of repair endonucleases[END_REF]

RESULTS

HeLa cells, chromatographic methods

Fourteen laboratories submitted results obtained with HPLC-ECD. Twelve used coulometric detection (laboratories 2, 3, 5, 6, 13, 14, 15, 16, 21, 23, 25, and 28), and two used amperometric detection (laboratories 4 and 9). Laboratory 3 also submitted results for HPLC-MS/MS. Participants 3, 4, and 23 used the ESCODD method only.

Figure 1 shows the concentrations of 8-oxodGuo found in HeLa cells by chromatographic methods (both laboratories' own methods, and the standard procedure recommended by ESCODD). The overall mean values were 3.38 and 2.62 8-oxodGuo per 10 6 dGuo for own and ESCODD procedures, respectively (not significantly different); overall median values were 2.78 and 2.73 8oxodGuo per 10 6 dGuo, respectively.

HeLa cells, enzymic methods

Eight laboratories submitted results for the comet assay 0.14 to 1.47 FPG-sensitive sites/10 6 guanines. Laboratory 23 was using the alkaline unwinding technique for the first time, which may account for the high variability in the results.

8-oxodGuo levels in human lymphocyte DNA

Eleven laboratories recruited volunteers (1, 2, 4, 5, 6, 13, 14, 15, 16, 21, and 28), although Participant 13 had only three subjects. Laboratory 1 carried out only the comet assay. and laboratories 13, 16, 21. and 28 carried out only HPLC-ECD. Laboratories 2, 4, 5, 6, 14. and 15 reported results for both HPLC-ECD and the comet assay. The results are displayed as mean values from each laboratory in Figure 3 (HPLC-ECD) and Figure 4 (comet assay). The "mean of means" (i.e., the mean of all individual laboratory means) for HPLC is 3.73 8-oxodGuo per 10 6 dGuo, whereas the median value is 4.24. With the comet assay, the mean of means and the median are both 0.34 FPG-sites per 10 6 guanines. Thus, the chromatographic and enzymic approaches give estimates of background 8-oxodGuo in lymphocytes that differ by factors of 11-12. Looking only at those laboratories that used both HPLC and the comet assay, the overall mean (and median) values of 8oxodGuo are 4.40 and 0.36 per 10 6 guanines, respectively-again, differing by a factor of ~12. The differences between laboratories may reflect national patterns, but there are too few results from different laboratories in the same country to be sure that the variations are reliable. There is no significant correlation between the mean values obtained by the two methods, compared across laboratories.

The CVs for HPLC determinations of HeLa (inter-and intratube) and lymphocytes (intrasubject) are shown in Table 3.

DISCUSSION

Our aims in this interlaboratory investigation were 1) to compare the performance of various assays in measuring background levels of 8-oxoGua or 8-oxodGuo, using standard samples of HeLa cells; 2) to establish the range of 8-oxoGua/8-oxodGuo levels to be expected in normal human lymphocyte DNA; and 3) to look for differences in mean background levels of 8-oxoGua/8oxodGuo in lymphocytes collected from healthy male volunteers in different countries.

Chromatographic determination of 8-oxodGuo

In a previous ESCODD trial, we examined the ability of the various assay methods to detect a doseresponse, using HeLa cells treated with the photosensitizer Ro 19-8022 and visible light to induce different amounts of 8-oxoGua. We concluded that HPLC-ECD was very accurate when measuring this experimentally induced base oxidation (7 of 8 laboratories found essentially the same slopes of dose-response curves). However, there was no agreement over the background level of 8-oxodGuo; estimates varied between 0.36 and 32 8-oxodGuo per 10 6 dGuo (5). This we attributed to the unsolved problem of adventitious oxidation of guanine during sample preparation.

Whether further improvement in technique (including adoption of a standard protocol for sample preparation) has improved the situation can be judged from the results reported here with standard samples of (untreated) HeLa cells. Excluding all results that are "not detected," there is a 32-fold variation in the mean values for 8-oxodGuo in HeLa DNA measured by HPLC when using own methods of DNA extraction and hydrolysis, and a 13-fold variation when using the ESCODD method. The latter represents a considerable improvement on the results from previous ESCODD trials. citation purposes)

However, the fact that variations still occur, even when the standard ESCODD protocol is in use, indicates that all the sources of variation or of spurious oxidation have still not been identified and controlled. It is seen from Fig. 1A that about half of the laboratories report values of ~4-5 8oxodGuo per 10 6 dGuo, and it is tempting to assume that the few very low figures must be the result of some experimental anomaly. But equally it is possible that these represent laboratories that have managed to minimize the spurious oxidation. Clearly, whatever the case, to account for the discrepant results there must be some divergence of experimental conditions between laboratories, in steps not covered by the standard protocol, that is, in the earliest stages of preparation, or during HPLC itself. In future, the use of immunoaffinity chromatography to preconcentrate 8-oxodGuo and so discard the excess of unmodified dGuo before HPLC (12) might limit adventitious oxidation occurring during HPLC.

Eight laboratories submitted results with HPLC using both their own, and the ESCODD methods. The degree of agreement between the methods can be assessed using a Bland and Altman plot of difference between values by the two methods against mean of the two values ( 13), as shown in Figure 5. Six laboratories showed reasonable agreement between the two methods. Two laboratories-those reporting the highest values with their own method-did not agree well. Participant 15's own method stands out in three respects. First, it produces a mean value about half that obtained in the same laboratory using the ESCODD method (though the difference is not statistically significant). Second, it gives the lowest values of all returned (excluding laboratories that reported no detectable peaks). Third, the method involves radical departures from the standard procedure, viz. the omission of RNase and proteinase treatments, keeping all solutions on ice (or colder) throughout, the use of TEMPO and guanidine thiocyanate (GTC), and dissolving the DNA precipitate in water containing catalase (and TEMPO) at the hydrolysis step. However, in a previous direct comparison of methods ( 6), the GTC-based method did not produce the lowest value for 8oxodGuo concentration, and so the value of these additional precautions is questionable.

Enzymic determination of 8-oxoGua

The alternative approach to chromatography is the use of enzymic conversion of 8-oxoGua to DNA breaks, which are then assayed by a variety of methods. Generally, in comparison with chromatographic methods, this approach has given lower estimates of background damage [START_REF] Collins | Problems in the measurement of 8oxoguanine in human DNA. Report of a workshop, DNA Oxidation[END_REF][START_REF] Escodd | Measurement of DNA oxidation in human cells by chromatographic and enzymic methods[END_REF][START_REF] Gedik | Oxidative stress in humans: validation of biomarkers of DNA damage[END_REF]. One possible explanation, of course, is that the methods using FPG, in general, underestimate 8oxoGua by several-fold, because of a failure of the enzyme FPG to detect all lesions, or because lesions occur in clusters so close together that they are detected as a single break (although this may occur only rarely). On the other hand, since FPG detects other oxidized purines in addition to 8oxoGua, our estimates of 8-oxoGua may be too high. It is also possible that the calibration of these assays, using ionizing radiation to induce known frequencies of breaks, is inaccurate. However, doseresponse experiments conducted with both alkaline elution [START_REF] Pflaum | Determination of steady-state levels of oxidative DNA base modifications in mammalian cells by means of repair endonucleases[END_REF] and the comet assay ( 14) in comparison with HPLC indicate that enzymic methods and HPLC measure experimentally induced 8-oxoGua (in cultured cells) with very similar efficiency (i.e., similar dose-response slopes), and that the main problem is therefore spurious oxidation inflating HPLC measurements.

The range of estimates of background 8-oxoGua in HeLa cells using the comet assay (Fig. 2) is 10fold. The overall range reflects extremes, and as is the case also with HPLC, most individual results agree much more closely than this range suggests. citation purposes)

The reason for distributing HeLa cells in this trial was to provide a reference standard, so that results for lymphocytes could be normalized, eliminating minor interlaboratory variations. However, because the values reported for HeLa cells (measured by chromatographic or enzymic method) showed large variations, it was considered that any attempt at normalization would simply add further variation to the results, rather than controlling for it.

8-oxodGuo (8-oxoGua) levels in human lymphocyte DNA

All laboratories applied their own DNA extraction method to the lymphocytes. In addition, laboratory 21 measured the same samples of lymphocyte DNA using their preferred method for DNA isolation and hydrolysis, alongside the standard ESCODD method. There was no significant correlation (r=-0.26) between the results for the two methods. The individual values obtained using the ESCODD protocol showed little variation. Results with the laboratory's own method were on average 1.67 times higher, and interindividual variation was more marked. With HeLa cells, in contrast, this laboratory found very similar 8-oxodGuo levels for the two methods.

Six laboratories measured both 8-oxodGuo and FPG sites on the same samples of lymphocytes. If levels of oxidized bases in DNA really do vary between subjects, we should expect individual values of 8-oxodGuo (by HPLC) and FPG-sensitive sites (by comet assay) to show a significant, positive correlation. (This is to assume that FPG-sites are predominantly 8-oxoGua, or at least that 8-oxoGua occurs as a fixed proportion of the total FPG-sensitive sites.) There are several possibilities. If the chromatographic and enzymic methods are measuring the lesion with equal efficiency and accuracy, there should be a numerical correspondence in the values. Clearly, because the overall mean values, comparing HPLC and enzymic lymphocyte determinations, differ by a factor of 12, this is not the case. But there could still be a correlation between the paired values, without numerical correspondence. One laboratory (participant 6) found such a correlation, significant at P < 0.001 (Table 4). This correlation does not, of course, indicate agreement, and in fact, HPLC values from this partner were again consistently higher than comet assay estimates, by a factor of 12. The other laboratories showed no significant correlation between the results with the two methods.

