Pauline Jacquet

David Daudé

Janek Bzdrenga

Patrick Masson

Mikael Elias

Eric Chabrière

Current and emerging strategies for organophosphate decontamination: special focus on hyperstable enzymes

Keywords: Bioremediation, Phosphotriesterase, Organophosphorus, Pesticide, Chemical warfare agent, SsoPox, Enzyme, Decontamination

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

Introduction

Man-made organophosphorus compounds (OPs) are highlytoxic chemicals that were produced as chemical warfare nerve agents (CWNAs) to harm, kill, or neutralize the opponent in a war strategy. The first massive production of OPs was developed by German scientists before and during the 2nd world war, with several molecules, dubbed agents G (for Germany), such as tabun (in 1936), sarin (in 1937), and soman (in 1944;[START_REF] Szinicz | History of chemical and biological warfare agents[END_REF]). After the 2nd world war, other toxic molecules, dubbed agents V (for Victory, Venomous, or Viscuous), were developed by other nations, such as VX (USA), RVX (Russia), or CVX (China; Fig. 1). Massive stocks of these compounds had been accumulated during the cold war [START_REF] Gupta | Handbook of toxicology of chemical warfare agents[END_REF]. In 1993, the international convention, signed by some 200 countries, called for the arrested development of these chemical weapons, and planned for the destruction of existing stocks before 2007 (Organisation for the Prohibition of Chemical Weapons 2005;Gupta2009). Large stocks, however, still exist today, partly because of the lack of low-cost, rapid, environmental friendly and safe solutions to destroy these chemicals. Additionally, such nerve agents (e.g., sarin) have been used during terrorist attacks in Matsumoto and Tokyo subway (Japan) in 1994 and 1995. Furthermore, organophosphate-based pesticides also constitute a serious threat as they are both highly toxic and widespread and could be fraudulently used for terrorist attacks or asymmetric conflicts.

OP compounds, and particularly V-agent derivatives, were also commercialized as insecticides, during the from any given surface, be it biotic or abiotic. It relies on mechanical action to absorb, remove, or flush the agent away, but does not usually destroy it. (ii) Chemical decontamination: It refers to any given method that allows neutralization of the agent by chemical reactions, such as: hydrolysis, oxidation, or reduction. (iii) Enzymatic decontamination: This method relates to the use of enzymes that are capable of hydrolyzing pesticides or CWNAs.

A possibly ideal decontamination device would use several methods that belong to the aforementioned categories, allowing for a surface cleaning combined with a rapid, efficient neutralization and/or destruction of the contaminating agent, thus, maximizing the effects and reducing the health and environmental hazards.

Large-scale decontamination

Large-scale decontamination is performed on surfaces after a massive OP exposure. The objectives are both to eliminate the contamination in situ as well as to reduce the potential of cross contamination, for example between the victim and the treating personnel. Currently available techniques are well suited for material decontamination; however, they often are aggressive and lack the ability to be fully safe for treating the victims. The solutions presented here are a non-exhaustive panel of available solutions.

-Hypochlorite Decontamination with a sodium hypochlorite solution may be used at the concentration of 0.5 % for personnel and 5 % for equipment [START_REF] Tuorinsky | Triage of chemical casualties[END_REF]. This solution proves to be useful both at removing the nerve agent from the surface and neutralizing it by oxidation. Be it for material and personnel, this method is still quite aggressive, due to its strong corroding potential towards eyes, skin, open wounds, and materials.

-Sodium hydroxide Sodium hydroxide is used to perform alkaline hydrolysis of OPs by a nucleophilic attack of the hydroxide ion on the phosphorus atom of OPs. It allows for the degradation of large amounts of nerve agents into their corresponding, non-toxic phosphonic acid. Combination of both aforementioned products, called super tropical bleach with 93 % calcium hypochlorite and 7 % sodium hydroxide, is used with high efficiency for material decontamination, with hypochlorite ion acting as a catalyst in the hydrolysis reaction [START_REF] Singh | Decontamination of chemical warfare agents (review article)[END_REF]). However, this method is highly corrosive and may neither be used for sensitive material nor personnel decontamination.

-DS2 DS2 or BDecontamination solution 2^is used for material decontamination. It contains 70 % diethylenetriamine (DETA), 28 % 2-methoxyethanol, and 2 % sodium hydroxide. The active ingredient, 2-methoxyethanol, is highly effective against sulfur mustard and acts by elimination reaction, thus, forming divinyl sulfide. The mode of action against CWNAs is somewhat different as 2methoxyethanol acts as a nucleophile and reacts with Gand V-agents [START_REF] Singh | Decontamination of chemical warfare agents (review article)[END_REF]). However, it is toxic and aggressive to treated surfaces as it even tends to degrade paints [START_REF] Fielding | Field decontamination studies with chemical warfare decontaminating solution DS2[END_REF][START_REF] Firmin | The future of decontamination operations-an analysis of decontamination foam 200[END_REF].

-BX-24 and BX-29

These decontamination solutions, used in NATO forces, act by oxidation and hydrolysis. Personnel decontamination solution is composed of amphoteric compounds, amides, and a surfactant. It is advertised to be a safe decontamination solution. The BX-24 is a non-corrosive product designed for material decontamination.

-Decontamination formulation This product is used as foam in order to decontaminate toxic chemical warfare material. An enhancement over the previous DF-100 product, DF-200 is composed of nucleophilic compounds and oxidizing products, a bleaching activator, a sorbent, and an inorganic base, dissolved in water. It is in three distinct parts to be mixed together prior to use, with part one being a proprietary mix of quaternary ammonium compounds and surfactants, part two is composed of 8 % hydrogen peroxide, and part three is composed of glycerol diacetate, which catalyzes the oxidation process [START_REF] Tucker | Decontamination formulation with sorbent additive[END_REF][START_REF] Tucker | Decontamination formulations for disinfection and sterilization[END_REF][START_REF] Tucker | Reduced weight decontamination formulation for neutralization of chemical and biological warfare agents[END_REF]). It is active against G-and V-agents with a 99 % decontamination of the treated surface after 30 min [START_REF] Tucker | Reduced weight decontamination formulation for neutralization of chemical and biological warfare agents[END_REF]. The fact that this product is divided into three parts might cause logistical difficulties on the onset of a massive event.

Personal-scale decontamination

The kinetics of nerve agent penetration in the body is so fast that decontamination and treatment have to be performed on site and as soon as possible, at the individual level. Indeed, transportation time to proper medical facilities without prior application of topical solutions would greatly reduce the chances of a patient's survival and would threaten medical personnel by spreading secondary contaminations. Currently available methods dealing with nerve agent exposure rely on external decontamination followed by medical treatment of the personnel exposed.

Ready-to-use kits for personal decontamination are primarily used by armies around the world. They have to comply with operational requirements and find the best trade-off in terms of size, weight, easiness of use, and of course efficiency.

-M291 SDK (Skin Decontamination Kit) Adopted by US Army in 1989, the M291 SDK is a readyto-use, non-woven fiber pad, filled with an absorbent and a resin. The absorbent is a high-contact surface carbonaceous compound, in order to remove the agent from the skin. The resin itself is composed of two ion exchange resins, an anionic and a cationic one, able to neutralize the contaminant through hydrolysis [START_REF] Khan | Recent advances in decontamination of chemical warfare agents[END_REF]. This technology was in use by the US Army until its progressive replacement by the Reactive Skin Decontamination Lotion (RSDL) around 2007. Its efficacy is very low on VX and very limited against soman as compared to other decontamination methods, such as 1 % soapy water and 0.5 % bleach (Braue Jr. et al. 2010a, b). It is not suitable for eye and wound decontamination.

-RSDL (Reactive Skin Decontamination Lotion)

The RSDL is the current solution provided to soldiers and civilian security personnel in order to deal with topical exposure to OPs. It consists of an individually packed sponge, impregnated with a solution containing Diacetylmonoxime (DAM) and Dekon 139-ap r o p r i etary mix-dissolved in polyethylene glycol monomethyl ether [START_REF] Bannard | Broad spectrum chemical decontaminant system[END_REF]. This formulation allows for desorption of the agent from the skin and its chemical degradation via a nucleophilic reaction. The time required to achieve decontamination is less than 3 min for G-and V-agents [START_REF] Elsinghorst | Detoxification of organophosphorus pesticides and nerve agents through RSDL: efficacy evaluation by 31P NMR spectroscopy[END_REF]. Its efficacy outperforms the one of M291 towards agents such as VX and soman (Braue Jr. et al. 2010a, b). The protective ratio (defined as the ratio between the LD 50 of treated versus untreated group) of RSDL on guinea pigs is about 60 times higher than the one of M291 against VX (66.4 and 1.1, respectively;Braue Jr. et al. 2010a). Regarding soman, the protective ratio of RSDL still is five times better than the one of M291 (14 and 2.7, respectively;Braue Jr. et al. 2010b). The RSDL is efficient at decontaminating the skin as long as it is applied within minutes post-exposure. Like the M291 SDK, RSDL is not compatible with eye and wound cleaning.

-Fuller'searth Used by several NATO countries, pads impregnated with Fuller's earth purely rely on a physical and passive absorbance of the contaminating OP. It is mainly composed of fine aluminum silicate powder that confers a high-surface area, allowing for a good topical absorbance of contaminants [START_REF] Taysse | Skin decontamination of mustards and organophosphates: comparative efficiency of RSDL and Fuller's earth in domestic swine[END_REF]). The major drawback of using such a pad comes from the generation of contaminated dusts presenting an inhalation hazard. There is a current lack of non-corrosive solutions that would prove to be adequate both for sensitive material and personnel decontamination. Indeed, oxidative and corrosive solutions for decontamination are harmful and destructive to most military and civilian equipment and basic supplies. They may eliminate the threat of CWNAs but will leave behind a chemical contamination that is disastrous. Moreover, it should be noted that personnel decontamination using surfactant containing solutions (i.e., soapy water) represents a risk of spreading the agent over a wider skin surface. Even though it helps solubilize the chemicals, it may negatively affect the very aim of decontamination. It is, thus, of uttermost importance to opt for methods enabling sequestration and/or swift destruction of said nerve agents.

Stockpile destruction

In order to comply the Chemical Weapon Convention, the means of disposal were studied that could lead to destruction of stockpiles without harmful effect on environment and associated health hazards, as previous dumping at sea and burying method were problematic (US National Research Council 1996). It encompasses large hydrolysis systems and incineration used by western countries. Russia uses another method for destruction of G-agents, relying on neutralization followed by bitumination [START_REF] Pearson | Critical evaluation of proven chemical weapon destruction technologies (IUPAC Technical Report)[END_REF].

-Incineration

The destruction by two-stage incineration is a process in use since the 1980s for the destruction of CWNA ammunitions, bulk agents, effluents, and equipment associated with stockpile disposal (US National Research Council 1994). First stage allows for the pulverization of the liquid agents and their heating to 1,480 °C. The subsequent gases serve as a fuel for processing by the second stage, an afterburner maintained at a lower, 1,100 °C temperature [START_REF] English | Design aspects of a low emission, two-stage incinerator[END_REF][START_REF] Pearson | Critical evaluation of proven chemical weapon destruction technologies (IUPAC Technical Report)[END_REF]. The incinerator is equipped with a pollution abatement system that mixes the gases with sodium hydroxide and water to remove acid gases and particles to reach safe standards [START_REF] Pearson | Critical evaluation of proven chemical weapon destruction technologies (IUPAC Technical Report)[END_REF].

