Hubert Casajus

Aurélie Lagarde

Lionel Nauton

Nazim Ocal

Martin Leremboure

Wolf-Dieter Fessner

Nicolas Duguet

Franck Charmantray

Laurence Hecquet

Cleavage of Aliphatic α-Hydroxy Ketones by Evolved Transketolase from Geobacillus stearothermophilus

Keywords: Biocatalysis, Transketolase, C-C bond cleavage, Mutagenesis, -hydroxy ketones

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INTRODUCTION

In their respective fields, chemists and biochemists have developed a range of synthetic strategies for the formation of C-C bonds in order to synthesize complex compounds from simple and abundant raw materials Recently, the use of biomass as a renewable feedstock has been recognized as one of the ways to face the depletion of fossil resources and to limit CO 2 emissions. The valorization of biomass implies, in that case, to split existing (natural) molecules to make compounds with higher added value (scheme 1). In this context, new synthetic methodologies for C-C bond cleavage, including enzymatic ones, should be developed.

Particularly, aliphatic -hydroxy ketones can be cleaved to yield free aldehydes of great interest for various applications such as aromas and intermediates for the synthesis of surfactants or for the preparation of polyesters and polyamides. [START_REF] Aranea | A Pharmacological Profile of the Aldehyde Receptor Repertoire in Rat Olfactory Epithelium[END_REF] The -hydroxyketone function can be found as such in several natural products such as dihydroxyacetone (DHA), acetoin, erythrulose or in the masked form in xylulose, sorbose and fructose. Moreover, -hydroxyketones can be formed by Benzoin condensation, enolate oxidation, C-H oxidation and several other means. [START_REF] Streuff | An Update on Catalytic Strategies for the Synthesis of α-Hydroxyketones[END_REF] More importantly, they can be readily prepared in one-step by enzymatic-, [START_REF] Moonmangmee | Purification and Properties of Two Different Dihydroxyacetone Reductases in Gluconobacter suboxydans Grown on Glycerol[END_REF] metallic-4 or electrochemical [START_REF] William | Electrochemical Oxidation of 1,2-Diols to α-Hydroxyketones in Water[END_REF] selective oxidation or by dehydrogenation [START_REF] Sato | Synthesis of α-Hydroxyketones from 1,2-Diols Over Cu-Based Catalyst[END_REF] of the corresponding 1,2-diols.

These methods allow the preparation of -hydroxyketones derived from renewable resources as the 1,2-diol moiety is widespread in Nature (e.g. glycerol, carbohydrates, etc.).

3

The retro-acyloin method allows the cleavage of aliphatic -hydroxy ketones to yield free aldehydes and could offer an alternative to reductive ozonolysis, which is a prominent reaction for the clean and efficient production of fatty aldehydes from a double bond (scheme 1). [START_REF] Bugaut | Organocatalytic Umpolung: N-Heterocyclic Carbenes and Beyond[END_REF] Scheme 1. Synthetic strategies for the preparation of aldehydes through C-C bond cleavage.