When the pairs of values from all six laboratories that measured 8-oxodGuo and FPG-sensitive sites are analyzed together, there is a significant but weak positive correlation. It is, however, more instructive to compare the two approaches after z-transforming the results to allow for the fact that values with HPLC are several times higher than values obtained using the enzymic approach. Figure 6 is a Bland and Altman plot of the difference between individual values against the mean of individual values, following z-transformation. Only one laboratory (participant 6) had results that are close to the line of agreement between the two methods.

Conclusions

We are left with a series of questions, rather than the answers we had hoped for. Is spurious oxidation under control? Apparently not, to judge by the very variable results obtained with identical samples of HeLa cells. If we could eliminate this artifact, would HPLC-ECD be sensitive enough to measure the low levels in real biological samples? Apart from the problem of accurate measurement of small peaks, there is the challenge of identifying 8-oxodGuo unequivocally. Normally a voltammogram can give a tentative identification, but there is insufficient DNA in the lymphocytes from blood samples citation purposes)

to do this. Thus, the only way to identify the 8-oxodGuo peak is by its retention time. HPLC-MS/MS can in principle provide a definitive identification, and measurement not just of 8-oxodGuo but also of other oxidative DNA lesions; but until now it has proved to be a demanding and complicated method and has not given reliable results in the analysis of 8-oxodGuo.

The enzymic approach, using FPG to convert oxidized purines to DNA breaks, appears to be relatively free from the problem of spurious oxidation. But to exploit this approach to full effect, we need to be able to use it quantitatively, and so we must address the question of which lesions FPG detects, and how efficiently. The calibration of the enzymic methods should not rely solely on comparison with the effects of ionizing radiation. It is worth pointing out that the use of repair endonucleases is not limited to FPG; endonuclease III can be used in a similar way to detect oxidized pyrimidines, providing another biomarker of oxidative damage.

Notwithstanding the remaining problems, note that we are talking now about variation between the two approaches of ~6-12-fold in measuring similar samples (HeLa cells or lymphocytes), compared with the far wider range that confronted us at the start of ESCODD. We are able to provide a realistic estimate of the actual background level of damage in lymphocytes; it is likely to lie somewhere between 4.2 and 0.3 8-oxodGuo per 10 6 guanines (i.e., the median values of the means from different laboratories, for HPLC and the comet assay).

The implications of this conclusion are far-reaching. Many trials have been carried out in which human lymphocytes, or tissues from normal or tumor tissue, have been analyzed for 8-oxodGuo using the techniques that we have tested (or GC-MS, which was represented in previous rounds of ESCODD). One topic under intensive investigation that is affected by this measurement problem is that of dietary antioxidants. In numerous studies, volunteers have taken supplements of antioxidants such as vitamin C, carotenoids, or flavonoids, or foods rich in antioxidants, and oxidized bases have been measured in the DNA of white blood cells. It is important to know whether these supplements are effective in decreasing the level of damage-which would support claims that they may help to prevent cancer. It will be necessary to reexamine those published reports that are based on claims of much higher values of 8-oxodGuo than 4 per 10 6 guanines. In future, with the information that we have provided, a more rigorous approach can be followed. a CVs for HeLa cells were calculated from triplicate determinations from each tube ("intra-tube") and from the mean values for three tubes ("intertube"). CVs for lymphocytes were calculated from the replicate determinations on each individual subject. Electrochemical detection (coulometric) was used except where indicated otherwise (amperometric or MS/MS). Laboratory 21 used the ESCODD method in addition to their own method for extraction/hydrolysis of DNA. b The value of 0.3 relates to a sample in which only duplicates were analyzed.

  (1, 2, 3, 4, 5, 6, 14, and 15). Participant 8 submitted results for alkaline elution, and participants 10 and 23 submitted results for alkaline unwinding. Results are shown in Figure2. The median value from all enzymic methods is 0.50 FPGsensitive sites/10 6 guanines in HeLa cell DNA, and the results show a 10-fold variation overall, from Page 8 of 25(page number not for citation purposes)

Fig. 1 Figure 1 .

 11 Fig. 1

Fig. 2

 2 Fig. 2

Figure 2 .

 2 Figure 2. Frequencies of FPG-sensitive sites in standard samples of HeLa cells, as measured by enzymic methods. Laboratory 8 used alkaline elution; laboratories 10 and 23 used alkaline unwinding; all others used the comet assay. Laboratory 4 carried out electrophoresis at 18ºC (4*) as well as at the standard temperature of 4ºC (4). Bars indicate SE.

Fig. 3 Figure 3 .

 33 Fig. 3

Fig. 4

 4 Fig. 4

Figure 4 .

 4 Figure 4. Mean frequencies of FPG-sensitive sites (measured with the comet assay) in lymphocytes from volunteers in different countries. Bars indicate SE. Below horizontal axis: first line, partner number; second line, country; third line, number of subjects. Abbreviations: Dk, Denmark; Sc, Scotland; Sw, Sweden; Sk, Slovakia; Be, Belgium; It, Italy.

Fig. 5 Figure 5 .

 55 Fig. 5

Fig. 6 Figure 6 .

 66 Fig. 6

  ml with cold RPMI medium containing 10% fetal calf serum. After centrifugation at 200 × g for 5 min at 4°C, cells were washed with cold PBS, counted, and centrifuged again. Samples were incubated with FPG, and then cells were subjected to alkaline treatment for a fixed period. The rate of unwinding in alkali is increased if the DNA contains breaks, and the break frequency is calculated from the fractions of single-and double-stranded DNA as measured by hydroxyapatite chromatography, calibrating with ionizing radiation. Full details are in refs 10 and 11.

FPG was prepared from Escherichia coli containing an overproducing vector. Laboratories 3 and 10 obtained the enzyme from Serge Boiteux (CNRS-CEA/Fontenay aux Roses, France); other laboratories used extracts prepared in the Rowett Research Institute or at the Institute of Pharmacy, University of Mainz.

Table 1 Variations in methods used for preparing DNA for analysis by HPLC. Laboratories that received samples but did not submit results are not listed here a

 1

	Purging of	Antioxidant/	Detergent	Antioxidant/	SDS treatment	RNase	Protein	Chloroform/3methyl-	Precipitation of
	solutions	chelator in	to lyse cells	chelator in nuclear	(temp., time)	treatment	digestion	1butanol extraction	remaining
		homogenizing		resuspension		(temp.,	(temp.,time)		protein with
		buffer		buffer		time)			salt
	2 No	5 mM DF	Triton X-100 5 mM DF	37ºC,	37ºC,	37ºC,	Yes	Yes
					10 min	30 min	30 min
	3 No	0.1 mM DF	Triton X-100 5 mM EDTA,	RT, 5 min	50ºC,	37ºC,	No	No
				0.15 mM DF		15 min	1 h
	4 No	0.1 mM DF	Triton X-100 5 mM EDTA,	RT, 5 min	50ºC,	37ºC,	No	No
				0.15 mM DF		15 min	1 h
	5 Ar	5 mM DF	Triton X-100 5 mM DF	37ºC,	37ºC,	37ºC,	Yes	Yes
					10 min	30 min	30 min
	6 N2	5 mM DF	Triton X-100 5 mM DF	37ºC,	37ºC,	37ºC,	Yes	Yes
					10 min	30 min	30 min
	9 No	1 mM DF	Triton X-100 1 mM DF	37ºC,	37ºC,	37ºC,	Yes	Yes
					10 min	30 min	30 min
	13 No	0.1 mM DF	Triton X-100 0.1 mM DF	37ºC,	37ºC,	37ºC,	No	No
					10 min	1 h	2 h
	14 No	0.1 mM DF	Triton X-100 5 mM EDTA,	37ºC,	50ºC,	37ºC,	No	No
				0.15 mM DF	15 min	15 min	1 h
	15 No: Chelex	1 mM DF	Tween-20 4 M GTC	None	None	None	Yes	No
	used
	16 No	1 mM DF, 3	Triton X-100 1 mM DF, 3 mM	37ºC,	37ºC,	37ºC,	Yes	No
		mM reduced		reduced GSH, 4	30 min	30 min	1 h
		GSH, 4 mM		mM his, 1 mM DTT
		his, 1mM DTT

(page number not for citation purposes)

Table 2 Variations in methods used for preparing DNA for analysis by HPLC (continued). Hydrolysis was with P1 nuclease/alkaline phosphatase in all cases. Electrochemical detection (coulometric) was used except where indicated (last column). Laboratories that received samples but did not submit results are not listed here a DNA precipitated with Salt added Removal of excess alcohol Method for dissolving DNA Treatment of hydrolysate

 2

	Variation
	of HPLC
	detection

a Abbreviations: RT, room temperature; DF, desferal (deferoxamine mesylate); TEMPO, 2,2,6,6-tetramethylpiperidine-N-oxyl.

Table 3 CVs (%) of determination of 8-oxodGuo in HeLa cells and lymphocytes by HPLC (with laboratories' own extraction methods) a

 3

	Participant

Table 4 Correlations between individual values for concentrations of 8-oxodGuo (HPLC) and FPG-sites (comet assay) in those laboratories performing both assays

 4

	Participant	Number of volunteers Correlation (r) P
	2	16	-0.30	NS a
	4	11	0.59	NS
	5	17	-0.14	NS
	6	20	0.93	<0.001
	14	8	-0.11	NS
	15	16	-0.41	NS

a NS; not significant

(page number not for citation purposes)

ACKNOWLEDGMENTS

Laboratory 4 would like to thank Professor P. Fondu (Laboratory of Hematology and Clinical Chemistry, Brugmann University Hospital, Brussels, Belgium) for help with collection of blood samples. This work was supported by the European Commission (QLK1-1999-00568) and the Scottish Executive Environment and Rural Affairs Department.


Vahagn Nersesyan
email: vahagn.nersesyan@math.uvsq.fr

Ergodicity for the randomly forced Navier-Stokes system in a two-dimensional unbounded domain

Keywords: AMS subject classifications: 35Q30, 37A25, 93B05 Navier-Stokes system, unbounded domain, exponential mixing, controllability, asymptotic compactness

Ergodicity for the randomly forced Navier-Stokes system in a two-dimensional unbounded domain

Vahagn Nersesyan

To cite this version:

Vahagn Nersesyan. Ergodicity for the randomly forced Navier-Stokes system in a two-dimensional unbounded domain.