-Hydrolysis OPs are hydrolyzed using sodium hydroxide. However, the way hydrolysis is performed depends on the nature of the agent. While G-agents can be hydrolyzed safely at ambient temperature, VX hydrolysis has to be performed at 90 °C, temperature at which P-Shydrolysisisfavored over P-Ob o n d( Y a n g1999;P e a r s o na n dM a g e e2002; [START_REF] Kim | Destruction and detection of chemical warfare agents[END_REF]).

-Neutralization-bitumination Mainly used by Russia, neutralization-bitumination is a two-stage method that allows for the destruction of Gnerve agents. CWNAs are first mixed with 80 % monoethanolamine for 1 h at 110 °C (neutralization), followed by a reaction with bitumen and calcium hydroxide for 1 h at 200 °C (bitumination). Russian VX (RVX) is disposed of by a reaction with potassium isobutylate dissolved in isobutanol and N-methylpyrrolidinone. This solution is heated at 95 °C for 30 min before being added in bitumen [START_REF] Pearson | Critical evaluation of proven chemical weapon destruction technologies (IUPAC Technical Report)[END_REF].

Emerging chemical and physical alternatives for OPs decontamination

Cyclodextrins (CDs)

CDs are cyclic oligosaccharides composed of six to eight Dglucopyranoside units. They form a hydrophobic cavity able to host hydrophobic and apolar molecules [START_REF] Sambrook | Supramolecular chemistry and chemical warfare agents: from fundamentals of recognition to catalysis and sensing[END_REF]. This biomimetic catalysis capacity allows CDs to scavenge OPs.

Native CDs (α-CD, β-CD, and γ-CD) were first demonstrated to inactivate CWNAs. Among these, β-CD was shown to be the most efficient [START_REF] Defrank | Screening of halophilic bacteria and Alteromonas species for organophosphorus hydrolyzing enzyme activity[END_REF][START_REF] Désiré | Inactivation of sarin and soman by cyclodextrins in vitro[END_REF]. Then, starting from β-CD as a precursor, substituted CDs such as iodosobenzoate-β-CD (IBA-β-CD) were developed and showed the ability to hydrolyze pesticides and CWNAs [START_REF] Hoskin | Substituted cyclodextrin as a model for a squid enzyme that hydrolyzes the nerve gas soman[END_REF]. The importance of the presence of an α-nucleophile such as IBA was underlined [START_REF] Masurier | Synthesis of 2substituted β-cyclodextrin derivatives with a hydrolytic activity against the organophosphorylester paraoxon[END_REF][START_REF] Wille | Detoxification of nerve agents by a substituted β-cyclodextrin: application of a modified biological assay[END_REF]M ü l l e r et al. 2011;[START_REF] Rougier | Effect of cyclodextrins on the reactivity of fenitrothion[END_REF]. Other derivatives were synthesized such as O-benzyl-β-CD, fully methylated β and α-CD (TRIMEB and TRIMEA, respectively), partially methylated β-CD (DIMEB), depicting a better solubility in water and organic solvents. CDs are promising compounds which could be used for skin, mucosa, wound, and also for material decontamination [START_REF] Letort | The first 2 IB,3 IAheterodifunctionalized β-cyclodextrin derivatives as artificial enzymes[END_REF]. Notwithstanding these benefits, some limitations have to be pointed out that circumvent the use of CDs, as a high-molecular weight (1, 134 g.mol -1 for β-CD) and a low turnover. Additionally, none of the current CDs are able to degrade VX and possibly any OP containing a P-S bond due to the remarkable stability of such a linkage [START_REF] Wille | Detoxification of nerve agents by a substituted β-cyclodextrin: application of a modified biological assay[END_REF][START_REF] Kalakuntla | New modified βcyclodextrin derivatives as detoxifying agents of chemical warfare agents (I). Synthesis and preliminary screening: Evaluation of the detoxification using a half-quantitative enzymatic assay[END_REF]). Furthermore, p-nitrophenol (degradation product of paraoxon) inhibits the degradation process efficiency of paraoxon by β-CD [START_REF] Masurier | Synthesis of 2substituted β-cyclodextrin derivatives with a hydrolytic activity against the organophosphorylester paraoxon[END_REF];E s t o u re ta l .2013). Presumably, other OPs degradation products could lead to the same effect. Moreover, CDs rate for P-F degradation is very low [START_REF] Hoskin | Substituted cyclodextrin as a model for a squid enzyme that hydrolyzes the nerve gas soman[END_REF].

Photochemical methods

Photochemical methods are based on light radiation to degrade OPs and organic substances in general. These methodologies have been well-reviewed elsewhere by [START_REF] Reddy | A review of photochemical approaches for the treatment of a wide range of pesticides[END_REF]. Photochemical methods are part of advanced oxidation processes (AOPs), which use the high reactivity of HO radicals to degrade pollutants [START_REF] Hossain | Degradation of chlorpyrifos, an organophosphorus insecticide in aqueous solution with gamma irradiation and natural sunlight[END_REF]. Briefly, photochemical-based techniques include five classes used for OPs degradation in soil, water, or air:

-Photolysis Photolytic degradation of targeted compound consists in the absorption of radiation in both direct and indirect ways, leading to the destruction of the compound. Several examples of OP photolysis (diazinon, methylparathion, or quinalphos) have been reported to date in both direct [START_REF] Wan | Comparative study on the quantum yields of direct photolysis of organophosphorus pesticides in aqueous solution[END_REF] [START_REF] Lam | PhotoFate: a new approach in accounting for the contribution of indirect photolysis of pesticides and pharmaceuticals in surface waters[END_REF]Garbinetal.2007).

-Photolysis with an oxidant

The association of classical photolysis with chemical oxidants like H 2 O 2 and O 3 enhances the efficiency of the process and seems to prevent formation of unfavorable products [START_REF] Wu | Phototransformation of selected organophosphorus pesticides: roles of hydroxyl and carbonate radicals[END_REF].

-Photo-Fenton method This method involves photo-Fenton process, relying on the photoreduction of dissolved Fe(III) complexes into Fe(II) ions, followed by the Fenton reaction and the oxidation of organic compounds [START_REF] Kim | Degradation of organic pollutants by the photo-fenton-process[END_REF][START_REF] Ikehata | Aqueous pesticide degradation by hydrogen peroxide/ultraviolet irradiation and Fenton-type advanced oxidation processes: a review[END_REF].

-Photocatalysis It refers to the photo-excitation of a semiconductor such as titanium dioxide (TiO 2 ), zinc oxide (ZnO), or tungsten trioxide (WO 3 ). Semiconductors are intermediates to reduction or oxidation of the OPs. This technique shows certain limits like electronic recombination [START_REF] Balkaya | A study of optimal experimental conditions in the photocatalytic degradation of an organophosphorous insecticide[END_REF][START_REF] Konstantinou | Photocatalytic degradation of selected s-Triazine herbicides and organophosphorus insecticides over aqueous TiO2 suspensions[END_REF];J o n i d i -J a f a r ie ta l . 2004;L ie ta l .2005; [START_REF] Evgenidou | Study of the removal of dichlorvos and dimethoate in a titanium dioxide mediated photocatalytic process through the examination of intermediates and the reaction mechanism[END_REF]Ch enetal . 2011).

-Photosensitized-induced process This process uses sensitizers able to absorb light radiation, then transfering the excess energy to a targeted compound. This technique is useful for OPs with a low absorption efficiency [START_REF] Kamiya | Effects of selected metal ions on photodegradation of organophosphorus pesticides sensitized by humic acids[END_REF][START_REF] Nowakowska | Photosensitized degradation of ethyl parathion pesticide in aqueous solution of anthracene modified photoactive dextran[END_REF]). Photochemical methods have been mainly used for water decontamination [START_REF] Reddy | A review of photochemical approaches for the treatment of a wide range of pesticides[END_REF]. A few studies on air and solid surface decontamination of CWNAs have been reported to date [START_REF] Zuo | Study on photolytic and photocatalytic decontamination of air polluted by chemical warfare agents (CWAs)[END_REF]K i me ta l .2007). Photochemical methods are easy to handle and rather inexpensive. However, the efficiency of those methods depends on numerous variables such as effluent flow rate, types, and concentrations of contaminants (like inorganic ions).

Others

Other techniques for OP decontamination exist. These methods remain marginal and would require further development. Hereafter is a non-exhaustive list of these techniques.

-Metal-Organic Framework (MOF) MOFs are crystalline materials, composed of metal ions or clusters linked together by polydentate organic linkers. Based on metal nodes, MOF could be tailored to work as an artificial enzyme [START_REF] Lee | Metal-organic framework materials as catalysts[END_REF]. For example, it could be used for CWNAs degradation through hydrolysis by adsorbed water using Cu-BTC (HKUST-1, [START_REF] Decoste | Metal-organic frameworks for air purification of toxic chemicals[END_REF][START_REF] Mondloch | Destruction of chemical warfare agents using metal-organic frameworks[END_REF].

Cu 3 (C 9 H 3 O 6 ) 2 )o rN U -1 0 0 0( Z r 6 ( μ 3 -O) 4 (μ 3 - OH) 4 (H 2 O) 4 (OH) 4 ) (DeCoste
-Degradation by non-thermal plasma (NTP) NTP (also called dielectric barrier discharge plasma) is able to destroy a broad spectrum of OPs as well as other chemicals and biological pathogens [START_REF] Kim | Decomposition reaction of organophosphorus nerve agents on solid surfaces with atmospheric radio frequency plasma generated gaseous species[END_REF]. When OPs are exposed to plasma, their chemical bonds are effectively broken; subsequently, generated products seem harmless [START_REF] Bai | Degradation of organophosphorus pesticide induced by oxygen plasma: Effects of operating parameters and reaction mechanisms[END_REF]). This technique is mainly used for wastewater decontamination [START_REF] Hu | Degradation of selected organophosphate pesticides in wastewater by dielectric barrier discharge plasma[END_REF]).

-Gamma irradiation process Gamma irradiation process is used for degradation of numerous pollutants. It is also part of AOPs such as photochemical methods. Gamma irradiation is used to degrade OPs such as chlorpyrifos, diazinon, or malathion in water. This technique is mainly used for water decontamination [START_REF] Basfar | Degradation of diazinon contaminated waters by ionizing radiation[END_REF][START_REF] Mohamed | Radiolytic degradation of malathion and lindane in aqueous solutions[END_REF][START_REF] Hossain | Degradation of chlorpyrifos, an organophosphorus insecticide in aqueous solution with gamma irradiation and natural sunlight[END_REF].

Organophosphate neutralizing enzymes Organophosphate hydrolase (OPH)

OPHs belong to the aryltriphosphate dialkylphosphohydrolase (EC 3.1.8.1) and are part of the amidohydrolase superfamily [START_REF] Schomburg | Aryldialkylphosphatase[END_REF][START_REF] Seibert | Structural and catalytic diversity within the amidohydrolase superfamily †[END_REF]. They are encoded by the OP degradation (opd) genes and have been initially found in Brevundimonas diminuta MG (previously Pseudomonas diminuta) and Sphingobium fuliginis ATCC 27551 (previously Flavobacterium sp.; [START_REF] Sethunathan | A Flavobacterium sp. that degrades diazinon and parathion[END_REF];S e r d a re ta l .1982; Mulbry et al. 1986;H a r p e r et al. 1988). Another enzyme, encoded by a gene closely related to opd,n a m e l yopdA, was isolated from Agrobacterium radiobacter P230 [START_REF] Horne | Identification of an opd (organophosphate degradation) gene in an Agrobacterium isolate[END_REF][START_REF] Horne | The phosphotriesterase gene opdA in Agrobacterium radiobacter P230 is transposable[END_REF]). Extensive efforts have been devoted to the characterization and engineering of these enzymes (Theriot and Grunden 2010;[START_REF] Wales | Organophosphorus hydrolase as an in vivo catalytic nerve agent bioscavenger[END_REF]. Relevant works are summarized hereafter.