Despite the economy atoms achieved during the implementation of the ozonolysis reaction, the use of ozone (O 3 ), which is both highly toxic and explosive presents the major drawback of this synthetic method. If the (asymmetric) acyloin condensation has been studied extensively for the synthesis of acyloins (-hydroxy ketones) by chemical and enzymatic ways, [START_REF] Müller | Thiamin Diphosphate in Biological Chemistry: Exploitation of Diverse Thiamin Diphosphate-Dependent Enzymes for Asymmetric Chemoenzymatic Synthesis[END_REF][START_REF] Li | Thiazolium-Derived N-Heterocyclic Carbene-Catalyzed Cross-Coupling of Aldehydes With Unactivated Imines[END_REF] its reversibility has been demonstrated in only a few cases [START_REF] Kuebrich | Mechanism of the Acyloin Condensation[END_REF][START_REF] Miyashita | Synthesis of Ketones by Retro-Acyloin Condensation Catalyzed by Potassium Cyanide[END_REF] and has been underexploited. [START_REF] Zhang | Catalytic Activation of Carbohydrates as Formaldehyde Equivalents for Stetter Reaction with Enones[END_REF][START_REF] Omonov | The Production of Biobased Nonanal by Ozonolysis of Fatty Acids[END_REF] Recently, an efficient chemical retro-acyloin method was reported for obtaining aliphatic free aldehydes by cleavage of fatty α-hydroxy ketones catalyzed by thiazolylidene. [START_REF] Deruer | Thiazolylidene-Catalyzed Cleavage of Methyl Oleate-Derived α-Hydroxy Ketone to the Corresponding Free Aldehydes[END_REF] The continuous distillation of the targeted aldehydes allowed the reaction equilibrium shift under anaerobic conditions. The main problems of this retro-acyloin strategy are the harsh temperature (180 °C) and the need to perform the reaction under reduced pressure (1 to 100 mbar). To overcome these limitations, enzymatic alternatives requiring mild conditions warrant investigation. A multienzymatic cascade was designed and evaluated for the cleavage of renewable fatty acids of plant origin (e.g., oleic acid, ricinoleic acid) successively catalyzed by alcohol dehydrogenase, Baeyer-Villiger monooxygenase and lipase, leading to the production of long-chain free α,ω-dicarboxylic acids and ω-hydroxycarboxylic acids (C 9 -C 13 ). [START_REF] Song | Multistep Enzymatic Synthesis of Long-Chain α,ω-Dicarboxylic and ω-Hydroxycarboxylic Acids From Renewable Fatty Acids and Plant Oils[END_REF] The enzymatic cleavage of aliphatic -hydroxy ketones to obtain the corresponding aldehydes in one step by a mimetic chemical retro-acyloin process remains a challenge (scheme 2B). The transketolase enzyme (TK) is involved in carbohydrate pathway metabolism. In biochemistry, its is able to catalyse this type of reaction since it uses thiamine diphosphate, the active form of vitamin B1 (natural carbene), as cofactor (Scheme 2A). [START_REF] Turner | Applications of Transketolases in Organic Synthesis[END_REF] Scheme 2. TK-catalyzed reaction with wild type in the pentose phosphate pathway A) and with variants for in vitro -hydroxyketone cleavage B) Incells, TK is engaged in the pentose phosphate pathway where this enzyme reversibly transfers a two-carbon unit, an α-hydroxy carbonyl (ketol) group from a ketose phosphate to an aldose phosphate.. Previous in vitro studies for biocatalytic applications showed that nonphosphorylated ketose and aldose compounds could also be used as TK substrates. In particular, the ketoacid hydroxypyruvate (HPA) as donor confers a major advantage on the TK-catalyzed reaction by rendering the reaction irreversible through the release of carbon dioxide. Thus, TK appears as a powerful enzyme for the asymmetric synthesis of -hydroxy ketones and related acyloins(Scheme 3). 17 When HPA was used as donor, the TK enzyme exhibited the best activities for nonphosphorylated short carbon chain -hydroxy aldehydes (C 2 -C 3 ). Regarding the TK stereoselectivity, , the newly-formed asymmetric carbon of the product has an (S) configuration.

In addition, TK preferentially accepts (2R)-hydroxy aldehydes, yielding (3S, 4R)-hydroxy ketones . For syntheticpurposes , mesophilic TKs from Saccharomyces cerevisiae (TK sce ) and Escherichia coli (TK eco ) have been widely used [START_REF] Zhang | Catalytic Activation of Carbohydrates as Formaldehyde Equivalents for Stetter Reaction with Enones[END_REF] and optimized by mutagenesis. [START_REF] Galman | α′-Dihydroxyketone Formation using Aromatic and Heteroaromatic Aldehydes with Evolved Transketolase Enzymes[END_REF][START_REF] Ranoux | Enhancement of the Substrate Scope of Transketolase[END_REF] We discovered and charaterized the first thermostable TK from Geobacillus stearothermophilus (TK gst ), displaying high stability at elevated temperature and robustness toward non-usual reaction conditions. [START_REF] Abdoul Zabar | Thermostable Transketolase from Geobacillus stearothermophilus: Characterization and Catalytic Properties[END_REF][START_REF] Abdoul Zabar | Engineering a Thermostable Transketolase for Unnatural Conversion of (2S)-Hydroxyaldehyde[END_REF][START_REF] Ali | Polyol Synthesis Catalyzed by a Thermostable Transketolase Immobilized on Layered Double Hydroxides in Ionic liquids[END_REF] The TK gst substrate specificity and enantiopreference were improved and broadened using rational mutagenesis approach toward hydroxylated or non-hydroxylated aldehydes as acceptors [START_REF] Lorillière | Evolved Thermostable Transketolase for Stereoselective Two-Carbon Elongation of Non-Phosphorylated Aldoses to Naturally Rare Ketoses[END_REF][START_REF] Yi | A Thermostable Transketolase Evolved for Aliphatic Aldehyde acceptors[END_REF] and aliphatic -ketoacids as donors. [START_REF] Saravanan | Donor Promiscuity of a Thermostable Transketolase by Directed Evolution: Efficient Complementation of 1-Deoxy-D-ylulose-5phosphate Synthase Activity[END_REF][START_REF] Casajus | Enzymatic Synthesis of Aliphatic Acyloins Catalyzed by Thermostable Transketolase[END_REF] We showed in particular that engineered TK gst , [START_REF] Casajus | Enzymatic Synthesis of Aliphatic Acyloins Catalyzed by Thermostable Transketolase[END_REF] like TK eco , 18d facilitated the synthesis of aliphatic -hydroxy ketones (C 5 -C 10 ) from combinations of aliphatic acceptor and and donor substrates.