Introduction

The ergodicity of randomly forced 2D Navier-Stokes (NS) system has been widely studied in the literature in the case of bounded domains (see the papers [FM95, KS00, EMS01, BKL02] for the first results and the book [START_REF] Kuksin | Mathematics of Two-Dimensional Turbulence[END_REF] and the reviews [START_REF] Flandoli | An introduction to 3D stochastic fluid dynamics[END_REF][START_REF] Debussche | Ergodicity results for the stochastic Navier-Stokes equations: an introduction[END_REF] for a detailed discussion of different methods and for further references). This paper is concerned with the ergodic behaviour of the NS system in an unbounded domain D in R 2 with smooth boundary ∂D: ∂ t u -ν∆u + u, ∇ u + ∇p = η(t, x), x ∈ D, (0.1) div u = 0, u| ∂D = 0, (0.2)

u(0) = u 0 . (0.3)
Here ν > 0 is the kinematic viscosity of the fluid, u = (u 1 (t, x), u 2 (t, x)) is the velocity field, p = p(t, x) is the pressure, and η is an external random force.

To have a suitable dissipativity property for solutions, we assume that D is a Poincaré domain, 1 i.e., there is a number λ 1 > 0 such that

D |v| 2 dx ≤ λ -1 1 D |∇v| 2 dx, v ∈ C ∞ 0 (D, R 2 ). (0.4)
The random force η is a process of the form

η(t, x) = ∞ k=1 I [k-1,k) (t)η k (t -k + 1, x), t ≥ 0, x ∈ D, (0.5)
where I [k-1,k) is the indicator function of the interval [k -1, k) and {η k } is a sequence of independent and identically distributed (i.i.d.) random variables in the space 2 E := L 2 ([0, 1], H). Moreover, the law of η k is assumed to be decomposable in the following sense.

Decomposability. There is an orthonormal basis {e j } in E such that

η k = ∞ j=1
b j ξ jk e j (0.6)

for some real-valued independent random variables ξ jk verifying |ξ jk | ≤ 1 and some positive numbers b j such that ∞ j=1 b 2 j < ∞. The law of the random variable ξ jk is absolutely continuous with respect to the Lebesgue measure and the corresponding density ρ j is C 1 -smooth and ρ j (0) > 0 for all j ≥ 1.

The restriction to integer times of the velocity field u t defines a family of Markov processes (u k , P u ) parametrised by the initial condition u 0 = u ∈ H. The associated Markov operators are denoted by P k and P * k . Recall that a measure µ ∈ P(H) is stationary for the family (u k , P u ) if P * 1 µ = µ. In this paper, we prove the following result.

Main Theorem. Under the above assumptions, the family (u k , P u ) has a unique stationary measure µ ∈ P(H). Moreover, it is exponentially mixing in the following sense: for any compact set H ⊂ H, there are numbers C > 0 and c > 0 such that

P * k λ -µ * L(H) ≤ Ce -ck , k ≥ 0 (0.7)
for any initial measure λ ∈ P(H) with supp λ ⊂ H. Here • * L(H) is the dual-Lipschitz metric defined by (0.11).

To the best of our knowledge, this is the first result that establishes uniqueness of stationary measure and exponential mixing for the NS system in an unbounded domain. In this unbounded setting there are at least two additional difficulties compared to the case of a bounded domain. First, the resolving operator of the equation is not compact and does not have a compact attracting set. The second and more important difficulty is related to the presence of a continuous component in the spectrum of the Stokes operator. Let us note that the existence of a stationary measure for the NS system perturbed by the random force (0.5) can be established by combining the Bogolyubov-Krylov argument and the asymptotic compactness of the dynamics. When the driving force is a white noise, existence results are obtained in [START_REF] Eckmann | Invariant measures for stochastic partial differential equations in unbounded domains[END_REF], for a real Ginzburg-Landau equation, in [START_REF] Rougemont | Space-time invariant measures, entropy, and dimension for stochastic Ginzburg-Landau equations[END_REF], for a complex Ginzburg-Landau equation, in [START_REF] Brzeźniak | Asymptotic compactness and absorbing sets for 2D stochastic Navier-Stokes equations on some unbounded domains[END_REF], for the NS system, and in [START_REF] Ekren | Existence of invariant measures for the stochastic damped Schrödinger equation[END_REF], for a damped Schrödinger equation.

The uniqueness of stationary measure and mixing properties for PDEs in unbounded domains have been studied previously only for the Burgers equation. The case of inviscid equation on the real line has been considered in the paper [START_REF] Bakhtin | Space-time stationary solutions for the Burgers equation[END_REF], under the assumption that the driving force is a space-time homogeneous Poisson point process. The proof is based on a combination of Lagrangian methods and first/last passage percolation theory. This result is generalised to the viscous case in the recent paper [START_REF] Bakhtin | Thermodynamic Limit for Directed Polymers and Stationary Solutions of the Burgers Equation[END_REF], where the perturbation is a space-time homogeneous random kick force. In both papers, the stationary measure is space translation invariant. In the case of the NS system perturbed by a homogeneous noise, the uniqueness of a space-time translation invariant measure (and in some cases also the existence) remains an open problem.

The proof of the Main Theorem uses a controllability approach of the papers [START_REF] Shirikyan | Control and mixing for 2D Navier-Stokes equations with space-time localised noise[END_REF][START_REF] Shirikyan | Controllability implies mixing II. Convergence in the dual-Lipschitz metric[END_REF], where exponential mixing is established for the NS system with a space-time and boundary localised forcing. In [START_REF] Nersesyan | Exponential mixing for a class of dissipative PDEs with bounded degenerate noise[END_REF][START_REF] Kuksin | Mixing via controllability for randomly forced nonlinear dissipative PDEs[END_REF], methods of control theory are used to study a family of parabolic PDEs with a random perturbation that is highly degenerate in the Fourier space; see also the paper [START_REF] Kuksin | Exponential mixing for dissipative PDEs with bounded non-degenerate noise[END_REF], where a non-degenerate version of these results is presented. In [START_REF] Jakšić | Large deviations and entropy production in viscous fluid flows[END_REF], controllability is used to derive large deviations principle for the Lagrangian trajectories of the NS system.

The main novelty of the controllability argument we use here is that the deterministic operator is not supposed to be regularising to a space that is compactly embedded into the main phase space (see Theorem 1.1). That regularisation condition is replaced by two weaker properties: we assume that the nonlinear dynamics is asymptotically compact and the linearised operator can be decomposed into a sum of two operators one of which is dissipative and the other is compact (see (1.3)). It is proved that these weaker conditions are still sufficient for the exponential mixing. We make an essential use of the form of the nonlinearity to verify these properties for the NS system. The reader is referred to Section 1 for further discussion of the abstract controllability criterion and to Section 2 for the verification of the conditions of this criterion for the NS system.

Let us close this section with two remarks. Note that the convergence rate c in (0.7) depends on the initial compact H. Indeed, this is related to the fact that the NS system in the unbounded case does not have a compact invariant attracting set. Without going into the details, let us mention that the existence of such attracting set (and a uniform convergence rate) can be recovered if we take initial condition and forcing in weighted Sobolev spaces as in [START_REF] Babin | The attractor of a Navier-Stokes system in an unbounded channel-like domain[END_REF].

The second remark is about the NS system with the Ekman damping. By literally repeating the arguments of the proof of the Main Theorem, one can establish exponential mixing in the case of the whole space D = R 2 or arbitrary unbounded3 domain D ⊂ R 2 with smooth boundary for the following system:

∂ t u -ν∆u + au + u, ∇ u + ∇p = η(t, x), div u = 0, u| ∂D = 0,
where a > 0 is the damping parameter. The damping ensures a dissipativity property for the solutions without any assumption on the domain. We shall not discuss the details of this generalisation in this paper.

Notation

Let D ⊂ R 2 be an unbounded Poincaré domain with smooth boundary. In this paper, we use the following functional spaces.

C ∞ 0 (D, R 2 ) is the space of compactly supported smooth functions u : D → R 2 , and V = {u ∈ C ∞ 0 (D, R 2 ) : div u = 0}. (0.8)
H s (D, R 2 ) and L p (D, R 2 ) are the Sobolev and Lebesgue spaces on D. We consider the NS system in the usual functional spaces:

H = closure of V in L 2 (D, R 2 ), (0.9) V = closure of V in H 1 (D, R 2 ) (0.10)
endowed with the scalar products

u, v = D u • v dx, u, v 1 = D (∇u 1 • ∇v 1 + ∇u 2 • ∇v 2 ) dx
and the corresponding norms

• = •, • and • 1 = •, • 1 . The Poincaré in- equality (0.4) implies that • 1 is equivalent to the norm inherited from H 1 (D, R 2 ).
The dual of V with respect to the scalar product in H is denoted by V ′ .

Let X be a Polish space with metric d. B(X) denotes the Borel σ-algebra on X. C b (X) is the space of continuous functions g : X → R endowed with the supnorm g ∞ . When X is compact, we write C(X). L b (X) is the space of functions g ∈ C b (X) for which the following norm is finite:

g L(X) = g ∞ + sup u =v |g(u) -g(v)| d(u, v) .
P(X) is the set of Borel probability measures on X endowed with the metric

µ 1 -µ 2 * L(X) = sup g L(X) ≤1 | g, µ 1 -g, µ 2 |, µ 1 , µ 2 ∈ P(X), (0.11)
where g, µ = X g(u)µ(du).