-OPH from Brevundimonas diminuta B. diminuta carrying the plasmid pCMS1 was found to hydrolyze the pesticide parathion [START_REF] Serdar | Plasmid involvement in parathion hydrolysis by Pseudomonas diminuta[END_REF]. From this plasmid, the gene opd coding for a 35-kDa phosphotriesterase was cloned and sequenced [START_REF] Mcdaniel | Cloning and sequencing of a plasmid-borne gene (opd) encoding a phosphotriesterase[END_REF]). This enzyme was purified and characterized against a wide panel of OP-based insecticides [START_REF] Dumas | Purification and properties of the phosphotriesterase from Pseudomonas diminuta[END_REF]). The catalytic efficiency (k cat / K M ) against paraoxon was found to be particularly high: 4.0 × 10 7 M -1 s -1 .Efficacyagainstsarinandsomanwasfurther highlighted, albeit with lower k cat /K M values as compared to paraoxon, 8.0 × 10 4 M -1 s -1 and9.3x10 3 M -1 s -1 , respectively [START_REF] Dumas | Crystallization and preliminary X-ray diffraction analysis of the hyperthermophilic Sulfolobus solfataricus phosphotriesterase[END_REF]). The 3D-structure of the apoenzyme was shown to consist of an (β/α) 8 fold [START_REF] Benning | Threedimensional structure of phosphotriesterase: an enzyme capable of detoxifying organophosphate nerve agents[END_REF]. The holoenzyme reconstituted with cadmium and a complex with the substrate analog diethyl 4-methylbenzylphosphate were also obtained [START_REF] Benning | Threedimensional structure of the binuclear metal center of phosphotriesterase[END_REF][START_REF] Vanhooke | Threedimensional structure of the zinc-containing phosphotriesterase with the bound substrate analog diethyl 4-methylbenzylphosphonate[END_REF]. Protein engineering strategies were further considered to increase OPH activity [START_REF] Iyer | Protein engineering of representative hydrolytic enzymes for remediation of organophosphates[END_REF]. Variants at positions W131, F132, and F306 with enhanced activity against diisopropylfluorophosphate (DFP) were obtained demonstrating for the first time that mutagenesis was relevant to increase the hydrolysis of P-F linkages [START_REF] Watkins | Augmented hydrolysis of diisopropyl fluorophosphate in engineered mutants of phosphotriesterase[END_REF]. Three hydrophobic binding pockets were identified in the structure and 14 residues forming these subsites were targeted for mutagenesis purposes (Chen-Goodspeed et al. 2001a).TheroleofG60residue on stereoselectivity was emphasized as G60A substitution drastically altered R P -enantiomers recognition leading to a 13000-fold greater k cat /K M in favor of methyl phenyl p-nitrophenyl phosphate S P -c o m p a r e dt oR P -enantiomer. Substitution H257Y was then associated with a reversal of chiral specificity with an increase by up to 500-fold regarding R P -enantiomers for a variant harboring 4 substitutions I106G/F132G/H257Y/S308G (Chen-Goodspeed et al. 2001b). DNA shuffling coupled with a bacterial cell surface display based screening, led to the isolation of variant 22A11 (A14T/A80V/K185R/H257Y/I274N) and a 25-fold increase in methylparathion hydrolysis [START_REF] Cho | Bacterial cell surface display of organophosphorus hydrolase for selective screening of improved hydrolysis of organophosphate nerve agents[END_REF]. A similar strategy led to the variant B3561 (A14T/ L17P/A80V/V116I/K185R/A203T/I274N/P342S) that exhibited a 725-fold increase in k cat /K M for chlorpyrifos and a 39-fold improvement in paraoxon hydrolysis [START_REF] Cho | Altering the substrate specificity of organophosphorus hydrolase for enhanced hydrolysis of chlorpyrifos[END_REF]). K185R and I274N substitutions were shown to increase overall hydrolysis rate by being involved in the formation of hydrogen bonds with surrounding residues leading to beneficial structural changes [START_REF] Cho | Functional analysis of organophosphorus hydrolase variants with high degradation activity towards organophosphate pesticides[END_REF]. Directed evolution of the phosphotriesterase from B. diminuta was also investigated and resulted in the selection of variant S5 (K185R/D208G/R319S) with a 20-fold enhancement in functional expression in E. coli [START_REF] Roodveldt | Directed evolution of phosphotriesterase from Pseudomonas diminuta for heterologous expression in Escherichia coli results in stabilization of the metal-free state[END_REF]. The degradation of V-type nerve agents was recently investigated through computationally focused library screening and two-site mutagenesis [START_REF] Cherny | Engineering V-type nerve agents detoxifying enzymes using computationally focused libraries[END_REF][START_REF] Bigley | Enzymatic neutralization of the chemical warfare agent VX: evolution of phosphotriesterase for phosphorothiolate hydrolysis[END_REF]. The first study led to various improved variants, among these C23 (K77A/A80V/F132E/T173N/G208D/ H254G/I274N) was particularly relevant with a k cat /K M value for the toxic S p isomer of VX of 8.3 x 10 4 M -1 s -1 [START_REF] Cherny | Engineering V-type nerve agents detoxifying enzymes using computationally focused libraries[END_REF]). In the second study, 12 active-site residues were targeted and led to a 26-fold improvement for the hydrolysis of the VX analogue DEVX as compared to the wild-type protein and a 230-fold improvement for the racemic nerve agent VX with catalytic efficiency up to 7 x 10 4 M -1 s -1 [START_REF] Bigley | Enzymatic neutralization of the chemical warfare agent VX: evolution of phosphotriesterase for phosphorothiolate hydrolysis[END_REF]. Very recently, PTE was engineered into a functional trimer showing enhanced activity when displayed onto semiconductor quantumdot [START_REF] Jr | Evaluation of RSDL, M291 SDK, 0.5% bleach, 1% soapy water and SERPACWA. Part 2. Challenge with soman[END_REF]. Altogether, these results underline that phosphotriesterase from B. diminuta is a good candidate for catalytic decontamination of organophosphorus chemicals that can be enhanced with enzyme engineering strategies.

-OPH from Sphingobium fuliginis. S. fuliginis (ATCC 27551) was isolated from paddy water and shown to degrade both diazinon and parathion [START_REF] Sethunathan | A Flavobacterium sp. that degrades diazinon and parathion[END_REF]). An enzymatic extract partially purified from S. fuliginis was further shown to possess a phosphotriesterase activity for the degradation of various OPs [START_REF] Brown | Phosphotriesterases of flavobacterium sp[END_REF]. The production of parathion hydrolase in S. fuliginis strain ATCC 2751 was associated to a 43-kb plasmid harboring a gene closely related with the opd sequence from B. diminuta (Mulbry et al. 1986), the encoded proteins sharing 99 % sequence identity. The recombinant expression of OPH from S. fuliginis was investigated [START_REF] Rowland | Purification and characterization of a secreted recombinant phosphotriesterase (parathion hydrolase) from Streptomyces lividans[END_REF]. Streptomyces lividans was targeted for this purpose as it expresses the enzyme as a secreted soluble enzyme at a milligram scale [START_REF] Steiert | A gene coding for a membrane-bound hydrolase is expressed as a secreted, soluble enzyme in Streptomyces lividans[END_REF]). The production yield was further optimized by investigating the effect of signal sequences [START_REF] Rowland | The effect of signal sequences on the efficiency of secretion of a heterologous phosphotriesterase by Streptomyces lividans[END_REF]). Preparation of OP-decontaminating solutions were considered and the enzyme was found to be stable for long term storage after freeze-drying in the presence of trehalose and its activity was enhanced with 0.01% sodium carboxyl polyoxyethylene tridecylether [START_REF] Sode | Compatibility of phosphotriesterase from Flavobacterium sp. with detergents[END_REF]. The influence of cobalt and zinc on both activity and expression of the enzyme was evaluated using opd-lacZ fusions [START_REF] Manavathi | Influence of zinc and cobalt on expression and activity of parathion hydrolase from Flavobacterium sp. ATCC27551[END_REF].

The capacity of the native enzyme to discriminate enantiomers was considered using various OPs and found to be limited [START_REF] Ohuchi | An optical resolution of racemic organophosphorous esters by phosphotriesterase-catalyzing hydrolysis[END_REF]. Protein engineering was thus considered to enhance its efficiency [START_REF] Iyer | Protein engineering of representative hydrolytic enzymes for remediation of organophosphates[END_REF]. Variant H254G/H257W/L303T was able to catalyze the hydrolysis of a chromogenic analogue of the most toxic stereoisomer of soman nearly 3 orders of magnitude faster than the native enzyme [START_REF] Hill | Enhanced degradation of chemical warfare agents through molecular engineering of the phosphotriesterase active site[END_REF]. Further mutagenesis enhanced the hydrolysis of VX and pesticides [START_REF] Gopal | Mutagenesis of organophosphorus hydrolase to enhance hydrolysis of the nerve agent VX[END_REF]. Variant L136Y displayed a 33% increase in VX hydrolysis rate. Rate hydrolysis of mutants W131F and F132Y was enhanced by 4fold and 5.8-fold with demeton-S methyl and DFP respectively. Further optimization lead to 25-fold increased in k cat /K M compared to the wild-type with demeton-S methyl for variant with nine mutations (A80V/I106V/F132D/K185R/D208G/ H257W/I274N/S308L/R319S) and up to 18-fold increase against malathion for variant harboring 10 amino-acid substitutions (G60V/A80V/I106V/F132D/K185R/D208G/H257W/ I274N/F306V/R319S) [START_REF] Schofield | Generation of a mutagenized organophosphorus hydrolase for the biodegradation of the organophosphate pesticides malathion and demeton-S[END_REF]. Protein engineering was also considered for enhancing CWNAs degradation. Position 254 was proved to be crucial for VX hydrolysis [START_REF] Nakayama | Improvement of organophosphorus hydrolase activity toward nerve agents by amino acid substitutions[END_REF], and catalytic efficiency of variant L271/Y309A was enhanced up to 150-fold compared to the wild-type enzyme [START_REF] Jeong | Rational design of organophosphorus hydrolase with high catalytic efficiency for detoxifying a V-type nerve agent[END_REF]).

-OPH from Agrobacterium radiobacter (OpdA)

The bacterial strain Agrobacterium radiobacter P230 was found capable of hydrolyzing OP-based pesticides. The gene, namely opdA, involved in OP hydrolysis was thus cloned and sequenced and was found closely related to the opd gene from Flavobacterium sp. strain ATCC 27551 [START_REF] Horne | Identification of an opd (organophosphate degradation) gene in an Agrobacterium isolate[END_REF].

OpdA shares 91% identity with OPH from Flavobacterium sp. strain ATCC 27551. The structure of OpdA was solved and complexes with ethylene glycol and dimethyl thiophosphate revealed similarities between OpdA mechanism and other previously reported metallophosphoesterases [START_REF] Jackson | The structure of an enzymeproduct complex reveals the critical role of a terminal hydroxide nucleophile in the bacterial phosphotriesterase mechanism[END_REF]. This enzyme showed 10-fold enhancement compared to homologous phosphotriesterase from Flavobacterium sp. strain ATCC 27551 for the degradation of dimethyl organophosphate based insecticides. Substitutions Y257H and F272L were mainly responsible for this difference [START_REF] Horne | Functional effects of amino acid substitutions within the large binding pocket of the phosphotriesterase OpdA from Agrobacterium sp. P230[END_REF].