Based on these last results, we studied the reversibility of the TK reaction (mimetic of retroacyloin condensation) for the cleavage of aliphatic-hydroxy ketones with appropriate TK gst variants, never hitherto explored to our knowledge (scheme 1B). This work is part of a research program on the valorization of vegetable oil derivatives in which these compounds will be used as model structures of fatty α-hydroxy ketones employed in the chemical retro-acyloin condensation reported earlier. [START_REF] Deruer | Thiazolylidene-Catalyzed Cleavage of Methyl Oleate-Derived α-Hydroxy Ketone to the Corresponding Free Aldehydes[END_REF] Here, we describeappropriate TK gst variants obtained by mutagenesis applied at targeted positions of the active site, the identification of best TK gst variants for the cleavage of aliphatic mono-,di-and tri-α-hydroxy ketones of various carbon chain length, and the quantification and characterization of the products.

RESULTS AND DISCUSSION

Construction of TK gst variant libraries. In the pentose phosphate pathway, -the phosphorylated aldoses (C 3 -C 5 ) and ketoses (C 5 -C 7 ) are the TK acceptor and donor substrates respectively. To improve TK gst activity toward non-phosphorylated -hydroxy ketones 1-7

(Table 1), we modified the TK gst active site at targeted positions by semi-rational mutagenesis (site saturation mutagenesis, SSM). The 3D structures of microbial TKs such as TK sce , [START_REF] Reetz | Addressing The Numbers Problem in Directed Evolution[END_REF] TK ec 28 and TK from Bacillus anthracis (TK ban ) [START_REF] Sundstrom | Yeast TKL1 Gene Encodes a Transketolase That is Required For Efficient Glycolysis and Biosynthesis of Aromatic Amino Acids[END_REF] have a strong protein sequence homology. [START_REF] Abdoul Zabar | Thermostable Transketolase from Geobacillus stearothermophilus: Characterization and Catalytic Properties[END_REF][START_REF] Abdoul Zabar | Engineering a Thermostable Transketolase for Unnatural Conversion of (2S)-Hydroxyaldehyde[END_REF] The key residues stabilizing the TK substrates are identical and have similar orientation to ThDP. The 3D structure of TK gst being unknown, a model of the TK gst active-site pocket containing its phosphorylated acceptor aldose, D-erythrose-4-phosphate (E4P), was constructed using the X-ray crystal structure of TK ban as a template, which belongs to the same microbial species and has a high percentage of identity (74%) (Figure 1). [START_REF] Abdoul Zabar | Thermostable Transketolase from Geobacillus stearothermophilus: Characterization and Catalytic Properties[END_REF][START_REF] Abdoul Zabar | Engineering a Thermostable Transketolase for Unnatural Conversion of (2S)-Hydroxyaldehyde[END_REF] To improve TK gst activity toward the cleavage of (3S)-1,3-dihydroxy ketones 1-5, the residues Leu191, Leu382, Phe435, and Asp470 were individually targeted. SSM applied earlier on these positions showed that D470I, L382F, F435I and L191I were the best variants offering a hydrophobic environment to substrate binding. These variants were used for the formation of these 1,3-dihydroxy ketones 1-5 obtained with hydroxypyruvate as donor and corresponding aliphatic acceptor substrates (Table 1). [START_REF] Yi | A Thermostable Transketolase Evolved for Aliphatic Aldehyde acceptors[END_REF][START_REF] Saravanan | Donor Promiscuity of a Thermostable Transketolase by Directed Evolution: Efficient Complementation of 1-Deoxy-D-ylulose-5phosphate Synthase Activity[END_REF][START_REF] Casajus | Enzymatic Synthesis of Aliphatic Acyloins Catalyzed by Thermostable Transketolase[END_REF] To identify plausible positions for improving TK gst activity toward the cleavage of aliphatic 4or 5-hydroxy ketones 6 or 7, we based our strategy on positions previously found