Let E be a Banach space endowed with a norm • E , and let

J T = [0, T ]. L p (J T , E), 1 ≤ p < ∞ is the space of measurable functions u : J T → E such that u L p (JT ,E) = T 0 u(s) p E ds 1 p < ∞. C(J T , E) is the space of continuous functions u : J T → E with the norm u C(JT ,E) = sup t∈JT u(t) E .
L(E, Y ) is the space of bounded linear operators from E to another Banach space Y . We write

L(E) when E = Y . B E (a, R) is the closed ball in E of radius R centered at a. We write B E (R) when a = 0. D(η) is the law of E-valued random variable η.

Abstract criterion

Let H be a separable Hilbert space, and E be a separable Banach space. In this section, we consider a random dynamical system of the form

u k = S(u k-1 , η k ), k ≥ 1, ( 1
u 0 = u ∈ H (1.2)
and the vectors η i = ζ i , i = 1, . . . , k. For any set H ⊂ H, we define the set of attainability in time k ≥ 1:

A k (H) := {S k (u; ζ 1 , . . . , ζ k ) : u ∈ H, ζ 1 , . . . , ζ k ∈ K}
and the set of attainability in infinite time:

A(H) := ∪ ∞ k=1 A k (H) H .
The following five conditions are assumed to be satisfied for the mapping S and the measure ℓ.

(i) Regularity. The mapping S : H × E → H is twice continuously differentiable, and its derivatives are bounded on bounded subsets. Moreover, for any (u, η) ∈ H × K, the derivative (D u S)(u, η) can be represented as

(D u S)(u, η) = Ψ 1 + Ψ 2 (u, η), (1.3)
where the operators4 Ψ 1 , Ψ 2 (u, η) ∈ L(H) are such that

Ψ 1 L(H) =: κ < 1 (1.4)
and Ψ 2 (u, η) is compact.

(ii) Asymptotic compactness. For any bounded sequence {u n 0 } in H, any integers l n ≥ 1 such that l n → ∞, and any family

{ζ n m : m, n ≥ 1} ⊂ K, the sequence {S ln (u n 0 ; ζ n 1 , . . . , ζ n ln )} is precompact in H. (iii)
Approximate controllability to a point. There is a point û ∈ H with the following property: for any ε > 0 and any compact H in H, there is an integer n ≥ 1 such that, for any initial point u ∈ H, there are vectors

ζ 1 , . . . , ζ n ∈ K satisfying S n (u; ζ 1 , . . . , ζ n ) -û ≤ ε.
(1.5) (iv) Approximate controllability of the linearisation. For any u ∈ H and η ∈ K, the image of the linear mapping (D η S)(u, η) : E → H is dense in H.

(v) Decomposability. The set K is compact in E. Moreover, there are sequences of closed subspaces {F n } and {G n } in E satisfying the following properties:

• F n are finite-dimensional, F n ⊂ F n+1 for any n ≥ 1, and E = ∪ n F n .
• E is the direct sum of the spaces F n and G n , and the norms of the corresponding projections P n and Q n are bounded in n ≥ 1.

• ℓ is the product of projections P n * ℓ and Q n * ℓ for any n ≥ 1. Moreover, the measures P n * ℓ have C 1 -smooth densities with respect to the Lebesgue measure on F n .

System (1.1) defines a family of Markov processes (u k , P u ) in H parametrised by the initial condition (1.2). Let P k : C b (H) → C b (H) and P * k : P(H) → P(H) be the associated Markov operators.

Theorem 1.1. Under Conditions (i)-(v), the family (u k , P u ) has a unique stationary measure µ ∈ P(H). Moreover, for any compact set H in H, there are numbers C > 0 and c > 0 such that

P * k λ -µ * L(H) ≤ Ce -ck , k ≥ 1 (1.6)
for any initial measure λ ∈ P(H) with supp λ ⊂ H.

See the papers [Shi15, Shi19, KNS18, KNS19, KZ18, JNPS19] for related abstract criteria for uniqueness of stationary measure and mixing. The formulation of Theorem 1.1 is close to the results in [KNS19, JNPS19], but the proof is based on a theorem obtained in [START_REF] Shirikyan | Controllability implies mixing II. Convergence in the dual-Lipschitz metric[END_REF]. There are two main differences in our formulation. First, in Condition (i), we do not suppose that the mapping S (or its linearisation) takes values in a space that is compactly embedded into H. Instead, we assume that (D u S)(u, η) is a sum of dissipative and compact operators. The second difference is Condition (ii), which allows to recover some compactness properties5 for the dynamics. These two new conditions make Theorem 1.1 applicable to the NS system in unbounded domains. Moreover, this theorem can be applied to dissipative PDEs without parabolic regularisation (this will be addressed in a subsequent publication).

As in this paper the random perturbation is non-degenerate, the image of the mapping (D η S)(u, η) is assumed to be dense for any η in K; degenerate versions of this criterion can also be envisaged (cf. [START_REF] Nersesyan | Exponential mixing for a class of dissipative PDEs with bounded degenerate noise[END_REF][START_REF] Kuksin | Mixing via controllability for randomly forced nonlinear dissipative PDEs[END_REF]).

Proof of Theorem 1.1. Step 1: Existence of stationary measure. The set X := A(H) is compact in H. Indeed, it suffices to show that any sequence {v n } of the form

v n = S ln (u n 0 ; ζ n 1 , . . . , ζ n ln )
with some u n 0 ∈ H, ζ n 1 , . . . , ζ n ln ∈ K, and l n ≥ 1, is precompact in H. If the sequence {l n } is bounded, then v n ∈ A m (H) for all n ≥ 1, where m = max{l n }. The fact that A m (H) is compact (as image of a compact set by a continuous mapping) implies that {v n } is precompact. In the case l n → ∞, the conclusion follows from Condition (ii).

The compactness of X, combined with the invariance property S(X×K) ⊂ X and the usual Bogolyubov-Krylov argument (e.g., see Section 2.5 in [START_REF] Kuksin | Mathematics of Two-Dimensional Turbulence[END_REF]), implies the existence of a stationary measure µ ∈ P(X).

Step 2: Limit (1.6). According to Theorem 1.1 in [START_REF] Shirikyan | Controllability implies mixing II. Convergence in the dual-Lipschitz metric[END_REF], limit (1.6) will be established if we verify the following property.

Local stabilisation. Let D δ := {(u, u ′ ) ∈ X × X : u -u ′ ≤ δ}.
For any R 0 and any compact K ⊂ E, there is a finite-dimensional subspace E ⊂ E, and a continuous mapping

Φ : D δ × B E (R) → E, (u, u ′ , η) → η ′ ,
which is continuously differentiable in η and satisfies the inequalities

sup η∈BE (R) Φ(u, u ′ , η) E + D η Φ(u, u ′ , η) L(E) ≤ C u -u ′ , (1.7) sup η∈K S(u, η) -S(u ′ , η + Φ(u, u ′ , η)) ≤ q u -u ′ , (u, u ′ ) ∈ D δ (1.8)
for some positive constants C, δ, and q < 1. Let us show that Conditions (i), (iv), and (v) imply this local stabilisation property. We use a construction of approximate right inverse for linear operators from Section 2.2 in [START_REF] Nersesyan | Exponential mixing for a class of dissipative PDEs with bounded degenerate noise[END_REF]. For any u ∈ X and η ∈ E, let A(u, η) : E → H be given by A(u, η) := (D η S)(u, η). Then G(u, η) := A(u, η) A(u, η) * : H → H is non-negative self-adjoint operator and Im(G(u, η)) is dense in H by Condition (iv). Thus (G(u, η) + γI) -1 is well defined for any γ > 0, and we have the limit

G(u, η)(G(u, η) + γI) -1 f → f as γ → 0 + , f ∈ H,
(1.9) by Lemma 2.4 in [START_REF] Nersesyan | Exponential mixing for a class of dissipative PDEs with bounded degenerate noise[END_REF]. This shows that A(u, η) * (G(u, η) + γI) -1 is an approximate right inverse for A(u, η). We truncate it to obtain an operator with finite-dimensional image:

R M,γ (u, η) := P M A(u, η) * (G(u, η) + γI) -1 ,
where P M is a projection as in Condition (v) and M ≥ 1 and γ > 0 are parameters that will be chosen later. It is straightforward to see that

R M,γ (u, η) L(H,E) + (D η R M,γ )(u, η) L(H×E,E) ≤ C 1 (R, M, γ) (1.10)
for any u ∈ X and η ∈ B E (R), where the constant C 1 (R, M, γ) > 0 does not depend on (u, η). By the Taylor formula, for any u, u ′ ∈ X and η, η ′ ∈ B E (R), we have

S(u ′ , η ′ ) -S(u, η) = (D u S)(u, η)(u ′ -u) + (D η S)(u, η)(η ′ -η) + r(u, u ′ , η, η ′ ) (1.11) where r(u, u ′ , η, η ′ ) ≤ C 2 (R) u -u ′ 2 + η -η ′ 2 E .
(1.12)

The mapping Φ is defined by

Φ(u, u ′ , η) := -R M,γ (u, η)Ψ 2 (u, η)(u ′ -u).
Then (1.7) is verified due to (1.10) and

C 3 := sup (u,η)∈X×BE (R) Ψ 2 (u, η) L(H) + D η Ψ 2 (u, η) L(H×E,H) < ∞. (1.13)
The fact that C 3 is finite follows from the boundedness on X × B E (R) of the norms of the derivatives of S(u, η) and the representation (1.3). Assume that, for any ε > 0, we are able to find numbers M ≥ 1 and γ > 0 such that

(D η S)(u, η)R M,γ (u, η)Ψ 2 (u, η)g -Ψ 2 (u, η)g ≤ ε g (1.14)
for any u ∈ X, η ∈ K, and g ∈ H. Then combining (1.3), (1.4), and (1.11)-(1.14), we obtain

S(u, η) -S(u ′ , η + Φ(u, u ′ , η)) ≤ κ u ′ -u + ε u ′ -u + r(u, u ′ , η, η + Φ(u, u ′ , η)) ≤ (κ + ε + C 4 (ε, R) u ′ -u ) u ′ -u ≤ q u ′ -u
for any (u, u ′ ) ∈ D δ and η ∈ K, where ε > 0 and δ > 0 are sufficiently small and q < 1. This completes the proof of the local stabilisation and limit (1.6). Let us prove inequality (1.14). By (1.9), the continuous dependence of the operators (D η S)(u, η) and Ψ 2 (u, η) on (u, η), and a simple compactness argument, we find a large integer M ≥ 1 and a small number γ > 0 such that6

(D η S)(u, η)R M,γ (u, η)f -f ≤ ε
for any u ∈ X, η ∈ K, and f ∈ Ψ 2 (u, η)(B H (1)). Then inequality (1.14) follows by homogeneity.