Protein engineering was used to enhance the degradation of OP-based pesticides. Rational design enabled to identify two successive residues, namely F131 and W132, involved in stereospecificity. Variant W131H/F132A displayed 480-fold and 8-fold enhancement in catalytic efficiency for stereoisomers Z and E of chlorfenvinphos respectively [START_REF] Jackson | Structure-based rational design of a phosphotriesterase[END_REF]). Combinatorial active site mutagenesis was also applied to OpdA and led to the selection of variant S308L/Y309A resulting in a 5000-fold increase in k cat /K M for the hydrolysis of malathion [START_REF] Naqvi | A 5000-fold increase in the specificity of a bacterial phosphotriesterase for malathion through combinatorial active site mutagenesis[END_REF].

The potential of OpdA for degrading CWNAs was also considered. The hydrolysis of both G-and V-type nerve agents by OpdA was investigated. The enzyme hydrolyzed cyclosarin and soman with catalytic efficiencies of 3.9 x 10 2 M -1 s -1 and 5.8 x 10 2 M -1 s -1 , respectively. The activities towards V-agents were lower by two orders of magnitude when compared to agents from G-series. Regarding VX, a k cat /K M value of 4.5 M -1 s -1 was measured [START_REF] Wille | Detoxification of G-and V-series nerve agents by the phosphotriesterase OpdA[END_REF]. Three OpdA variants were investigated and shown k cat /K M values up to 1.7 x 10 5 M -1 s -1 for tabun. Conversely, soman was poorly hydrolyzed [START_REF] Dawson | Degradation of nerve agents by an organophosphate-degrading agent (OpdA)[END_REF].

-Examples of applications of OPH for OP decontamination Insofar as OPH are efficient biocatalysts for OP hydrolysis, they have been considered for decontamination purposes. In order to promote the use of these enzymes and to address their stability limitations, immobilization strategies were envisaged. Trityl agarose was used for the immobilization of a phosphotriesterase in a fixed bed reactor maintaining its ability to hydrolyze organophosphate pesticides (Caldwell and Raushel 1991a). The thermostability of the immobilized enzyme was moreover enhanced compared to the free form with a 6-fold increase in half-life at 55°C. A process for the immobilization onto nylon was also described (Caldwell and Raushel 1991b). The K M was increased by 5-6 times compared to the soluble enzyme but reduced the maximum rate against paraoxon to less than 10%. Long term storage was also enhanced for the immobilized enzyme. The enzyme was immobilized into a decontaminating sponge [START_REF] Havens | Reusable immobilized enzyme/polyurethane sponge for removal and detoxification of localized organophosphate pesticide spills[END_REF], and covalently linked within polyurethane foams [START_REF] Lejeune | Dramatically stabilized phosphotriesterase-polymers for nerve agent degradation[END_REF]. The stability of the enzyme was tremendously enhanced and enzymatic foams became attractive for large scale nerve agents decontamination [START_REF] Lejeune | Dramatically stabilized phosphotriesterase-polymers for nerve agent degradation[END_REF][START_REF] Lejeune | Dramatically stabilized phosphotriesterase-polymers for nerve agent degradation[END_REF][START_REF] Lejeune | Dramatically stabilized phosphotriesterase-polymers for nerve agent degradation[END_REF]. However, limitations of the technology were due to insufficient activity of OPH for the most toxic organophosphorus chemicals [START_REF] Donarski | Structure-activity relationships in the hydrolysis of substrates by the phosphotriesterase from Pseudomonas diminuta[END_REF]). Surfaceexpression coupled with immobilization was further considered to develop low-cost decontamination systems [START_REF] Richins | Biodegradation of organophosphorus pesticides by surface-expressed organophosphorus hydrolase[END_REF]Ch enetal .2000;[START_REF] Richins | Expression, immobilization, and enzymatic characterization of cellulose-binding domainorganophosphorus hydrolase fusion enzymes[END_REF][START_REF] Cho | Bacterial cell surface display of organophosphorus hydrolase for selective screening of improved hydrolysis of organophosphate nerve agents[END_REF]Manseeetal.2005). Recently, covalent immobilization of OpdA onto polyester textiles was reported offering various perspectives for environmental decontamination [START_REF] Gao | Bioremediation of pesticide contaminated water using an organophosphate degrading enzyme immobilized on nonwoven polyester textiles[END_REF]. Other examples of OPH immobilization on amyloid fibril nanoscaffold or amphiphilic block copolymer were also described [START_REF] Raynes | Immobilization of organophosphate hydrolase on an amyloid fibril nanoscaffold: towards bioremediation and chemical detoxification[END_REF][START_REF] Suthiwangcharoen | Enhancing enzyme stability by construction of polymer-enzyme conjugate micelles for decontamination of organophosphate agents[END_REF] as well as an original approach consisting in the immobilization of OPH onto carbon nanotube paper [START_REF] Mechrez | Biocatalytic carbon nanotube paper: a Bone-pot^route for fabrication of enzymeimmobilized membranes for organophosphate bioremediation[END_REF].

DFPase from Loligo vulgaris

A special attention was raised to the Diisopropylfluorophosphate fluorohydrolase (DFPase, EC 3.1.8.2) from Loligo vulgaris.This squid-type enzyme was initally found to efficiently hydrolyze DFP and tabun [START_REF] Hoskin | Diisopropylphosphorofluoridate and tabun: enzymatic hydrolysis and nerve function[END_REF][START_REF] Hoskin | Hydrolysis of nerve gas by squid-type diisopropyl phosphorofluoridate hydrolyzing enzyme on agarose resin[END_REF]. Its catalytic mechanism was considered and the 3D-structure was solved and described as a 6-fold β propeller (Scharff et al. 2001a;Scharff et al. 2001b). Kinetics studies and site-directed mutagenesis were performed to decipher the reaction mechanism of the enzyme and identify the essential residues in the active site (Hartleib and Rüterjans 2001a;Scharff et al. 2001a;Scharff et al. 2001b;[START_REF] Katsemi | Mutational and structural studies of the diisopropylfluorophosphatase from Loligo vulgaris shed new light on the catalytic mechanism of the enzyme †[END_REF]. Residue H287 was shown to act as a general base catalyst whereas two other histidine residues, namely H181 and H274, appeared to have a stabilizing effect. Roles of the two high-affinity Ca 2+ -binding sites were also described as being strongly involved in the overall stabilization of the DFPase structure and probably in the catalysis (Hartleib et al. 2001). The heterelogous production of the enzyme was considered using an E. coli expression system and yielded to large amounts of active and soluble protein that could be further purified using an His-tag and appeared to be stable for 1 year at 4 °C (Hartleib and Rüterjans 2001b).

Organophosphate acid anhydrolase (OPAA) and related prolidases

A moderately halophilic bacterial isolate identified as a species of Alteromonas was found able to degrade several OPs [START_REF] Defrank | Purification and properties of an organophosphorus acid anhydrase from a halophilic bacterial isolate[END_REF]. From this extract, the main enzyme, OPAA-2, was purified to homogeneity and found to be a60-kDapolypeptide.Thisenzymewasshowntohydrolyze DFP (k cat / K M = 7 . 7x1 0 4 M -1 s -1 )a sw e l la s phosphofluoridate nerve agents including sarin (k cat / K M =2.8x10 5 M -1 s -1 ), soman (k cat /K M =6.1×10 4 M -1 s -1 ) or cyclosarin (k cat /K M = 9.6 x 10 5 M -1 s -1 ) and was poorly active with paraoxon [START_REF] Cheng | Cloning and expression of a gene encoding a bacterial enzyme for decontamination of organophosphorus nerve agents and nucleotide sequence of the enzyme[END_REF]Theriot and Grunden 2010). A special attention was thus paid to Alteromonas strains that were screened for their OPhydrolyzing activity [START_REF] Defrank | Screening of halophilic bacteria and Alteromonas species for organophosphorus hydrolyzing enzyme activity[END_REF][START_REF] Cheng | Purification and properties of a highly active organophosphorus acid anhydrolase from Alteromonas undina[END_REF]. Many genes encoding OPAAs were reported and the enzymes were shown to be closely related to prolidases (EC 3.4.13.9). Their promiscuous ability to hydrolyze dipeptides with C-terminal proline (Xaa-Pro) was then underlined [START_REF] Cheng | Cloning and expression of a gene encoding a bacterial enzyme for decontamination of organophosphorus nerve agents and nucleotide sequence of the enzyme[END_REF][START_REF] Cheng | Nucleotide sequence of a gene encoding an organophosphorus nerve agent degrading enzyme from Alteromonas haloplanktis[END_REF]. Nevertheless, OPAAs are particularly efficient for hydrolyzing OP compounds and are also able to degrade G-type CWNAs. Their large-scale production has moreover been demonstrated [START_REF] Cheng | G-type nerve agent decontamination by Alteromonas prolidase[END_REF][START_REF] Cheng | Alteromonas prolidase for organophosphorus G-agent decontamination[END_REF][START_REF] Cheng | Hydrolysis of Organophosphorus Compounds by Bacterial Prolidases[END_REF]. Both substrate and stereochemical specificity of OPAA from Alteromonas sp. JDS6.5 were analyzed and a preference for S p enantiomers was demonstrated [START_REF] Hill | Substrate and stereochemical specificity of the organophosphorus acid anhydrolase from Alteromonas sp. JD6.5 toward p-nitrophenyl phosphotriesters[END_REF]. The Xray structure of the enzyme revealed two distinct domains, Nand C-terminus. The C-terminus domain contains Mn 2+ divalent cations and displayed a Bpita-bread^fold [START_REF] Vyas | Structural insights into the dual activities of the nerve agent degrading organophosphate anhydrolase/prolidase[END_REF]. This work highlighted relationships between prolidases and OPAAs that were completed by the 3D structure of OPAA from the marine bacterium Alteromonas macleodii (Štěpánková et al. 2013). Prolidases from hyperthermophilic sources were also attractive and Pyrococcus sp. strains were investigated. Enzymes from Pyrococcus furiosus or Pyrococcus horikoshii [START_REF] Maher | Structure of the prolidase from Pyrococcus furiosus[END_REF][START_REF] Theriot | Characterization of two proline dipeptidases (prolidases) from the hyperthermophilic archaeon Pyrococcus horikoshii[END_REF]Theriot et al. 2010), appeared to be good candidates for the development of decontamination solutions and were further considered for random mutagenesis investigations. Improved variants against nerve agents and Xaa-Pro dipeptides over a broad range of temperature were described (Theriot et al. 2010;[START_REF] Theriot | Improving the catalytic activity of hyperthermophilic Pyrococcus horikoshii prolidase for detoxification of organophosphorus nerve agents over a broad range of temperatures[END_REF]. The dual activity prolidase/OPAA was also observed with prolidase from human origin [START_REF] Wang | Production of human liver prolidase by Saccharomyces cerevisiae as host cells[END_REF]W a n ge ta l .2006;[START_REF] Ditargiani | In search of a catalytic bioscavenger for the prophylaxis of nerve agent toxicity[END_REF]Costanteetal.2012). This is particularly interesting as human enzymes are really appealing for the development of catalytic bioscavengers.