to increase TK gst activity for the formation of the same aliphatic -hydroxy ketones obtained with pyruvate analogs as donors and aliphatic aldehydes as acceptors. Previous results showed that the double variant H102L/H474S was the best variant to improve the substrate affinity toward pyruvate analogs including oxobutyrate. [START_REF] Saravanan | Donor Promiscuity of a Thermostable Transketolase by Directed Evolution: Efficient Complementation of 1-Deoxy-D-ylulose-5phosphate Synthase Activity[END_REF] The presence of smaller and less polar side chains in place of the two histidine allowed enlargement of active site for biger hydrophobic donor substrates and kept the catalytic mechanism intact. The two positions were combined successively with mutations on Asp470, Leu382, Phe435 or Leu191 as the best positions reported earlier for aliphatic acceptor substrates. [START_REF] Yi | A Thermostable Transketolase Evolved for Aliphatic Aldehyde acceptors[END_REF][START_REF] Casajus | Enzymatic Synthesis of Aliphatic Acyloins Catalyzed by Thermostable Transketolase[END_REF] In addition, we selected position L118, which is involved in the stabilization of the thiazolium cycle of ThDP and which could also stabilize the aliphatic substrates by interacting with an appropriate longer carbon chain residue than Leu. For the design of focused TK gst variant libraries, SSM was applied on each of the five positions using NDT codon degeneracy to reduce the screening efforts.screening efforts. Indeed, [START_REF] Reetz | Addressing The Numbers Problem in Directed Evolution[END_REF] this strategy involves 12 codons (in place of 20 with NNK systems) corresponding to 12 amino acids (Phe, Leu, Ile, Val, Tyr, His, Asn, Asp, Cys, Arg, Ser, Gly) which are representatives of polar and nonpolar, aliphatic and aromatic, and negatively and positively charged side chains.Five libraries were thereby created, H102L/H474X/D470X (or L382X, or F435X or L191X or L118X) (Table 1).

Screening of TK gst variant libraries with aliphatic -hydroxy ketones 1-5. To determine the specific activities of TK gst variants (enzymatic lysates or purified enzymes) toward cleavage of aliphatic -hydroxy ketones 1-5, the release of aldehydes 1′′-5′′ was detected by our spectrophotometric assay at 340 nm via consumption of the reduced form of nicotinamide adenine dinucleotide (NADH) and using yeast alcohol dehydrogenase (ADH) as an auxiliary enzyme. [START_REF] Hecquet | New Assays for Transketolase[END_REF] This method offers a continuous measurement of TK initial velocity by shifting the TK gst equilibrium. TK being a transferase, the overall process requires an aldehyde as acceptor substrate. The absence of acceptor drastically decreases TK gst activity (data not shown). The phosphorylated aldose, D-ribose-5-phosphate (D-R5P), the TK acceptor in the pentose phosphate pathway or D-erythrose, one of the best non-phosphorylated acceptors for TK, were chosen as co-substrates because they are not substrates of ADH (Scheme 4). To select the best variants against cleavage of (3S)-di-or trihydroxy ketones 1-5, the single TK gst variants D470I, L382F, F435I and L191I were expressed in E. coli BL21(DE3)pLysS strain and purified by Ni 2+ chelating affinity column chromatography. TK gst variant activities with (3S)-hydroxy ketones 1-3 were compared to those obtaianed with wild-type TK gst (Chart 1).

Chart 1. Determination of TK gst variant activities with (3S)-hydroxy ketones 1-3 and wild-type

 , L382F , D470I , L191I , F435I  1 2 3
Enzyme reactions were carried out in triplicate with L-erythrulose 1, 1,3-dihydroxypentan-2one 2, 1,3-dihydroxyhexan-2-one 3 (50 mM), D-ribose-5-phosphate (2.5 mM), purified enzyme (1-5 mg•mL -1 ),  wt TK gst  L382F,  D470I,  L191I,  F435I at pH 8 and 25 °C.