Step 3: Uniqueness of stationary measure. To complete the proof, it remains to show the uniqueness of stationary measure in P(H). Assume that µ is the stationary measure supported7 in A({û}), and let λ 1 be any stationary measure for (u k , P u ) in P(H). As λ 1 (A(H)) = 1, there is a sequence

{H n } of compacts in H such that λ 1 (A(H n )) > 1 -1/n for any n ≥ 1.
By Condition (iii), we have A({û}) ⊂ A(H n ), and µ and

λ 1 /λ 1 (A(H n )) are stationary measures for (u k , P u ) in P(A(H n )). So µ = λ 1 /λ 1 (A(H n ))
by the uniqueness of stationary measure on X = A(H n ). Therefore,

λ 1 (Γ) = lim n→∞ λ 1 (Γ ∩ A(H n )) = lim n→∞ λ 1 (Γ ∩ A(H n ))/λ 1 (A(H n )) = µ(Γ)
for any Γ ∈ B(H). Thus λ 1 = µ.

Proof of the Main Theorem

In this section, we prove the Main Theorem by applying Theorem 1.1. We begin with a short discussion of the deterministic NS system, then turn to the verification of Conditions (i)-(v) in an appropriate functional setting.

Preliminaries

Applying the Leray projection 8 Π to Eq. (0.1), we eliminate the pressure term and consider the evolution system

u + νLu + B(u) = η, (2.1)
where L = -Π∆ is the Stokes operator and B(u

) = Π( u, ∇ u). Let us define a bilinear symmetric form [•, •] : V × V → R by [u, v] := u, v 1 - λ 1 2 u, v , u, v ∈ V.
The Poincaré inequality (see (0.4))

u 2 ≤ λ -1 1 u 2 1 , u ∈ V (2.2) implies that 1 2 u 2 1 ≤ [u] 2 := [u, u] ≤ u 2 1 . (2.3) Thus [•, •] defines a scalar product on V with norm [•] equivalent to • 1 .
Proposition 2.1. For any T > 0, u 0 ∈ H, and η ∈ L 2 (J T , H), there is a unique solution u ∈ C(J T , H) ∩ L 2 (J T , V ) of problem (2.1), (0.3). It satisfies the following inequalities

u(t) 2 ≤ e -νλ1t u 0 2 + ν -2 λ -1 1 η 2 L 2 (JT ,V ′ ) , t ∈ J T , (2.4)
u 2 L 2 (JT ,V ) ≤ ν -1 u 0 2 + ν -2 η 2 L 2 (JT ,V ′ ) , (2.5)
and the equality

u(t) 2 = e -νλ1t u 0 2 + 2 t 0 e -νλ1(t-s) η(s), u(s) -ν[u(s)] 2 ds, t ∈ J T .
(2.6)

Let us define the mapping

S t : H × L 2 (J T , H) → H, (u 0 , η) → u(t), t ∈ J T .
The following stability result holds.

8 That is the orthogonal projection in L 2 (D, R 2 ) onto H.

Proposition 2.2. Assume that the sequence {u n 0 } converges weakly to u 0 in H and the sequence {η n } converges strongly to η in L 2 (J T , H). Then

S t (u n 0 , η n ) ⇀ S t (u 0 , η) weakly in H for t ∈ J T , (2.7)
S • (u n 0 , η n ) ⇀ S • (u 0 , η) weakly in L 2 (J T , V ).
(2.8)

The proofs of these two propositions are carried out by standard methods and are given in the Appendix.

Let us now describe the functional setting in which Theorem 1.1 is applied. The space H is defined by (0.9), E := L 2 ([0, 1], H), and the mapping

S := S 1 : H × E → H, (u 0 , η) → u(1)
is the time-one shift along trajectories of Eq. (2.1). Then the restriction to integer times of the solution of (2.1), (0.3), (0.5) satisfies

u k = S(u k-1 , η k ), k ≥ 1, (2.9)
where {η k } is a sequence of i.i.d. random variables as in the decomposability condition in the Introduction. It is straightforward to see that Condition (v) satisfied.9 In the next three subsections, we check Conditions (i)-(iv).

Regularity condition

The smoothness of the mapping S : H × E → H and the boundedness of its derivatives on bounded subsets of H × E are proved using well-known methods (e.g., see Chapters I and VII in [START_REF] Babin | Attractors of Evolution Equations[END_REF] for the case of bounded domain D).

Let us consider the linearisation of Eq. (2.1) around the trajectory ũ(t) = S t (u, η) corresponding to an initial condition ũ(0) = u ∈ H and control η ∈ K:

ẇ + νLw + Q(ũ, w) = 0,
(2.10)

w(0) = w 0 , (2.11)
where

Q(a, b) = Π( a, ∇ b) + Π( b, ∇ a).
(2.12)

Then (D u S)(u, η)w 0 = w(1) for any w 0 ∈ H, and we can write w = v 1 + v 2 , where v 1 and v 2 are the solutions of the problems

v1 + νLv 1 = 0, v 1 (0) = w 0 , (2.13) v2 + νLv 2 + Q(ũ, w) = 0, v 2 (0) = 0.
(2.14)

The representation (1.3) holds with the linear operators

Ψ 1 : H → H, w 0 → v 1 (1), Ψ 2 (u, η) : H → H, w 0 → v 2 (1).
Inequality v 1 (t) ≤ e -νλ1t/2 w 0 implies (1.4) with κ := e -νλ1/2 . The following lemma (whose proof is given in the Appendix) completes the verification of Condition (i).

Lemma 2.3. For any (u, η) ∈ H × K, the operator Ψ 2 (u, η) : H → H is compact.

Asymptotic compactness

For any integer k ≥ 1 and any function η ∈ L 2 (J k , H) of the form (0.5) with some sequence {η n } ⊂ E, we write u(t) = S t (u 0 , η) =: S t (u 0 ; η 1 , . . . , η k ), t ∈ J k .

(2.15)

Proposition 2.4. For any bounded sequence {u n 0 } in H, any sequence of integers l n ≥ 1 such that l n → ∞, and any family {ζ n m : m, n

≥ 1} ⊂ K, the sequence v n = S ln (u n 0 ; ζ n 1 , . . . , ζ n ln ) is precompact in H.
Proof. Let us first explain the scheme of the proof. Using the dissipativity of the system, we show that {v n } is bounded in H. As H is a Hilbert space, there is a subsequence {v kn } such that v kn ⇀ w weakly in H.

( Step 1: Boundedness. Let us show that, for any bounded set H ⊂ H, the attainability set A(H) ⊂ H is bounded. Indeed, let

M 1 := sup u∈H u 2 < ∞.
For any u 0 ∈ H and ζ 1 , . . . , ζ k ∈ K, let u k := S k (u 0 ; ζ 1 , . . . , ζ k ). Then by inequality (2.4), we have

u k 2 ≤ κ u k-1 2 + M 2 ,
where κ := e -νλ1 < 1 and M

2 := ν -2 λ -1 1 sup η∈K η 2 L 2 ([0,1],V ′ ) . Iterating this, we get u k 2 ≤ κ k u 0 2 + M 2 (1 -κ) -1 ≤ M 1 + M 2 (1 -κ) -1 =: M.

This shows that sup

u∈A(H)

u 2 ≤ M.

Step

w 2 = u m (m) 2 = e -νλ1m w m 2 + 2 m 0 e -νλ1(m-s) ξ m (s), u m (s) -ν[u m (s)] 2 ds, (2.20)
where

u m (t) = S t (w m , ξ m ), ξ m (t) = m k=1 I [k-1,k) (t)ξ m k (t -k + 1), t ∈ J m .
On the other hand, again by (2.6), we have As m ≥ 1 is arbitrary, we arrive at (2.18).

Approximate controllability

By the decomposability assumption, we have 0 ∈ K. In view of (2.4), Condition (iii) is verified with û = 0, ζ 1 = . . . = ζ n = 0, and sufficiently large n ≥ 1.

To check Condition (iv), we consider the following linearisation of Eq. (2.1) around the same trajectory ũ as in Section 2.2:

ẇ + νLw + Q(ũ, w) = ζ(t),
(2.23)

w(0) = 0,
where Q is given by (2.12). Then where the constant C 3 > 0 does not depend on the numbers n, m, ε. Combining this with (3.7) and choosing n and m sufficiently large, we see that Φ ũ(w kn 0 ) -Φ ũ(w km 0 ) L 2 (J,H -1 ) < 2C 3 ε.

This shows that Φ ũ is compact. Thus the map Ψ 2 : H → H is compact as a composition of Φ ũ with some linear continuous map.

  . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10 2.2 Regularity condition . . . . . . . . . . . . . . . . . . . . . . . . . 11 2.3 Asymptotic compactness . . . . . . . . . . . . . . . . . . . . . . . 12 2.4 Approximate controllability . . . . . . . . . . . . . . . . . . . . . 14

v kn 2 =+2 m 0 e 0 e 0 e 0 e

 20000 u n,m (m) 2 = e -νλ1m w n,m 2 -νλ1(m-s) η n,m (s), u n,m (s) -ν[u n,m (s)] 2 ds,(2.21)whereu n,m (t) = S t (w n,m , η n,m ), η n,m (t) = m k=1 I [k-1,k) (t)η n kn-m+k (t-k+1), t ∈ J m . Note that m -νλ1(m-s) η n,m (s), u n,m (s) ds → m -νλ1(m-s) ξ m (s), u m (s) ds, (2.22) as η n,m → ξ m strongly in L 2 (J m , H) and u n,m ⇀ u m weakly in L 2 (J m , V ). Since m 0 e -νλ1(m-s) [•] 2 ds 1/2is a norm in L 2 (J m , V ) which is equivalent to the original one, the following inequality holdsm 0 e -νλ1(m-s) [u m (s)] 2 ds ≤ lim infn→∞ m -νλ1(m-s) [u n,m (s)] 2 ds. Combining this with (2.20)-(2.22), we get lim sup n→∞ v kn 2 ≤ lim sup n→∞ e -νλ1m w n,m 2 + w 2 -e -νλ1m w m 2 ≤ e -νλ1m M + w 2 .

  (D η S)(u, η)ζ = w(1) for any ζ ∈ E. For any smooth function w 1 ∈ H, we can find a smooth function w : [0, 1] × D → R 2 such that w(0) = 0, w(1) = w 1 , and w(t) ∈ H for all t ∈ [0, 1]. Replacing w into Eq. (2.23), we find explicitly a control ζ ∈ E such that (D η S)(u, η)ζ = w 1 . This shows that the image of the mapping (D η S)(u, η) : E → H is dense in H for any (u, η) ∈ H ×K. Thus Conditions (i)-(v) are verified. Applying Theorem 1.1, we complete the proof of the Main Theorem.

  .1) where S : H × E → H is a continuous mapping and {η k } is a sequence of i.i.d. random variables in E. Let K ⊂ E be the support of the law ℓ := D(η k ). For any sequence {ζ k } in K, let us denote by S k (u; ζ 1 , . . . , ζ k ) the trajectory of (1.1) corresponding to the initial condition

  2: Proof of (2.18). From the previous step it follows that {v n } is bounded in H. Let {v kn } be a subsequence verifying (2.16). It is of the formv kn = S ℓn (u n 0 ; η n 1 , . . . , η n ℓn )for some vectors η n 1 , . . . , η n ℓn ∈ K and integers ℓ n ≥ 1 such that ℓ n → ∞. Using the boundedness of the set A({u n 0 , n ≥ 1}) and passing to a subsequence if necessary (applying the diagonal process), we can assume that

	w n,m := S ℓn-m (u n 0 ; η n 1 , . . . , η n ℓn-m ) ⇀ w m weakly in H	(2.19)
	From (2.16) we derive the equality
	w = S m (w m ; ξ m 1 , . . . , ξ m m ) for any m ≥ 1.
	This and (2.6) imply that

for any m ≥ 1 and some w m ∈ H. Using the compactness of K and again passing to a subsequence if necessary, we can assume that

η n ℓn-i → ξ m m-i strongly in E, i = 0, . . . m

-1. Combining this with (2.19) and Proposition 2.2, we get v kn = S m (w n,m ; η n ℓn-m+1 , . . . , η n ℓn ) ⇀ S m (w m ; ξ m 1 , . . . , ξ m m ) weakly in H.

I.e., domain D which does not necessarily satisfy the Poincaré inequality.

Ψ 1 does not depend on (u, η).

The asymptotic compactness is a well-known property in the study of the attractors for deterministic PDEs (e.g., see[START_REF] Ladyzhenskaya | Attractors for semigroups and evolution equations[END_REF][START_REF] Rosa | The global attractor for the 2D Navier-Stokes flow on some unbounded domains[END_REF]).

Here we use also the boundedness of the projections P M (see Condition (v)), which implies the limit P M → I as M → ∞ in the operator topology.

Using Condition (iii), it is easy to see that supp µ = A({û}).

Note that K := supp D(η k ) is compact in E, since it is contained in a Hilbert cube.

Acknowledgments

The author thanks Armen Shirikyan for helpful discussions and suggestions. This research was supported by Agence Nationale de la Recherche grant ANR-17-CE40-0006-02 and CNRS PICS grant Fluctuation theorems in stochastic systems.

Appendix

3.1 Proof of Proposition 2.1

The existence and uniqueness of solution is proved, e.g., in Chapter III of [START_REF] Temam | Navier-Stokes equations. Theory and numerical analysis[END_REF].

Here we give a formal derivation of inequalities (2.4) and (2.5) and equality (2.6).

Taking the scalar product in H of Eq. (2.1) with 2u and using the identity

Combining this with the Poincaré inequality, we obtain (2.4). From (3.1) we also derive the inequality

which implies (2.5). To prove (2.6), we rewrite the equality in (3.1) in the form

and apply the variation of constants formula.

Proof of Proposition 2.2

By Proposition 2.1, the sequence

) and numbers α 0 = α 1 = 2, and using the diagonal process, we find a subsequence {u kn } such that

for any R > 0. Passing to the limit in the equation for u kn (t), we conclude that u(t) is the solution S t (u 0 , η), t ∈ J T . Moreover, by the uniqueness of the limit, it is easy to see that limits (3.4)-(3.6) hold for the full sequence {u n }. This proves (2.8).

Let us take any ϕ ∈ V (see (0.8)). By inequality (2.4), we have

and by inequality (3.3),

for any t ∈ J T and τ ∈ (0, T -t). Thus the Arzelà-Ascoli theorem implies that

uniformly in t ∈ J T . Using the fact that V is dense in H, we get (2.7).

Proof of Lemma 2.3

Let us set J := [0, 1]. By Proposition 2.1, we have ũ = S • (u, η) ∈ L 2 (J, V ) for any u ∈ H and η ∈ K. Using standard methods, one can show that the mapping Ψ : w 0 → w (i.e., the resolving operator of problem (2.10), (2.11)) is continuous from

). Let us show that the linear mapping Φ ũ : H → L 2 (J, H -1 ), w 0 → ũ, ∇ w + w, ∇ ũ is compact. Indeed, let {w n 0 } be a bounded sequence in H. Then the sequence {Ψ(w n 0 )} is bounded in X . As in the previous subsection, we apply Theorem 2.1 of Chapter III in [START_REF] Temam | Navier-Stokes equations. Theory and numerical analysis[END_REF] with spaces

) and numbers α 0 = α 1 = 2, and use the diagonal process to find a subsequence {Ψ(w kn 0 )} converging strongly in L 2 (J, L 2 (D R )) for any R > 0. On the other hand, for any ε > 0, there is a number R > 0 and a smooth function ϕ : J × D → R 2 with supp ϕ(t, •) ⊂ D R for any t ∈ J and ũ -ϕ L 2 (J,V ) < ε.

(3.7)

Using the boundedness of the sequence {Φ ũ(w kn 0 )} in L 2 (J, V ) and the estimate a

we obtain Φ ũ(w kn 0 ) -Φ ũ(w km 0 ) L 2 (J,H -1 ) ≤ Φ ϕ (w kn 0 ) -Φ ϕ (w km 0 ) L 2 (J,H -1 ) + Φ ũ-ϕ (w kn 0 ) -Φ ũ-ϕ (w km 0 ) L 2 (J,H -1 ) ≤ C 3 Ψ(w kn 0 ) -Ψ(w km 0 )

1/2 L 2 (J,L 2 (DR))

+ C 3 ũ -ϕ L 2 (J,V ) ,


Abhishek Baral

Aromal Asokan

Valentin Bauer

Bruno Kieffer

Vladimir Torbeev
email: torbeev@unistra.fr

Chemical synthesis of transactivation domain (TAD) of tumor suppressor protein p53 by native chemical ligation of three peptide segments

Keywords:

published or not. The documents may come

Chemical synthesis of transactivation domain (TAD) of tumor suppressor protein p53 by native chemical ligation of three peptide segments

Introduction

Tumor protein 53 (Tp53 or p53) continued to be a subject of active research since its first identification in the late 1970s. [1][2][3][4][5][6][7][8][9][10] The protein p53 first thought to be an oncoprotein has eventually emerged as a tumor suppressor playing central role in the maintenance of genome stability. 1,[START_REF] Olivier | Cold Spring Harb[END_REF] In unstressed cells, the negative regulator human double minute 2 (HDM2) protein polyubiquitinates p53 leading to its continuous proteosomal degradation, therefore, keeping p53 concentration at a low level. [12][13][14] Whereas, upon cell stress or DNA damage, p53 gets activated via disruption of the p53-HDM2 interaction, which results in accumulation of p53 leading to inhibition of cell growth and apoptosis. [12][13][14] In many human cancer cells p53 behaves incorrectly with up to 50% of tumors containing cells possessing p53 mutations. [START_REF] Olivier | Cold Spring Harb[END_REF] Furthermore, cancer-inducing viruses can inactivate p53, for example, E6 protein of human papilloma virus (HPV) induces p53 degradation. 15 The protein p53 consists of five major domains, such as transactivation domain (TAD), proline-rich domain, DNA binding core domain, tetramerization domain and regulatory domain. 13 The N-terminal TAD is acidic, intrinsically disordered and responsible for binding to various transcription factors. 13 The activity of p53 is controlled by its interaction with HDM2 and transcriptional co-activators CREB-binding protein (CBP) and p300. 14 The TAD of p53 has binding sites to interact with transcription adaptor putative zinc finger 1 (TAZ1), TAZ2, kinase-inducible domain (KID) interacting domain (KIX), and nuclear receptor co-activator binding domain (NCBD) of CBP or p300 along with the p53-binding domain of HDM2. 14 TAD-p53 interacts via its two active regions AD1 (1-42) and AD2 (43-63). 2 In the unstressed cells, AD2 binds to CBP domains while AD1 binds to HDM2. 14 Thus, through AD1 and AD2, TAD forms a ternary complex that leads to the degradation of p53 by ubiquitination. 14 On the other hand, genotoxic stress results in the phosphorylation at Thr and Ser residues of AD1 which severely weakens its interaction with HDM2, 14 which leads to preferential binding of TAD to the CBP domains and activation of p53. 13,14 There are not many proteins in molecular oncology that have so many sites for post-translational modifications (PTMs) as p53. [START_REF] Meek | Cold Spring Harb[END_REF] Importantly, the tumor suppressor p53 was shown to be a phosphoprotein just after few years after its discovery. 17,18 Thus, phosphorylation turned into the most recognized PTM and was extensively studied over the years. 3 The TAD alone contains seven Ser and two Thr residues available for phophorylation. 14 A single phosphorylation at Thr18 results in two-times increase of p53 affinity towards TAZ1 of CBP. 14 A more significant 10-fold enhancement occurs when Ser15 and Ser20 are also phosphorylated along with Thr18. 14 Studies by several authors have shown that individual phosphorylations of Ser15, Ser20,