Paraoxonase

Another class of enzyme from human origin, namely paraoxonase (PON, EC 3.1.8.1.), has found considerable interest. Since the earliest report describing the ability of a mammalian serum esterase to degrade paraoxon [START_REF] Aldridge | Serum esterases[END_REF]) and nerve agents [START_REF] Broomfield | Hydrolysis of nerve gases by plasma enzymes[END_REF]Daviesetal.1996), numerous data were collected regarding the potential of PON for OP bioscavenging (Rochu et al. 2007). PON is capable of hydrolyzing a wide spectrum of OP pesticides [START_REF] Mackness | Substrate specificity of human serum paraoxonase[END_REF][START_REF] Furlong | Purification of rabbit and human serum paraoxonase[END_REF]Lietal.1995). PON from rabbit and human serum were purified to homogeneity and were shown to be similar, though differing in the amino terminus sequence [START_REF] Gan | Purification of human serum paraoxonase/arylesterase. Evidence for one esterase catalyzing both activities[END_REF][START_REF] Furlong | Purification of rabbit and human serum paraoxonase[END_REF]. However, human PON remains difficult to purify from plasma or heterologously express in high yield. Baculovirus and adenovirus infections of Trichoplusia ni larvae allowed to produce large quantities of functional recombinant HuPON1 [START_REF] Otto | Purification and characterization of functional human paraoxonase-1 expressed in Trichoplusia ni larvae[END_REF][START_REF] Hodgins | Assessing protection against OP pesticides and nerve agents provided by wild-type HuPON1 purified from Trichoplusia ni larvae or induced via adenoviral infection[END_REF]. Enzyme engineering was deeply investigated to generate improved variants. Site-directed mutagenesis and group-selective labeling modifications were applied to decipher the role of molecular determinants of catalysis (Josse et al. 1999a;Josse et al. 1999b). Supplementary essential residues were also identified with a DFPase-like homology model [START_REF] Yeung | Structure/function analyses of human serum paraoxonase (HuPON1) mutants designed from a DFPase-like homology model[END_REF]). Complete description of PON1's active site was achieved by solving the 3D structure of a recombinant PON1 variant (rePON1-G2E6) expressed in E. coli and obtained through directed evolution [START_REF] Harel | Structure and evolution of the serum paraoxonase family of detoxifying and antiatherosclerotic enzymes[END_REF][START_REF] Aharoni | Directed evolution of mammalian paraoxonases PON1 and PON3 for bacterial expression and catalytic specialization[END_REF]). Mammalian paraoxonase PON1 was found to be more efficient for the detoxification of soman and cyclosarin than OPH from P. diminuta and DFPase from Lolio vulgaris. Variants obtained from PON1 were further improved by about 10-fold against cyclosarin and soman as well as DFP, parathiol, or chlorpyrifios-oxon with up to 380-fold increase [START_REF] Amitai | Enhanced stereoselective hydrolysis of toxic organophosphates by directly evolved variants of mammalian serum paraoxonase[END_REF]). An engineered recombinant human PON1 (rHuPON1 K192 ) was also generated and highly purified. The stereospecificity of both recombinant and wild-type PON1 were evaluated using fluorogenic surrogates of nerve agents and were showntobeinfavorofR p isomers [START_REF] Ashani | Stereo-specific synthesis of analogs of nerve agents and their utilization for selection and characterization of paraoxonase (PON1) catalytic scavengers[END_REF]. Directed evolution experiments were, thus, utilized to increase the efficiency of PON1 towards G-agents [START_REF] Gupta | Directed evolution of hydrolases for prevention of G-type nerve agent intoxication[END_REF]. Rational and random mutagenesis were combined using a highthroughput methodology based on the utilization of fluorogenic analogs in emulsion compartments and a low-throughput plate screening. The activity against the coumarin analog of S p cyclosarin was enhanced by ≈10 5 -fold for variant 4E9 (L69G/ S111T/H115W/H134R/F222S/T332) as compared to the wildtype like variant rePON1-G3C9 reaching catalytic efficiency >10 5 M -1 s -1 . Another work led to the isolation of significantly improved variants for degrading G-Agents [START_REF] Goldsmith | Evolved stereoselective hydrolases for broad-spectrum G-type nerve agent detoxification[END_REF]. Among these, variant VIID2 (L55I/L69V/H115A/ H134R/H197/R/F222M/I291L/T332S) was of special interest. Although significant enhancements for the decontamination of G-agents have been achieved, the hydrolysis of V-type agents still remains critical. Computational experiments were applied to understand the way PONs hydrolyze VX and drew new perspectives for their engineering [START_REF] Peterson | VX Hydrolysis by human serum paraoxonase 1: a comparison of experimental and computational results[END_REF]. Doublesubstituted variants of human PON1 (H115W/Y71A and H115W/F347W) were constructed and showed up to 2-fold increase in catalytic efficiency as compared to the wild-type enzyme [START_REF] Kirby | Human paraoxonase double mutants hydrolyze V and G class organophosphorus nerve agents[END_REF]. Noteworthy, PON were also deeply studied for evolutionary biology considerations. Original strategies including back-to-ancestor/consensus mutations or neutralgenetic drift were employed to evolve PON in non-conventional manners [START_REF] Amitai | Latent evolutionary potentials under the neutral mutational drift of an enzyme[END_REF][START_REF] Khersonsky | Directed evolution of serum paraoxonase PON3 by family shuffling and ancestor/ consensus mutagenesis, and its biochemical characterization[END_REF][START_REF] Alcolombri | Directed evolution of sulfotransferases and paraoxonases by ancestral libraries[END_REF][START_REF] Bar-Rogovsky | The evolutionary origins of detoxifying enzymes: the mammalian serum paraoxonases (PONs) relate to bacterial homoserine lactonases[END_REF]. Although rarely investigated, PON3 paraoxonase was also purified [START_REF] Draganov | Rabbit serum paraoxonase 3 (PON3) is a high density lipoprotein-associated lactonase and protects low density lipoprotein against oxidation[END_REF][START_REF] Reddy | Human paraoxonase-3 is an HDL-associated enzyme with biological activity similar to paraoxonase-1 protein but is not regulated by oxidized lipids[END_REF]) and considered for directed evolution purposes. The catalytic efficiency of recombinant variants of PON3 against paraoxon was enhanced by up to 240-fold [START_REF] Aharoni | Directed evolution of mammalian paraoxonases PON1 and PON3 for bacterial expression and catalytic specialization[END_REF]). Both activity and stability of PON3 were shown to be tunable through combination of family shuffling and phylogeny-based mutagenesis [START_REF] Khersonsky | Directed evolution of serum paraoxonase PON3 by family shuffling and ancestor/ consensus mutagenesis, and its biochemical characterization[END_REF]. Of note, another enzyme, namely, the Senescence marker protein 30 (SMP30), has been showed to display a 3D structure closely related to PON1, although containing a single metal-binding site. SMP30 is a lactonase able to hydrolyze various aldolactones and is also promiscuous towards OPs sush as DFP [START_REF] Kondo | Senescence marker protein-30 is a unique enzyme that hydrolyzes diisopropyl phosphorofluoridate in the liver[END_REF][START_REF] Scott | Senescence marker protein 30: functional and structural insights to its unknown physiological function[END_REF].

Methylparathion hydrolase

Methylparathion hydrolase (MPH, EC 3.1.8.1) is an aryldialkylphosphatase belonging to metallo-β-lactamase family. Gene mpd coding for MPH was first identified in Plesiomonas sp. strain M6 [START_REF] Zhongli | Isolation of methyl parathiondegrading strain M6 and cloning of the methyl parathion hydrolase gene[END_REF]). The gene mpd was then sequenced, effectively expressed in E. coli and further purified as a monomer [START_REF] Fu | Expression, purification, and characterization of a novel methyl parathion hydrolase[END_REF]). Another closely related gene was identified in Pseudomonas sp. strain WBC-3 [START_REF] Liu | Plasmid-borne catabolism of methyl parathion and p-nitrophenol in Pseudomonas sp. strain WBC-3[END_REF]). The encoded enzyme was purified and crystallized and the 3D structure revealed a dimeric enzyme each subunit containing a mixed hybrid binuclear zinc center in which the more solventexposed β-metal cation was replaced by cadmium [START_REF] Dong | Crystal structure of methyl parathion hydrolase from Pseudomonas sp. WBC-3[END_REF]. Interestingly, MPH was shown to be homologous to other metallo-β-lactamases but did not display any similarity with phosphotriesterases. Recombinant MPH expressed in E. coli was characterized and was shown to be active on methylparathion as well as other OP pesticides including malathion or dichlorvos [START_REF] Yang | Overexpression of methyl parathion hydrolase and its application in detoxification of organophosphates[END_REF]. Another MPH, dubbed OPHC2, was identified in Pseudomonas pseudoalcaligenes strain C2-1 [START_REF] Ningfeng | Cloning and expression of ophc2, a new organphosphorus hydrolase gene[END_REF])a n dStenotrophomonas sp. strain SMSP-1 [START_REF] Shen | Isolation of a methyl parathiondegrading strain Stenotrophomonas sp. SMSP-1 and cloning of the ophc2 gene[END_REF]. The former was characterized and its 3D structure was solved (Gotthard et al. 2013a;Gotthard et al. 2013b). Recently, an OPHC2 analogue, namely PoOPH from Pseudomonas oleovorans, was discovered [START_REF] Luo | Switching a newly discovered lactonase into an efficient and thermostable phosphotriesterase by simple double mutations His250Ile/Ile263Trp[END_REF]. This enzyme displays lactonase and arylesterase activities as well as a promiscuous OPH activity. Variant PoOPH M2 (H250I/I263W) was designed and found improved by 6,962and 106-fold against methylparathion and ethylparaoxon, respectively. Random mutagenesis of MPH from Pseudomonas stutzeri further led to a 5-fold increase in the hydrolysis of chlorpyrifos [START_REF] Xie | Improving methyl parathion hydrolase to enhance its chlorpyrifos-hydrolysing efficiency[END_REF]. Additional mpd genes harbored by seven strains isolated from methylparathion contaminated sites were subcloned and expressed in E. coli [START_REF] Zhang | Cloning of the organophosphorus pesticide hydrolase gene clusters of seven degradative bacteria isolated from a methyl parathion contaminated site and evidence of their horizontal gene transfer[END_REF]). Phylogenetic analysis supported by GC-content comparisons between mpd genes and respective host's chromosome suggested that horizontal gene transfer may have occurred, playing an important role for bacterial adaptation to methylparathion contamination.

Phosphotriesterase-Like Lactonase

Among the enzymes able to degrade OPs, phosphotriesteraselike lactonases (PLLs) constitute a promising protein family closely related to bacterial phosphotriesterases [START_REF] Afriat | The latent promiscuity of newly identified microbial lactonases is linked to a recently diverged phosphotriesterase †[END_REF][START_REF] Draganov | Lactonases with oragnophosphatase activity: structural and evolutionary perspectives[END_REF]. PLLs are natural lactonases catalyzing the hydrolysis of N-acyl-homoserine lactones (AHLs) involved in the quorum sensing system of bacterial species, and additionally, exhibit a promiscuous OPH activity. Regarding both the predominant metabolic role of lactones in bacterial communication and the relatively recent occurrence of OPs, widely used as pesticides during the last decades, it has been assumed that the phosphotriesterase activity of PLL has emerged from a lactonase template harboring a promiscuous activity for OP hydrolysis. Interestingly, modern phosphotriesterase have probably evolved from a lactonase ancestor that lost its capacity to hydrolyze AHL to the benefit of higher OP-degradation rates. Natural evolution has led to the selection of enzymes with phosphotriesterase activity close to catalytic perfection to the detriment of the natural lactonase activity. Afriat-Jurnou and coworkers have, for example, demonstrated that ancestral reconstruction, starting from OPH from B. diminuta restore its lactonase catalytic ability, providing an experimental evidence of the evolutionary linkage between these enzymatic activities [START_REF] Afriat-Jurnou | Reconstructing a missing link in the evolution of a recently diverged phosphotriesterase by active-site loop remodeling[END_REF]. PLLs have been isolated from various hosts including hyperthermophile bacteria [START_REF] Xiang | Functional annotation and three-dimensional structure of Dr0930 from Deinococcus radiodurans, a close relative of phosphotriesterase in the amidohydrolase superfamily † ‡[END_REF][START_REF] Zhang | Enhancing the promiscuous phosphotriesterase activity of a thermostable lactonase (GkaP) for the efficient degradation of organophosphate pesticides[END_REF], or archaea [START_REF] Merone | A thermostable phosphotriesterase from the archaeon Sulfolobus solfataricus: cloning, overexpression and properties[END_REF];H i b l o te ta l .2012a;G o t t h a r de ta l . 2013b; [START_REF] Bzdrenga | SacPox from the thermoacidophilic crenarchaeon Sulfolobus acidocaldarius is a proficient lactonase[END_REF][START_REF] Kallnik | Characterization of a phosphotriesterase-like lactonase from the hyperthermoacidophilic crenarchaeon Vulcanisaeta moutnovskia[END_REF]). These enzymes display remarkable stability to temperature, pH, detergent, solvent, or storage, offering good perspectives for utilizations in external decontamination of OPs. The enzyme SsoPox from the archeon Sulfolobus solfataricus is particularly interesting and is one of the most promising PLL for OP countermeasures. Major results and perspectives are detailed hereafter.