(3S)-1,3,4-trihydroxybutan-2-one 1 (L-erythrulose), one of the best non-phosphorylated substrates for wild-type TKs was chosen as a reference. [START_REF] Hecquet | New Assays for Transketolase[END_REF] As expected, with wild type and variants, 1 gave the best activities, except with D470I, the only polar residue replaced by an aliphatic chain. We note the strong influence of the nonpolar residue Ile in place of Asp, leading to a marked decrease in activity due probably to the lack of interaction with the hydroxyl group of 1 C4. The mutations of other nonpolar residues by the aliphatic chain had no influence on activity. The (3S)-1,3-dihydroxy ketones 2 and 3 with increasing non-hydroxylated carbon chain The percentages of hits in each library were first estimated on sight by thin layer chromatography (TLC) based on the spot intensity of products 6′ or 7′. We did not observe the formation of 7′ with any TK gst variant libraries, whereas product 6′ was observed on TLC with several TK gst variant lysates (SI). We observed that the library H102L/H474S/L118X gave the greatest number of active TK gst variants. Considering these results, the 22 best hits were sequenced, of which 17 revealed the presence of an Ile in place of Leu118. Ile induces a new position of ThDP favoring the interaction of the unipolar head of 6 with H102L (Figure 2). For further improvement, we set out to create another library keeping L118I and also H102L and creating diversity by SSM on the H474 position, expecting higher activity with a nonpolar residue in place of Ser.

All the best candidates were finally purified by affinity chromatography and their specific activities toward 6 were determined using the spectrophotometric assay described earlier (Chart 2). For comparison purpose, wild-type TK gst , and H102L/H474S TK gst variant were used as controls since they were already described [START_REF] Saravanan | Donor Promiscuity of a Thermostable Transketolase by Directed Evolution: Efficient Complementation of 1-Deoxy-D-ylulose-5phosphate Synthase Activity[END_REF][START_REF] Casajus | Enzymatic Synthesis of Aliphatic Acyloins Catalyzed by Thermostable Transketolase[END_REF] to improve TK gst activity for the formation of C-C with pyruvate homologues as donors and aliphatic acceptors.

Chart 2. Determination of the TK gst variant activities with (±)-4-hydroxyhexan-3-one 6 compared to wild type.

Enzyme reactions were carried out in triplicate with 6 (200 mM), D-erythrose (50 mM), purified enzyme (5 mg . mL -1 ), at pH 8 and 25 °C.

While wild-type TK gst displayed no activity toward 6, all variants containing Ile in place of Leu at 118 position gave higher activities compared to H102L/H474S showing the cooperative effect afforded by this mutation for the C3-C4 bond cleavage of 6. As expected, the results (Chart 2)

showed that replacing His 474 by a nonpolar residue such as Gly (H102L/H474G/L118I) notably improved activity (7, 5, 3, and 1,7 times higher than that observed with H102L/H474S, H102L/L118I, H102L/H474A/L118I and H102L/H474S/L118I respectively).

The hydrophobic environment given by Gly and the two other aliphatic residues favors the stabilization of the nonpolar head (ethyl group) of 6 (Figure 3). The nonpolar Ala or Val residues in place of His 474 (H102L/H474A/L118I and H102L/H474V/L118I) gave lower activities compared to Gly owing to their higher steric hindrances, which impede the interaction of the nonpolar head of 6 with H102L. The polar and/or charged Ser and Glu in place of H474 (H102L/H474E/L118I) also led to lower activities compared to Gly, explained by the polar side chain of Ser and by a negative charge of Glu in conflict with D470. This environment destabilizes the position of the nonpolar head of 6 toward H102L.

Preparative scale study of aliphatic -hydroxyketone cleavage with the best TK gst variants

The cleavage reactions of (3S)-1,3-dihydroxyhexan-2-one 3 and (±)-4-hydroxyhexan-3-one 6 was investigated at preparative scale in the presence of glycolaldehyde. This cheap, non-volatile, and highly reactive co-substrate is the most plausible candidate for further aliphatic aldehyde 3 " or 6 " separation by in situ distillation of reaction mixture.