A R T I C L E I N F O A B S T R A C T M A N U S C R I P T A C C E P T E D ACCEPTED MANUSCRIPT Tetrahedron 2
Ser33, Ser37, Ser46, Thr18 and Thr55 modulate interactions of p53 to each of the four binding domains of CBP/p300 but to different extents. [START_REF] Meek | Cold Spring Harb[END_REF]19,20 However, Thr18 has been found having the most potent influence. 3,13,[START_REF] Meek | Cold Spring Harb[END_REF] Moreover, it has been observed that depending on the type and intensity of the stress and cell type, differential phosphorylation and thereby fine modulation of p53-CBP/p300 interaction and transcriptional outcome occurs. 3,[START_REF] Meek | Cold Spring Harb[END_REF] Thus, these events collectively provide a regulatory switch that facilitates TAD-p53 binding to CBP/p300 over its negative regulator HDM2. [START_REF] Meek | Cold Spring Harb[END_REF] Apart from phosphorylation, there are several acetylation sites in the C-terminal domains of p53 which are crucial for p53 to maintain its function. The lysine residues, Lys305, Lys370, Lys372, Lys373, Lys381, Lys382, and Lys386 are acetylated during different forms of genotoxic and non-genotoxic stress. [START_REF] Meek | Cold Spring Harb[END_REF]21,22 Significance of acetylation lies in the fact that it is mutually exclusive to ubiquitination for p53 regulation. [START_REF] Meek | Cold Spring Harb[END_REF] In unstressed cells ubiquitination leads to the degradation of p53 while acetylation ensures non-ubiquitination of the p53 residues in stressed cells which leads to p53 stabilization. Thus, both phosphorylation and acetylation are critical factors for p53 stabilization. [START_REF] Meek | Cold Spring Harb[END_REF] Although there are multiple studies that provided valuable insights into the role of phosphorylation of individual residues and other PTMs, there are still many questions that need to be addressed to fully understand the biological significance of p53. The TAD-p53 (wild-type or phosphorylated) used so far in most of the studies were either prepared biosynthetically or biologically expressed 14 or in some cases a small part of it has been synthesized chemically. 13 The efficient total chemical synthesis or semi-synthesis of full-length p53 would thus be particularly attractive to address the questions of the role of the combinations of PTMs in p53 function because such methods can allow for preparation of modified analogues with precise patterns of the desired modifications. 23 Towards this goal, in this manuscript a detailed synthetic procedure for preparation of TAD-p53 is described based on the combination of Fmoc/tBusolid phase peptide synthesis (SPPS) and native chemical ligation (NCL), which complements previously published synthetic protocol. 24

Results and Discussion

In this work the 61-residue transactivation domain of p53 (TAD-p53) has been synthesized using a three peptide segment ligation strategy (Fig. 1). The peptides were joined by native chemical ligation (NCL) using unprotected peptide-α thioesters and Cys-containing peptide segments. [25][26][27][28][29] The TAD-p53 has no native Cys residues in its sequence, therefore, Gln16 and Gln38 were replaced by Cys residues in which the thiol side chains were alkylated using bromoacetamide to achieve a side chain chemically similar to glutamine. Another modification was made to prevent the problem of oxidation of the methionines, which was shown to complicate the synthesis and isolation. 24 Thus, Met1, Met40 and Met44 were replaced with nearly isosteric norleucine (Nle) residues. As a result, the three segments in the synthesis were Nle1-Ser15 (segment-1), Cys16-Ser37 (segment-2) and Cys38-Asp61 (segment-3) (Fig. 1).

The ligation of three segments can be advantageous compared to previously reported synthesis based on a two-segment approach 24 for the following reasons: (i) peptide segments needed for ligation are shorter and easier to synthesize; (ii) elucidation of the role of PTMs requires synthesis of protein libraries where the three-segment strategy is more modular and flexible. PTMs can occur along the entire sequence of TAD-p53. 3 For this reason larger combinatorial diversity of the library will be easier to achieve with three-segment ligation approach, which will require lesser number of peptide building blocks. For example, to prepare all 512 possible phospho-variants of TAD-p53 (seven Ser and two Thr phosphorylation sites lead to 2 9 = 512 combinations), it will require synthesis of 132 peptide segments (2 7 +2 2 ) via a two-segment ligation strategy (approach in ref. 24) and only 28 peptide building blocks (2 3 +2 4 +2 2 ) using a threesegment ligation method (Fig. 1).

The ligation junctions (residues (15-16) and (37-38), respectively) were chosen in the regions of the sequence that remain unstructured in the known complex with nuclear coactivator binding domain (NCBD) of CBP, 2 therefore modification of Gln16 and Gln38 into pseudo-glutamines is unlikely to affect the properties of the synthetic TAD-p53.

Both the C-to-N and N-to-C one-pot ligation strategies were tested in our study with different thioesters of the middle segment used in the N-to-C native chemical ligation and their comparative yields are presented in this article.

Synthesis of the peptide segments

For the synthesis of the segment-1 (Nle1-Ser15), manual conventional Fmoc/tBu-SPPS was applied using hydrazide resin and HBTU (hexafluorophosphate benzotriazole tetramethyl uronium) as coupling agent and N,N-diisopropylethylamine (DIEA) as activator base (details are in the Appendix A). Subsequent cleavage from the resin and HPLC purification resulted in a yield of 56%. Segment-1 (Nle1-Ser15)-α NHNH 2 was then transformed into the corresponding α thioester (Fig. S1). To achieve such conversion, the C-terminal α hydrazide functional group was first oxidized to α azide by sodium nitrite and in situ thioesterified with MeSNa (sodium 2-mercaptoethanesulphonate) making it suitable for native chemical ligation. 30 Meanwhile, the middle segment-2 (Cys16-Ser37) was synthesized in two forms (see Table S1 and Fig. S2). In the first approach, using the hydrazide resin, (Cys16-Ser37)-α NHNH 2 (segment-2A) was synthesized using microwave-assisted SPPS using N,N′-diisopropylcarbodiimide (DIC) and Oxyma (2-cyano-2-(hydroxyimino)acetic acid ethyl ester) as coupling reagents. After cleavage and HPLC purification the yield was found to be 32%. Such (Cys16-Ser37)-α NHNH 2 containing N-terminal Cys residue was used in N-to-C ligations.

In another method, employed for C-to-N ligations, N-acylurea linker was used for the middle segment synthesis. 31 In this process, a linker 3-amino-4-(methylamino)benzoic acid was prepared with the primary amino group Fmoc-protected. 31 The Fmoc-Rink amide resin was used in this case and amino acid glycine was coupled to it followed by the linker and then the Cterminal amino acids starting from Ser37 were coupled using Fmoc-SPPS. As required in this method the last coupled Nterminal amino acid was Boc-Thz (Boc-thiazolidine). 31 It has to be noted that in the synthesis of (Thz16-Ser37)-α MeNbz (segment-2B) a more efficient coupling reagent HATU (hexafluorophosphate azabenzotriazole tetramethyl uronium) needs to be used instead of HBTU in conjunction with double couplings of the five C-terminal residues to avoid deletions. The peptide was prepared at room temperature using Syro I automatic peptide synthesizer. After the last coupling, the N-methylaniline Multiple peaks for each charge state are due to Na + and K + ion adducts. A repeating of the synthesis (summarized in Fig. S4) using this procedure with several modifications (see Appendix A) confirms complications in N-to-C strategy. moiety (linker) was subjected to cyclization on resin using chloroformate derivative and DIEA base in successive steps to convert it into an N-acylurea (MeNbz) moiety. 31 The peptide was cleaved and purified with a yield of 25% before MeSNa was used for thioesterification of (Thz16-Ser37)-α MeNbz by substituting the MeNbz moiety (Fig. S2). The resultant (Thz16-Ser37)α thioester building block was used for C-to-N ligations.

The C-terminal (Cys38-Asp61)-α amide segment was synthesized on a Rink-amide resin using microwave-assisted SPPS using DIC and Oxyma with an isolated yield of 16% (Fig. S3) and was used in both N-to-C and C-to-N ligations.

Native chemical ligation of peptide segments 2.2.1. N-to-C one-pot and sequential ligations

The two segments, (Nle1-Ser15)-α SR (MeSNa thioester) and (Cys16-Ser37)-α NHNH 2 were ligated in a denaturing buffer (6 M guanidine-HCl, 200 mM phosphate, pH 7) within 3 h with a good conversion as evident from analytical HPLC (Fig. 2). Subsequently, (Nle1-Ser15)-(Cys16-Ser37)-α NHNH 2 was converted to α thioester upon treatment with NaNO 2 and in situ reaction with MeSNa furnishing the corresponding (Nle1-Ser15)-(Cys16-Ser37)-α thioester ready for the second NCL with (Cys38-Asp61)-α amide segment. However, we found that the second NCL was complicated by hydrolysis of (Nle1-Ser15)-(Cys16-Ser37)-α thioester. The reaction was continued for extended period of time (10 h) to reach maximum possible conversion. In Fig. 2 ( The next step was the alkylation of the two Cys residues to convert them into pseudo-glutamines (ψ-Gln16 and ψ-Gln38). To achieve this, 1 M bromoacetamide was prepared in 6 M guanidine-HCl, 200 mM phosphate, pH 7 buffer and added to the reaction mixture in three-fold excess over total thiol amount in the solution and kept at 40 °C after adjusting the pH to 6.6. This reaction was fast and was stopped after 10 min since longer reaction times can result in undesirable alkylation of residues other than Cys. Reduction of pH to around 4 after the reaction prevents any undesired alkylation. Purification of the reaction mixture resulted in the final 61-residue TAD-p53 analogue in 13% overall yield (based on the (Nle1-Ser16)-α SR).