SsoPox: a highly stable, OP-degrading capable enzyme

An archaeon-sourced enzyme with tremendous stability

The archea Sulfolobus solfataricus was discovered within the Ve s u vi o 's sulfatare and found to possess a gene homologous to the B. diminuta PTE. The encoded enzyme was expressed and its phosphotriesterase ability, albeit low, was highlighted. Among the hydrolyzed substrates was paraoxon, hence, was it called SsoPox: Sulfolobus solftataricus Paraoxonase [START_REF] Merone | A thermostable phosphotriesterase from the archaeon Sulfolobus solfataricus: cloning, overexpression and properties[END_REF]. It was also found to efficiently hydrolyze various lactones, including N-acyl homoserine lactones, known to be involved in bacterial communication (quorum sensing). Actually, the lactonase activity of SsoPox was found to be two orders of magnitude higher than its phosphotriesterase activity, underlining that it is most probably a natural lactonase, whose capacity to hydrolyze OPs chemicals stemmed from a promiscuous enzymatic template.

Due to its hypertermophilic origin the enzyme exhibits an incredible stability to temperature, being active from 10 to 100 °C, with an impressive denaturation temperature of 106 °C as well as a tremendous resistance to the denaturing effect of urea. Furthermore, SsoPox was active from pH 5.0 to pH 9.0 confirming the unusual properties of this enzyme. SsoPox was crystallized and its 3D structure was solved in both apo and complex forms (Elias et al. 2007;[START_REF] Elias | Structural basis for natural lactonase and promiscuous phosphotriesterase activities[END_REF]. The structure was reported as a distorted (β/α) 8 barrelfold and was found to be close to mesophilic OPH from both B. diminuta and A. radiobacter.

The major structural discrepancies between SsoPox and OPHs consist in the shortening of loop 7 as well as both extremities of the polypeptide chain. The arrangement of loop 8 is also variable between the structures and extra loops are observed within SsoPox structure (Fig. 2). These conformational changes are probably involved in the overall enzyme stability by either rigidifying the structure or favoring dimer formation. An investigation of the structural determinants of SsoPox as compared to its mesophilic counterparts was also performed [START_REF] Vecchio | Structural determinants of the high thermal stability of SsoPox from the hyperthermophilic archaeon Sulfolobus solfataricus[END_REF]. Surface salt bridge network and a tight quaternary structure were shown to be involved in complex networks and optimization of dimer subunit interfaces, respectively. From this observation came the idea to benefit from the outstanding structural stability of this enzyme, in order to transfer the efficient B. diminuta PTE active site within SsoPox scaffold.

Engineering SsoPox to enhance promiscuous phosphotriesterase activity Thus, protein engineering strategies have been considered in order to increase its potential for bioremediation. Indeed, protein stability has been proved to promote evolvability by minimizing the effect of destabilizing mutations and buffering their deleterious damages [START_REF] Tokuriki | Stability effects of mutations and protein evolvability[END_REF]. Hyperthermophile scaffolds are usually good candidates for evolutionary considerations as their tremendous stability may confer an enhanced tolerance to mutation-induced stability changes. Protein polymorphism may thus be generated without drastic effect on protein fitness. Moreover, substrate promiscuity has been identified as a potential origin of functional divergences that could be used as starting point in evolutionary trajectories [START_REF] Khersonsky | Enzyme promiscuity: evolutionary and mechanistic aspects[END_REF].

The rationale was that mesophilic phosphotriesterases probably emerged from PLLs, so their respective sequences and structures were compared and led to the identification of residues probably involved in the enhancement of OP-degradation. Mutational databases listing potential beneficial substitutions for phosphotriesterase activity were designed [START_REF] Chabriere | Sulfolobal phosphotriesterase-like (pll) lactonases activity having enhanced properties and the uses thereof[END_REF][START_REF] Chabriere | Vulcanisaetal phosphotriesterase-like lactonases (pll) having enhanced properties and the uses thereof[END_REF]. These databases combined with in silico analyses were used as starting point for engineering SsoPox toward OP based pesticide degradation (Fig. 3). Variants with up to 2000-fold increased activity against certain pesticides and enlarged substrate scope were obtained (unpublished results). Mutations R223H and Y97W were first shown to impact phosphotriesterase activity [START_REF] Elias | Structural basis for natural lactonase and promiscuous phosphotriesterase activities[END_REF]. Residue W263 was also deeply investigated (Hiblot et al. 2012b). This residue is located in the active site and was shown to be involved in enzyme conformational flexibility, mediating substrate promiscuity. Saturation mutagenesis was performed at this position using NNS-degenerated primers. Two subsets of mutations were distinguished as regard to either their lactonase or phosphotriesterase activity. Within this latter, variants W263F, W263L and W263M were particularly attractive. Catalytic parameters of SsoPox variants towards OPs are reported in Table 1. SsoPox and its variants were shown to be active on a wide range of OPs including pesticides (e.g. paraoxon, parathion, malathion) and chemical warfare agent analogs (IMP-, PinP-, and CMPcoumarin). The stimulation of SsoPox activity by anionic detergent at ambient temperature was also demonstrated and is of prime interest for external decontamination purpose as it could help spreading the enzyme on soiled surfaces (Hiblot et al. 2012b). Future engineering experiments on SsoPox are in progress in order to provide a stable, proficient catalyst capable of detoxifying a broad spectrum of toxic OPs.

Concluding remarks

Decontamination methods against OP poisoning have been exhaustively studied during the past decades. Efficient physico-chemical methods including hypochlorite and the atomic partial charges (Gasteiger method) of the receptor protein models were added using ADT 4.2. A +2 charge was attributed to the two-cation ions that comprise the active site of SsoPox. The docking simulations were performed using Autodock 4.0 [START_REF] Morris | Identification of a plasmid-borne parathion hydrolase gene from Flavobacterium sp. by southern hybridization with opd from Pseudomonas diminuta[END_REF][START_REF] Huey | A semiempirical free energy force field with charge-based desolvation[END_REF]) and the Lamarckian genetic algorithm. Forty runs of genetic algorithm were performed for each couple ligand-receptor pair using the default parameters. The output ligand configurations were clustered and selected by their binding energy scores and their chemical relevance sodium hydroxide have emerged for the efficient external decontamination of OPs or stockpile elimination and have already been used for this purpose. These methods usually involve harsh chemical conditions and are not compatible with the decontamination of personnel or sensitive materials. Other methods were developed such as RSDL or Fuller's earth. Whereas RSDL is efficient at decontaminating the skin, it is not compatible with eye and wound cleaning and would be hardly adaptable to a large number of contaminated people. Fuller's earth shows good topical absorbance of contaminants but do not degrade OPs, possibly resulting in large amounts of secondary pollution. Moreover, toxic dusts are generated during the decontamination process, complicating the use of the method at a large scale. Alternatively, photochemical methods are under development for the mineralization of OP chemical in environmental media but would not be relevant for people decontamination. Methods that can be used for both material and personel are under intensive investigations. CDs were proved to be efficient bioscavengers but because of their stoichiometric action, they may not be cost-effective in a case of large scale pollution. Investigations also focus on another promising solution, the use of OP-degrading enzymes. Indeed, the use of enzymes is appealing because they offer a non-corrosive, safe and catalytic way for decontaminating OPs. Protein engineering has been deeply applied for increasing the catalytic efficiencies of recombinant enzymes.

Whereas the stability and the production costs of enzymes may be an issue, the discovery of highly thermostable OPsdegrading enzymes such as SsoPox, and their engineering for higher activity is expected to turn them into competitive and economically-attractive OP-biodecontaminants.

  Photolytic degradation of targeted compound consists in the absorption of radiation in both direct and indirect ways, leading to the destruction of the compound. Several examples of OP photolysis (diazinon, methylparathion, or quinalphos) have been reported to date in both direct[START_REF] Wan | Comparative study on the quantum yields of direct photolysis of organophosphorus pesticides in aqueous solution[END_REF];S i n d e r h a u fa n d Schwack 2003;G o n ç a l v e se ta l .2006; Segal-Rosenheimer and Dubowski 2010; Muñoz et al. 2011) and indirect manner[START_REF] Lam | PhotoFate: a new approach in accounting for the contribution of indirect photolysis of pesticides and pharmaceuticals in surface waters[END_REF]Garbinetal.2007).

Fig. 2

 2 Fig.2Overall 3D structure of SsoPox in complex with fensulfothion (PDB ID: 3uf9). Protein backbone is showed as cartoon; N-and C-ter are highlighted in purple and green, respectively. Loop-7 and loop-8 that govern substrate promiscuity are emphasized in yellow and red.Divalent active-site cations are shown as spheres

Fig. 3

 3 Fig. 3 Molecular docking of paraoxon to SsoPox structure. The docking of paraoxon (ground state, GS; green sticks; a) and of its corresponding putative oxyanionic pentavalent hydrolytic transition state (TS; cyan sticks; b) was performed in SsoPox (PDB ID: 2vc5). Metal cations and active-site water molecules (in red) are shown as spheres. Distances are indicated in Ångstroms. Briefly, the molecular docking was performed using torsion and charge parameters generated using ADT 4.2 (http:// autodock.scripps.edu/resources/adt) and the Gasteiger method. A single-negative charge was attributed to the oxyanion atom of the pentavalent paraoxon hydrolysis intermediate. Hydrogen atom positions and

Table 1

 1 Catalytic parameters of

	SsoPox wild-type and variants	Substrate	Enzyme	Condition	k cat (s -1 )	K M (μM)	k cat /K M (M -1 s -1 )
	towards organophosphate pesticides and nerve agents with	Ethylparaoxon	WT	70 °C	3.98	3,270	1.22 × 10 3
	respective reactions conditions (Elias et al. 2008;Hiblotetal. 2012b;Hiblotetal.2013)			25 °C SDS 0.1 %, 25 °C SDS 0.01%, 25 °C	12.59 40.72 24.59	24,250 12,340 3,830	5.19 × 10 2 3.30 × 10 3 6.42 × 10 3
				DOC 0.1 %, 25 °C	6.30	570	1.10 × 10 4
				DOC 0.05 %, 25 °C	4.72	270	1.22 × 10 3
				DOC 0.01 %, 25 °C	10.51	730	1.72 × 10 4
			W263F	25 °C	8.47	700	1.21 × 10 4
				SDS 0.1 %, 25 °C	117.70	2,462	4.78 × 10 4
				SDS 0.01 %, 25 °C	85.85	1,168	7.35 × 10 4
			W263M	25 °C	6.82	931	7.33 × 10 3
			W263L	25 °C	ND	ND	2.37 × 10 3
			W263I	25 °C	ND	ND	1.21 × 10 3
			W263V	25 °C	ND	ND	8.83 × 10 3
			W263T	25 °C	ND	ND	1.06 × 10 3
		Methylparaoxon	WT	25 °C	2.71	2,142	1.27 × 10 3
		Methylparathion	WT	25 °C	1.1 × 10 -3	121	9.09
		Malathion	WT	25 °C	8.9 × 10 -4	160	5.56
		CMP-coumarin	WT	25 °C	ND	ND	8.13 × 10 3
			WT	SDS 0.01 %, 25 °C	25.47	137	1.86 × 10 5
			W263F	25 °C	9.41	114	8.23 × 10 4
			W263F	SDS 0.01 %, 25 °C	8.64	60	1.44 × 10 5
		IMP-coumarin	WT	25 °C	ND	ND	1.67 × 10 3
			W263F	25 °C	8.39	95	8.85 × 10 4
		PinP-coumarin	W263F	25 °C	0.11	16	7.08 × 10 3

Acknowledgments P.J. and J.B. are Ph.D. students granted by Direction Générale de l'Armement (DGA).