The reactions were carried out at 50 °C and at pH 7.5 with purified WT and variants identified earlier (Table 2). All substrates, reagents and variants were fully stable at 50 °C. The reactions were regularly monitored by 1 H NMR analysis with TSP-d4 as reference. We followed the disappearance of 3 and 6 concomitantly with the appearance of products 3′ and 6′ to prove the transfer of the hydroxyacetyl group of 3 or acyl group of 6 on the glycolaldehyde (Table 2). The in situ quantification of aldehyde products 3′′ and 6′′ would be less reliable because of their volatility. To shift the equilibrium toward the products, four equivalents (200 mM) of -hydroxy ketones 3 or 6 were required against one equivalent (50 mM) of glycolaldehyde.

The TK gst variant F435I identified earlier at analytical scale as the best candidate showed a total recovery of activity after 5 days at 50 °C and 7 fold higher activity than at 25 °C, thus allowing appreciably shorter reaction times. The in situ 1 H NMR study of the reaction with 3, the best substrate for this variant (Chart 1) showed the conversion of 92% (46 mM) of glycolaldehyde after 2 h and the formation of product 3′ with an in situ yield of 92% (46 mM), demonstrating the cleavage of the C2-C3 bond of (3S)-1,3-hydroxyhexan-2-one 3 by F453I (Table 2).

The triple variants H102L/H474 (S, G, or A) /L118I were selected to cleave (±)-4hydroxyhexan-3-one 6. As expected, considering the lower specific activities obtained previously (Chart 2 compared to Chart 1), a total or almost total conversion of glycolaldehyde was obtained after 24 h. a Enzyme reactions were carried out with purified enzyme (1 mg•mL -1 with 3 and 5 mg•mL -1 with 6), and incubated from 2 h to 24 h at pH 7.5 and 50 °C. b Concentrations determined by in situ NMR (SI). c Enantiomeric excess (ee) determined by chiral GC analysis after derivatization (SI). 20a H102L/H474S/L118I gave the best results with 90% yield of product 6′ (45 mM) against 82% (41 mM) with H102L/H474A/L118I and 78% (39 mM) with H102L/H474G/L118I, whereas the conversion of glycolaldehyde was total. This can be explained by the self-condensation of glycolaldehyde giving L-erythrulose already reported and observed on 1 H NMR spectra (SI).

This by-product had not previously been identified with F435I and 3, the reaction time being shorter. It can be noted that we did not observe the compound corresponding to the condensation of the ketol group of L-erythrulose which could act as TK gst donor substrate on propanal 6''. In addition, the results revealed that H102L/H474S/L118I was slightly more efficient that H102L/H474 (G or A) /L118I, in line with the thermostability study of these variants giving a higher thermostability over time (at 50 °C) compared to H102L/H474 (G or A) / L118. The molecular modeling of the H102L/H474S/L118I active site showed that the presence of Ser in position 474 favors the interaction with Asp 470, which seems important to stabilize the bridge between Asp 470 from chain A or B and Arg 93 from chain B or A, explaining the greater thermostability than H102L/H474 (G or A) /L118. The importance of a nonpolar residue at the 474 position (particularly Gly or Val) in place of Ser showed a weaker influence on product 6′ formation when reactions were conducted at 50 °C for a long reaction time compared to the effect of these mutations on specific activities measured earlier at analytical scale at 25 °C.

Finally, (±)-4-hydroxyhexan-3-one 6 being racemic, the stereoselectivity of the three triple variants were determined. The enantiomeric excess of product 6′ (ee > 95 %) revealed the presence of one predominant enantiomer, suggesting a high stereoselective control of the C3 formation by TK gst variants. Given the stereoselectivity of H102L/H474S reported earlier in the reverse reaction (formation of aliphatic -hydroxy ketones), [START_REF] Casajus | Enzymatic Synthesis of Aliphatic Acyloins Catalyzed by Thermostable Transketolase[END_REF][START_REF] Reetz | Addressing The Numbers Problem in Directed Evolution[END_REF] a 3S configuration is expected.

We also investigated the removal of aldehyde product by continuous distillation of the reaction mixture heated at 50°C under reduced pressure. This experiment was performed with H102L/H474S/L118I from (±)-4-hydroxyhexan-3-one 6 at 4 mmole scale, this compound being cheap and commercially available compared to (3S)-1,3-dihydroxyhexan-2-one 3 which was obtained in lower quantities by enzymatic synthesis. [START_REF] Casajus | Enzymatic Synthesis of Aliphatic Acyloins Catalyzed by Thermostable Transketolase[END_REF] In addition, considering that this work is taking part of a research program on the valorization of vegetable oil derivatives, aliphatic hydroxyketone 6 was chosen as it is more structurally related to fatty α-hydroxyketones than 3.