Instead of adding MeSNa to convert (Nle1-Ser15)-(Cys16-Ser37)-α azide into the corresponding α thioester, ligation with (Cys38-Asp61)-α amide was also tried with 50 mM or 100 mM MPAA (4-mercaptophenylacetic acid) through in situ thioester generation and subsequent ligation but in each case large amounts of hydrolyzed (Nle1-Ser15)-(Cys16-Ser37)-α OH were observed concomitant with minute amounts of the ligation product with (Cys38-Asp61)-α amide segment (data not shown). It appears that activation in the form of aryl-thioester produces highly reactive peptide-α thioester particularly labile to hydrolysis. In the presence of a different alkyl-thiol such as 2-(dimethylamino)ethanethiol hydrochloride used previously in NCL 32 introduced for transforming (Nle1-Ser15)-(Cys16-Ser37)α azide into a thioester (while maintaining the same concentrations and conditions as of MeSNa) the desired full length TAD-p53 could be obtained in a 10% yield (based on (Nle1-Ser15)-α SR) which is similar to the results with MeSNa thiol (Fig. S5).

The sequential three segment ligation in N-to-C direction rather than one-pot procedure was also attempted. In this case the product of the first ligation (Nle1-Ser15)-(Cys16-Ser37)α NHNH 2 was alkylated using 2-bromoacetamide and converted into the corresponding α thioester followed by HPLC purification. Subsequently, second ligation was performed (see Fig. S6). After Cys alkylation and second HPLC purification, the desired product was obtained, however, with the overall yield (6%) less

C-to-N one-pot ligation approach

To improve the efficiency of the sequential native chemical ligations and reduce the amount of by-products arising from the hydrolysis observed in the N-to-C ligation approach, a reverse Cto-N ligation route was adopted (Fig. 3). In this case the thioesterification of the middle segment was performed separately and unlike in N-to-C ligation the (Thz16-Ser37)-α SR (MeSNa thioester) segment was used as the starting peptide containing Thz (thiazolidine) moiety at the N-terminus instead of Cys, which is necessary to avoid undesired head-to-tail cyclization. Thus, (Thz16-Ser37)-α SR (MeSNa thioester) and (Cys38-Asp61)-α amide were ligated in 6 M guanidine-HCl, 200 mM phosphate buffer at pH 6.8 (Fig. 3). Subsequently, the thiazolidine ring was cleaved by treatment with 0.2 M methoxyamine-HCl at pH 4. After thiazolidine deprotection, second ligation with (Nle1-Ser15)-α thioester segment was performed in the presence of 50 mM MPAA. The MPAA arylthiol catalyst was used to increase the efficiency of the second ligation in the presence of an internal Cys residue. 28 It took 4 h at 40 °C for the reaction to reach near completion indicated by unchanged relative intensity of the analytical HPLC peaks (Fig. 3). Then, 2-bromoacetamide was used for alkylation similar to the procedure described earlier for N-to-C ligations. An overall yield of 26% was obtained after HPLC purification of the final peptide (based on (Cys38-Asp61)-α amide segment).

Circular dichroism study

TAD-p53 is an intrinsically disorder protein domain, i.e. in isolated state it does not have well-defined secondary or tertiary structure. 13 We confirmed that synthetic TAD-p53 is unstructured, which can be seen in its CD spectrum corresponding to a random coil (Fig. 4). Intrinsic disorder plays a key role in the biological function of p53 by providing structural plasticity and enabling this protein to bind to a variety of other proteins. 33 In this way, p53 can serve as a hub in protein-protein interaction networks regulating gene expression. 33 One of such interacting partners is nuclear co-activator binding domain (NCBD) of CREB-binding protein (CBP). We therefore inspected structural changes in synthetic TAD-p53 which occur upon interaction with NCBD by recording the CD spectra of TAD-p53/NCBD complex. We found that the helicity in the complex increases significantly (Fig. 4) compared to free TAD-p53 or NCBD alone judged by the enhancement of ellipticity at 222 nm. The presence of α-helices in the TAD-p53/NCBD complex is in accordance with previously reported NMR structure of the complex. 2 Thus, the CD experiments confirm that the chemically synthesized TAD-p53 retains the characteristics of biologically expressed protein domain by forming a complex with NCBD.

NMR study

In addition, 1D 1 H NMR and 2D 1 H-13 C heteronuclear single quantum coherence (HSQC) spectroscopy were also used to monitor structural changes occurring upon binding of TAD-p53 to NCBD (Fig. 5). Changes upon binding are clearly visible in 1 H NMR corresponding to aromatic protons (spectra were recorded in D 2 O), where sharp resonances are present for free TAD-p53, which is in agreement with intrinsically disordered properties of this protein domain, whereas signals become broader upon addition of NCBD corresponding to a larger size of the complex and different environment of aromatic side-chains (Fig. 5a). The 1 H- 13 C HSQC is also supportive of complex formation: overlay of the methyl regions of the spectra for free TAD-p53 and its complex with NCBD show distinct resonances indicative of different chemical surroundings of the corresponding methyl groups (Fig. 5b).

Conclusions

In this article we reported total chemical synthesis of the (1-61)-form of transactivation domain (TAD) of p53 protein based on native chemical ligation of three peptide segments. The peptide building blocks are relatively short (less than 24 residues) and can be efficiently synthesized using Fmoc/tBu-SPPS approach accessible in many academic laboratories using commercially available building blocks. Two approaches, N-to-C and C-to-N have been used to ligate the peptide segments to achieve the full-length TAD-p53 and it has been found that C-to-N approach results in 2-to 2.5-fold higher yield than N-to-C method. The non-canonical residues introduced in the sequence to facilitate the synthetic protocols such as three norleucines (at sites 1, 40 and 44) and two pseudo-glutamines (at sites 16 and 38) do not affect coupled folding and binding to NCBD which is a known binding partner of p53. The reported synthetic approach based on the ligation of three segments will be particularly advantageous for the combinatorial synthesis of the posttranslationally modified variants of TAD-p53 (e.g. phosphorylated) in order to perform proteome-wide interaction profiling as a function of phosphorylation pattern using protein arrays and other approaches to study protein binding interactions.

M A N U S C R I P T A C C E P T E D ACCEPTED MANUSCRIPT

3 Fig. 1 .

 31 Fig. 1. Schematic representation of the N-to-C and C-to-N peptide ligation strategies for the total chemical synthesis of transactivation domain (TAD) of p53.

Fig. 2 .

 2 Fig. 2. (a) One-pot native chemical ligation towards TAD-p53 in N-to-C direction (using MeSNa for preparing active thioester after first NCL) monitored by analytical HPLC. By-products in panel 6 (from the top) are (1) (Cys38-Asp61)-α amide, (2) (Nle1-Ser15)-(Cys16-Ser37)-α OH, (3) (Nle1-Ser15)-(Cys38-Asp61)α amide. (b) ESI-MS-Orbitrap analysis of the pure TAD p53.Multiple peaks for each charge state are due to Na + and K + ion adducts. A repeating of the synthesis (summarized in Fig.S4) using this procedure with several modifications (see Appendix A) confirms complications in N-to-C strategy.

  6 th panel from the top) the peaks 1, 2, 3 correspond to unreacted (Cys38-Asp61)-α amide, (Nle1-Ser15)-(Cys16-Ser37)-α OH and (Nle1-Ser15)-(Cys38-Asp61)-α amide, respectively, along with the desired (Nle1-Ser15)-(Cys16-Ser37)-(Cys38-Asp61)-α amide product. High quantities of unreacted (Cys38-Asp61)-α amide are due to undesired depletion of (Nle1-Ser15)-(Cys16-Ser37)-α thioester. The by-product 3 arises from the undesired ligation between (Nle1-Ser15)-α SR (remaining from the first ligation) and (Cys38-Asp61)-α amide, which is present in excess.

  what was obtained from one-pot procedure because of losses in the additional HPLC purification.

Fig. 3 .

 3 Fig. 3. (a) One-pot native chemical ligation of three peptide segments in C-to-N direction monitored by analytical HPLC. (b) ESI-MS-Orbitrap analysis of the pure TAD-p53.

2. 3 .

 3 Characterization of binding of chemically synthesized TAD-p53 to nuclear co-activator binding domain(NCBD)

Fig. 4 .

 4 Fig. 4. Circular dichroism (CD) spectra of free TAD-p53, NCBD and their 1:1 complex in 20 mM phosphate buffer at pH 7.4 at 25 °C (concentration of protein = 25 µM).

Fig. 5 .

 5 Fig. 5. (a) Overlay of 1 H NMR (aromatic region) and (b) 1 H-13 C HSQC (methyl region) spectra of free TAD-p53 (in blue) and 1:1 complex with NCBD (in red) recorded in D 2 O.

Acknowledgements

The research was supported by the European Research Council (ERC) under the European Union's Horizon 2020 research and innovation programme (ERC-2016-StG, grant number 715062-HiChemSynPro).

Appendix A. Supplementary data

Detailed experimental procedures and characterization of synthesized compounds can be found as the supplementary data related to this article at https://