Marco Congedo
email: marco.congedo@gipsa-lab.fr

A tribute to Alan Turing: Symmetric Gaussian Elimination

Keywords: Cholesky Decomposition, LDL T Decomposition, Triangularization, Gaussian Elimination, symmetric positive definite, Alan Turing

The triangular decomposition of a square matrix is the "key interpretation" of Gaussian elimination and is recognized universally as one of the most important and ubiquitous numerical tool for science and industry. It is also one of the most ancient, being known in algebraic form over 2000 years ago in China.

The standard algorithm currently in use for decomposing a symmetric positive definite (SPD) matrices is credited to André-Louis Cholesky (1910). However, the first elegant and modern treatment of Gaussian elimination as a matrix decomposition process has been given by Alan Turing (1948). Such Gaussian elimination still today provides the most economical numerical algorithm for decomposing an SPD matrix, requiring as few as n 2 /2-n/2 divisions, n 3 /6-n/6 multiplications and n 3 /6-n/6 subtractions. In this note we give a synopsis of symmetric Gaussian elimination, including the description of the two major conceptual procedures, one being the one offered by Turing in 1948, an historical overview and exact count of arithmetic operations, identifying the most advantageous algorithms.

Introduction

Nobody knows how many times per day Gaussian elimination is invoked all over the world, but certainly this number is formidable and is destined to increase with the proliferation of electronic devices.

Currently, Gaussian elimination is the method of choice for solving systems of linear equations (the Ax=b problem), which are used routinely to obtain the result of algebraic expression involving A and its inverse A -1 . It is also the method of choice for computing the inverse of a matrix explicitly, for evaluating its determinant, to ascertain if a square matrix is symmetric positive-definite (SPD) and to reveal the rank of a symmetric positive semi-definite matrix (SPSD). These tasks cover already an important portion of the well-established arsenal of numerical analysis [START_REF] Golub | Matrix Computations (4 th Eds)[END_REF]. Somehow less known is that Gaussian elimination provides the fastest option to reduce a matrix in tridiagonal form [START_REF] Trefethen | Three mysteries of Gaussian Elimination[END_REF], to reduce a generalized eigenvalue problem to a symmetric eigenvalue problem [START_REF] Fox | Notes on the solution of algebraic linear simultaneous equations[END_REF], to compute inverse power iterations (Parlett, 1980, p. 62) and to obtain low-rank approximations of a matrix (Townsend and Trefethem, 2013), although more accurate and stable alternatives exist. Gaussian elimination can also recover the eigenvalues of a matrix [START_REF] Fernando | Implicit Cholesky algorithms for singular values and vectors of triangular matrices[END_REF], although not in an efficient way. Remarkably, all this is offered by one of the oldest numerical algorithm worked out by the mankind.

Gaussian elimination was known in algebraic form over 2000 years ago in China and in other ancient civilizations such as the Greek, Roman, Arab and Hindu [START_REF] Grcar | How Ordinary Elimination Became Gaussian Elimination[END_REF]. In his lecture notes Arithmetica Univerʃalis, written in Latin starting about 1670, Isaac Newton called the action of eliminating variables from a system of equations an extermination ("exterminatio"). Leonhard Euler in a German writing of 1771 refers to the elimination process as the most natural way to proceed ("der natürlichtste Weg") [START_REF] Grcar | How Ordinary Elimination Became Gaussian Elimination[END_REF]. Adrien-Marie Legendre in 1806 and Carl Friedrich Gauss in 1807 write about it as ordinary methods, respectively in French ("méthode ordinaire") and Latin ("eliminatio vulgaris") [START_REF] Legendre | Nouvelles méthodes pour la détermination des orbites des comètes; avec un supplement contenant divers perfectionnemens de ces méthodes et leur application aux deux Comètes de 1805[END_REF][START_REF] Stewart | Gauss, Statistics, and Gaussian Elimination[END_REF]. The merit of Gauss has been to introduce a notation helping the computation of linear systems by pencil and paper, which spread out rapidly, before as a professional method for geodesists in the (governmental) field of cartography and then as the solution to Gauss's normal equation. His work has fostered the large demand for solving least-squares problems, such as linear regression in statistics [START_REF] Dwyer | A Matrix Presentation of Least Squares and Correlation Theory with Matrix Justification of Improved Methods of Solution[END_REF] and data fitting in experimental physics, astronomy and engineering [START_REF] Grcar | How Ordinary Elimination Became Gaussian Elimination[END_REF][START_REF] Stewart | Gauss, Statistics, and Gaussian Elimination[END_REF]. The handbook of geodesy published in 1888 by Wilhelm Jordan reports results on his studies on triangularization and triangular diagonalization, what today is known as reduction to row (or column) echelon form and the Gauss-Jordan method for matrix inversion [START_REF] Althoen | Gauss-Jordan Reduction: A Brief History[END_REF]. With remarkable penetration, at the beginning of the twentieth century André-Louis Cholesky discovered a recurrence for the symmetric case dominated by sums of products [START_REF] Brezinski | The life and work of André Cholesky[END_REF][START_REF] Cholesky | Sur la résolution numérique des systèmes d'équations linéaires, hand-writing document[END_REF]. This algorithm was invented to help a computer, which at the time of Cholesky was human or mechanical at best, but turned out to suits electronic ones as well [START_REF] Fox | Notes on the solution of algebraic linear simultaneous equations[END_REF]. By the beginning of the 1940's Gaussian elimination is still found mostly described in algebraic terms [START_REF] Crout | A short method for evaluating determinants and solving systems of linear equations with real or complex coefficients[END_REF][START_REF] Dwyer | The Solution of Simultaneous Equations[END_REF]. A rapid advance in the research on Gaussian elimination is observed in the 40's, stimulated by the imminent realization of the first electronic computer, yielding the widespread conception of Gaussian elimination as one of the big six decomposition problems [START_REF] Stewart | The Decompositional Approach to Matrix Computation[END_REF]. This has provided a unified framework demonstrating the equivalence of the several available algebraic methods and turned out to be inspiring, both for the comprehension of the method and for designing algorithms [START_REF] Higham | Gaussian Elimination[END_REF]. [START_REF] Frazer | Elementary Matrices and Some Applications to Dynamics and Differential Equations[END_REF] appear to be the first to show the process of matrix reduction to triangular and diagonal form under the action of triangular matrices. A similar development is seen in [START_REF] Banachiewicz | An outline of the Cracovian algorithm of the method of least squares[END_REF]. [START_REF] Dwyer | A Matrix Presentation of Least Squares and Correlation Theory with Matrix Justification of Improved Methods of Solution[END_REF] obtained a version of the symmetric decomposition based on outer products, showing the equivalence with the Doolittle algorithm. Gaussian elimination was investigated by von [START_REF] Neumann | Numerical inverting of matrices of high order[END_REF] in the perspective of numerical computing. They anticipated the prolific literature on error bounds in finite precision arithmetic, to flourish soon with the availability of first electronic computers. The Cholesky method, known at the time as the square root method, spread in UK after a talk given by John Todd at King's College of London in 1946 [START_REF] Taussky | Cholesky, Toeplitz and triangular factorization of symmetric matrices[END_REF], where it was taken up by the group at the National Physical Laboratory [START_REF] Fox | Notes on the solution of algebraic linear simultaneous equations[END_REF]. It was Alan [START_REF] Turing | Rounding-off-errors in Matrix Processes[END_REF], also working in this group, who gave the first elegant and simple description of Gaussian elimination as the LU decomposition. He used a strikingly modern notation and his presentation of Gaussian elimination in matrix form is still actual, both conceptually and algorithmically.

Here we give an account of it for the special case when the matrix to be triangularized is symmetric, allowing a valid alternative to the Cholesky method. Turing did not work out the details for this case (Turing, 1948, remark 8, p. 292). They are relevant in that they allow the most parsimonious recurrences for decomposing a matrix. This report complements and links the excellent accounts of Higham on the LU algorithm (2011), i.e., asymmetrical Gaussian elimination, and on the Cholesky algorithm (2008b).

Turing's decomposition for the symmetric case

The symmetric version of Turing's LU decomposition (1948) are the Cholesky, or triangular decomposition [START_REF] Toeplitz | Die Jacobische Transformation der quadratischen Formen von unendlichvielen Veränderlichen[END_REF][START_REF] Cholesky | Sur la résolution numérique des systèmes d'équations linéaires, hand-writing document[END_REF]

) T R R A   1 T R A R I   (1)
and the equivalent LDL T , or unitriangular decomposition [START_REF] Dwyer | A Matrix Presentation of Least Squares and Correlation Theory with Matrix Justification of Improved Methods of Solution[END_REF][START_REF] Satterthwaite | Error control in matrix calculation[END_REF])

T LDL A   1 T L AL D   , ( 2
)
where L is a lower unitriangular matrix, that is, a triangular matrix with 1s on the diagonal, D a positive diagonal matrix and R T = LD 1/2 a lower triangular matrix. A lower atomic matrix of dimension nxn is a unit-determinant lower unitriangular matrix with form

    1 1 1 1 1 1 1 1 n rr qr r qr nr LI Γ                   , 1 ≤ r < n (3)
where Γqr =qrEqr, with Eqr denoting the nxn elementary matrix filled with 1 at entry (q,r) and zero elsewhere. Atomic matrices and their elementary constituents I+Γqr can be also found under the name of addition elementary matrices [START_REF] Frazer | Elementary Matrices and Some Applications to Dynamics and Differential Equations[END_REF], Gaussian transformations (Golub and van Loan, 2013, p. 112), or elementary lower triangular matrices (Stewart, 1998, p. 154). The elements  are named the Gaussian or shear coefficients since for n=2 they are used in computer graphics to perform area-preserving shearing (skewing) of images (e.g., [START_REF] Tsuchida | Hardware for image rotation by twice skew transformations[END_REF]. The inverse of an atomic matrix is readily obtained changing the sign of the Gaussian coefficients, that is,

  1 1 11 nn qr qr r q r q r LI Γ I Γ               . ( 4
)
Atomic matrices do not commute in general, however the shear coefficients simply add when multiplied in whatever increasing order of subscript, that is

    11 , 1 . nn qi q j ij q i q j L L I Γ Γ i j n                     (5)
Multiplication ( 5) is what makes Gaussian elimination so efficient and for this reason it has been referred to as one of the more happy "coincidences" in matrix computations [START_REF] Golub | Matrix Computations (2 th Eds)[END_REF].