After 48h of reaction time, only a very small quantity of aldehyde 6" was recovered, the boiling points of 6 and 6" being close (64°C and 49°C respectively). Nevertheless, the other product 6'

was purified by extraction with diethyl ether and obtained with 8 % yield, without any optimization. The compound 6' was characterized by NMR and GC analysis given an enantiomeric excess of 95% (SI).

CONCLUSION

This study shows for the first time that rationally designed TK gst variants can cleave carboncarbon bond of aliphatic -hydroxy ketones 1, 2, 3, 6 with a carbon chain from 4 to 8 atoms to release the corresponding products. The simple variant F435I displayed high activity with (3S)-1,3-dihydroxyhexan-2-one 3 leading to the formation of product 3′ with 92% yield after only 2 h reaction time. The combination of L118I with the double-site mutation H102L/H474S previously identified for improving activity toward pyruvate homologues for synthetic purposes [START_REF] Saravanan | Donor Promiscuity of a Thermostable Transketolase by Directed Evolution: Efficient Complementation of 1-Deoxy-D-ylulose-5phosphate Synthase Activity[END_REF] gave three new triple variants TK gst H102L/H474 (S, G and A) /L118I that were found to cleave (±)-4hydroxyhexan-3-one 6 to give product 6′ with 90, 82 and 79% yield respectively, while wildtype and H102L/H474S were respectively ineffective and only weakly effective. The improvement of TK gst activity through rational design enables us to propose an enzymatic onestep reaction which offers an attractive alternative to other multi-enzyme or chemical strategies developed for the cleavage of aliphatic -hydroxy ketones with longer carbon backbone lengths so far. by.

Experimental Section

General. All chemicals were purchased from Sigma-Aldrich, Alfa-Aesar, TCI Chemicals, and CarboSynth, Bio-Rad and QIAGEN. A QuikChange Lightning site-directed mutagenesis kit (Agilent, USA) was used to construct the TK gst variant libraries. Oligonucleotides synthesis and gene sequencing were performed by Eurofins Genomics (Ebersberg, Germany). The GenElute™ Plasmid Miniprep kit or with the GenElute™ Plasmid Midiprep kit (Sigma-Aldrich) was used for Plasmid DNA extraction. Chiral (3S)-1,3-hydroxy ketones 1 was purchased from Sigma 2, 3, 4 and 5 were synthesized using the procedure described in the literature. [START_REF] Casajus | Enzymatic Synthesis of Aliphatic Acyloins Catalyzed by Thermostable Transketolase[END_REF] (±)-hydroxy ketones 6 or 7 were purchased from TCI Europe. NMR spectra were recorded in D 2 O or CDCl 3 on a 400 MHz Bruker Avance III HD spectrometer. GC analyses were performed on a ThermoScientificTM Trace GC Ultra instrument coupled with a Trace DSQ mass spectrometry detector (scanning between m/z 30 and 400). Enantiomeric excess was determined using a RestekTM chiral column RtTM-bDEXsm (length 30 m, inside diameter 0.25 mm, film thickness 0.25 µ).

Site-directed mutagenesis. The QuikChange XL II site-directed mutagenesis kit (Agilent Technologies)was used . Single variants (D470I, L382F, F435I and L191I) and triple variants (H102L/H474S/D470I or H102L/H474S/L382F or H102L/H474S/F435I or H102L/H474S/L191I) were previously described. [START_REF] Casajus | Enzymatic Synthesis of Aliphatic Acyloins Catalyzed by Thermostable Transketolase[END_REF] For the construction of triple variant NDT libraries, the double variant TK gst H102L/H474S 25 was used as template using appropriate oligonucleotides (SI). The PCR products were transformed by heat shock into XL-10 Gold competent cells which were cultured in Luria-Bertani (LB) broth medium containing 30 μg . mL - 1 of kanamycin overnight. Plasmids were extracted and sequenced to evaluate the quality of mutagenesis and to check the correct substitutions. E. coli BL21(DE3)pLysS competent cells were used for TK gst protein expression.