Given an SPD matrix X, Touring's method consists in a sequence of n-1 transformations X←L(r) -1 X,

where at each step L(r) -1 is given by setting qr = xrq/xrr, q=r+1…n. Elements xrr are named the pivots and elements xrq the multipliers [START_REF] Higham | Gaussian Elimination[END_REF]. In section 4 (Turing, 1948, p. 290) Turing explains the elimination process as forming the products L(1) -1 A, "recording it", then forming product L(2) -1 (L(1) -1 A), etc., using all matrices L(r) -1 as per (3) for r = 1…n-1. In remark (9) at p. 292 he notes that the application of such recurrence is equivalent to successive conjugations …(L(2

) -1 (L(1) -1 AL(1) -T )L(2) -T )…
, which in Gaussian elimination terminate with the diagonalization of A, hence directly with the inversion of A. In fact, conjugation L(1) -1 AL(1) -T zeroes all non-diagonal elements in the first row and column of A. Next conjugation L(2) -1 (L(1) -1 AL(1) -T )L(2) -T does the same to the second row and column. On completion we have

L(n-1) -1 L(n-2) -1 … L(1) -1 =L -1
, resulting in L -1 A=DL T , and by virtue of ( 4) and ( 5)

              1 1 1 1 1 2 1 1 2 1 1 . qr n n n r q n L L L L L L L I Γ                (6)
We hence obtain the following general symmetric Gaussian elimination algorithm,

      1 1 1 , for 1< do , with n r rr qr qr r qr r X L X XA r n L I x x E YI Y Y L            . ( 7
)
Proposition (8) (Touring, 1948).

Sequence ( 7) terminates with

YL  ,   diag X D  ,   T upper X DL  ,   1 strictly lower X I L   .
 For initialization XA  , only the upper elements of A are needed.

 The line Y←YL(r) amounts to the simple affectations yqr ← xrq/xrr, q=r+1…n, as per (5).

 If L is needed, but not L -1 , as for instance in linear system solvers, only the upper elements of X needs to be updated in X←L(r) -1 X. In fact, therein the strictly lower elements of X are concerned only with the recurrence for computing L -1 .

 Computing D and L can be done in-place and requires n 2 /2-n/2 divisions, n 3 /6-n/6 multiplications and n 3 /6-n/6 subtractions, as detailed by the following algorithm, given here in pseudo-code:

Algorithm LDL T (9). Given an SPD matrix A of order n, of which only the upper elements are needed, overwrites its strictly lower part with the strictly lower part of L, its diagonal part with D and its upper part with DL T . In many situations it will be preferred an equivalent division-free algorithm trading off the divisions for an equal amount of multiplications plus n-1 reciprocations. Also, it is current practice to compute L -1 by computing the LDL T decomposition and then inverting L. A direct method for computing both L, D and L -1 is procedure (7), which adds n 3 /6-n 2 /2+n/3 multiplications and n 3 /6-n 2 /2+n/3 subtractions to the count in (9), those additional operations being needed to update the strictly lower part of X. The computation flow of algorithm (9) for a matrix of dimension 4 is shown in Fig. 1. An important alternative view of the Gaussian elimination process is as it follows: from Fig. 1 we see that the first row of the input matrix is not touched at all by the algorithm. Set r=1 and denote such row ar T . We know that upon completion of the algorithm row ar T holds drrlr T , the first row of DL T , and that its first element is drr, the first element of D. As a consequence / TT r r rr r rr r a a a l d l  , thus rr rr da  and / r r rr l a a  . To find the second row of DL T we compute /

Line Instruction

T r r rr

A A a a a  , which has zeros in the first column and row, and we proceed in the same way with the second row setting r=2. Proceeding this way for r=1…n-1 steps yields the so-called outer-product Cholesky algorithm [START_REF] Golub | Matrix Computations (2 th Eds)[END_REF], also named the "iterative view" of Gaussian elimination (Townsend and Trefethem, 2013). If at each new step we find the permutation matrix P bringing the maximum diagonal element in the r th position by congruence PAP T , which for a symmetric matrix corresponds to complete pivoting, then the matrix 1 / , Townsend and Trefethem, 2013). In the same way, stopping when no more positive pivot can be found, we obtain rank-revealing versions of triangular decompositions, enabling the application of the procedure to SPSD matrices as well (Golub and Van Loan, 2013, p. 166;[START_REF] Higham | Analysis of the Cholesky Decomposition of a Semidefinite Matrix[END_REF]. Since the outer-product scheme we have just seen is equivalent to a symmetric Gaussian elimination, one can apply directly Gaussian elimination as per algorithm ( 9) to obtain a rank-revealing LDL T decomposition, as detailed by the following algorithm:

p T r r rr r a a a p n    is a p-rank approximation of A (
Algorithm PAP T =LDL T (10). Given an SPSD matrix A of order n, overwrites its strictly lower part with the strictly lower part of L and its diagonal part with D.

Procedure Swap (r, m). Swap rows r and m and elements r+1…n of columns r and m of A (if P is needed, update it here).  As stopping criterion we are taking the machine epsilon (macheps) x n x the first pivot, as suggested by [START_REF] Higham | Analysis of the Cholesky Decomposition of a Semidefinite Matrix[END_REF].

Line Instruction

 As compared to algorithm (9), algorithm (10) adds a stopping criterion, pivoting and copies elements aqk into element akq in the main loop (line 11), since those elements may be later on involved in swapping, whereas they are never needed in algorithm (9). Otherwise, the two algorithms are identical and performs the same number of arithmetic operations.

 As it stands algorithm (10) needs the whole matrix A as input and not just the upper triangle as for algorithm (9). This can be avoided with a smart coding of the swap procedure, in which case also the writing of elements akq would no longer be necessary.

 The reason why in the Swap procedure the row of A are swapped entirely while only the elements r+1…n of the columns are swapped is that the elements 1…r of the columns are never used, while the swapping of elements 1…r of the rows belongs to L. They also need to be swapped (see [START_REF] Higham | Analysis of the Cholesky Decomposition of a Semidefinite Matrix[END_REF], while in the algorithm given in Golub and Van Loan, 2013, p. 166, this detail is omitted).

 Unless one is sure that matrix A is SPD and maximal efficiency is of concern, algorithm (10) is to be preferred because is stable regardless the condition number of A and more accurate thanks to complete pivoting.

Operation Count

In table 1 the operation used by LDL T algorithm (9) are compared to state-of-the art algorithms with minimal operation count we have found in the literature, namely, LDL T algorithm 4.1.1 of Golub and Van Loan (2013, p. 158) and the following two R T R algorithms:

 algorithm CholSax of Van Loan (2000, p. 262) and the  algorithm given by Higham (2008b, p. 2).

The most economical algorithms are (9) for the LDL T decomposition and the one given by Higham (2008b, p. 2) for the R T R decomposition, which in addition requires n square roots. Furthermore, the count for the latter increases by n 2 -n multiplications if we try to eliminate the n 2 /2-n/2 divisions as it is possible to do for algorithm (9).

Discussion

It has been said that "if numerical analysts understand anything, surely it must be Gaussian Elimination", yet, "the closer one looks, the more subtle and remarkable Gaussian elimination appears" [START_REF] Trefethen | Three mysteries of Gaussian Elimination[END_REF]. In fact, Gaussian elimination is not an established knowledge, as demonstrated by the fact that research on it is currently flourishing more than ever; a research using keyword "Gaussian Elimination" in the BIB digital scientific library of CNRS returns 785 results in the period 1980-2000 and more than twice as much since 2000. Current research topics include reduction of complexity for sparse matrices, pivoting strategies, eigenvalue localization and estimation, singular-value decomposition, generalized inverses, parallelization, hardware computing, matrix partitioning, block form, preconditioning, iterative refinement of the solution, equalization, rank-revealing and randomized versions, among others. Above all, the renowned numerical stability of Gaussian elimination is still a puzzling theoretical problem [START_REF] Higham | Gaussian Elimination[END_REF][START_REF] Trefethen | Three mysteries of Gaussian Elimination[END_REF], despite the advances achieved by mathematicians such as Hotelling

Figure 1 .

 1 Figure 1. Computation flow of algorithm (9). The flow proceeds from top to bottom and from left to right, that is, in the order (r=1, q=2), (r=1, q=3), (r=1, q=4), (r=2, q=3)... At each step the entry marked by  indicates the affectation of element aqr at line 3, corresponding to the q th element of Y←YL(r) in procedure (7), while the curved arrows indicates the computations of line 4, corresponding to the upper part of X←L(r) -1 X . Note that the flow may equivalently proceed from left to right and top to bottom, that is, in the order (r=1, q=2), (r=1, q=3), (r=2, q=3), (r=1, q=4)…, yielding different versions of Gaussian elimination algorithms.

|

  If amm > ε then| | If r = 1 then ε = amm n macheps | | If m > r then Swap (r, m) aqk  aqk -(aqr  ark) | | | | akq  aqk | Else EXIT

Table 1 .

 1 Flops count of two LDL T and two R T R algorithms. Legend: sr=square roots, d=divisions, r=reciprocations, m=multiplications; a/s=additions or subtractions.

 (1943), von [START_REF] Neumann | Numerical inverting of matrices of high order[END_REF], [START_REF] Wilkinson | Error Analysis of Direct Methods of Matrix Inversion[END_REF], Higham (2008a) and [START_REF] Golub | Matrix Computations (4 th Eds)[END_REF].

Both for solving systems of linear equations and for explicit inversion, the LDL T decomposition is advantageous over the R T R decomposition. [START_REF] Cho | Efficient implementation of linear system solution block using LDL T factorization[END_REF] give some figures for a highly efficient FPGA (Field Programmable Gate Array) implementation with fixed-point precision. The LDL T decomposition is also preferred because it avoids the computation of the n square roots required by the R T R decomposition, thus it is stable in practice even if the condition number of A is low and suits symbolic computing. In this note we have focused on algorithms for standard processing units, however all Gaussian elimination procedures are recurrences based on triple loops, thus they all can be parallelized in a similar way [START_REF] Higham | Gaussian Elimination[END_REF]. The algorithms we have reported can easily be derived as division-free algorithms, relying only on subtractions, multiplications and reciprocations; this suits x86 processing units, for which divisions necessities more clock cycles than multiplications. Moreover, reciprocations and square roots, the former needed only for the R T R decomposition, can be obtained from a look-up table stored in memory, given normalized input matrices [START_REF] Burian | A fixed-point implementation of matrix inversion using Cholesky decomposition[END_REF]) and refinement can used as needed to match the required precision [START_REF] Golub | Matrix Computations (4 th Eds)[END_REF][START_REF] Higham | Gaussian Elimination[END_REF]. This in turn allows the possibility to exploit fast hardware computing environments operating exclusively subtractions and multiplications.

It is often reported that partitioned decompositions are advantageous over recurrences applied to the entire matrix as treated here, because the former can make use of efficient matrix multiplication routines [START_REF] Higham | Gaussian Elimination[END_REF]. However, triangularization in such algorithms is still for some diagonal partitions. The number of arithmetic operations used by the LDL T algorithm (9) and the R T R algorithm given by [START_REF] Higham | Functions of Matrices: Theory and Computation[END_REF] are minimal (Table 1); n 2 /2-n/2 divisions are needed to compute the Gaussian coefficients qr and the n 3 /6-n/6 multiplications and subtractions are needed for the main recurrence, regardless how it is carried out. These operations appears incompressible, however the question whether one can obtain an equivalent numerical result with a smaller number of operations is still open [START_REF] Trefethen | Three mysteries of Gaussian Elimination[END_REF].