Expression of TK gst . E. coli BL21(DE3)pLysS strain was used for TK gst expression. This strain was transformed by heat shock with the TK gst plasmids expressing synthetic wild-type TK gst , [START_REF] Song | Multistep Enzymatic Synthesis of Long-Chain α,ω-Dicarboxylic and ω-Hydroxycarboxylic Acids From Renewable Fatty Acids and Plant Oils[END_REF] µg . mL -1 kanamycin. The plates were sealed with breathable sealing film to prevent evaporation, and then incubated at 37 °C, 900 rpm overnight. Aliquots of 50 µl were added from each well to 96-deepwell plates containing 800 µl per well of LB-kanamycin growth medium. The plates were sealed with breathable sealing film and then incubated at 37 °C, 900 rpm for 3 h. 20 µL of IPTG solution (4 mM) was added to each well to start the induction phase.. The plates were sealed with breathable sealing film and then incubated at 30 °C, 900 rpm for 21 h. After centrifugation at 3000 rpm for 20 min, the recovered cell pellets were stored at -80 °C for at least 30 min. For cell lysis, the cell pellets were added in 50 µl of buffer (1/10 BugBuster solution (Novagen), 0.5 mg . mL -1 lysozyme (Roth)) and 4 U.mL -1 benzonase endonuclease (Novagen), and incubated in a shaker for 1.5 h at 37 °C. The cell lysate was then centrifuged at 3000 rpm for 1 h and stored at 4 °C before screening.

Purification of TK gst (wild type and variants).

The pellets were intoduced in 35 mL of phosphate buffer (50 mM) containing NaCl (300 mM) at pH 8.0 and then disrupted by sonication on ice for 30 min. After centrifuging (14,000 rpm at 4 °C, 15 min), Insoluble pellets were discarded. Crude extracts containing His6-tagged TKs were purified using Ni-NTA column equilibrated with phosphate buffer. After washing with phosphate buffer, the elution of His6tagged TKs was performed with phosphate buffer (50 mM) containing NaCl (300 mM) and imidazole (300 mM) at pH 8.0. The fractions containing the eluted proteins were collected and dialyzed against triethanolamine buffer (2 mM, pH 7.5) and then against water (pH 7.5) through dialysis tubing (cutoff 14,000 g•mol -1 ) at 4 °C. Finally, the protein solutions were lyophilized.

The Bradford method using bovine serum albumin (BSA) as the standard was used for the determination of protein concentration. The specific activity of lyophilized wild-type and variant TK gst were measured at 25 °C and gave 0.01 U.mg -1 to 1. . mL -1 ) were added in a well of a 96-well multiplate and adjusted to 250 µL with with HEPES buffer (pH 8, 50 mM). The disappearance of NADH was followed by spectrophotometry at 340 nm (value of ε NADH at 340 nm is 6220 M -1 . cm -1 ). All measurements were performed in triplicate.

Screening of TK gst variants with TLC. Screening reactions were performed in 96-well microplates at room temperature and 400 rpm overnight. First, a premix containing 10 mL of (4S,4R)-hydroxyhexan-3-one 6 (500 mM), 2 mL of glycolaldehyde (625 mM), 250 µL of MgCl 2 (100 mM), 250 µL of ThDP (10 mM) and 2.5 mL of distilled water was prepared and pH was adjusted to 7.5 before use. In a well, the reaction was started by adding 68 µL of premix with 45 µL of cellular lysate. After one night of incubation, product 6′ was visualized by TLC using p-anisaldehyde stain (solvent system: ethyl acetate/cyclohexane 9/1, R f 6′ = 0.44).

Determination of the best TK gst variant activities.

Protocol with dihydroxy ketones (2, 3, 4 and 5). One unit of TK gst activity was defined as the amount of enzyme that catalyzes the formation of 1 µmol of -hydroxy ketone product per minute at 25 °C in HEPES buffer (50 mM, pH 8). TK gst enzymatic assay was carried out with dihydroxy ketone (2, 3, 4 or 5) and DR5Pleading DS7Pand the corresponding aldehyde (propanal, butanal, pentanal or hexanal respectively). ADH was used to reduce the released aldehyde to the corresponding alcohol with NADH. 50 µL dihydroxy ketone (250 mM), 50 µL D-R5P (12.5 mM), 10 µL ThDP (2.5 mM), 10 µL MgCl 2 (25 mM), 10 µL NADH (10 mg . mL -1 ), 10 µL of ADH (25 U.mL -1 ) and 10 µL TK gst solution (1-5 mg.mL -1 ) were added in a well of a 96-well microplate and adjusted to 250 µL with HEPES buffer (pH 8, 50 mM). The disappearance of NADH was followed by spectrophotometry at 340 nm (value of ε NADH at 340 nm is 6220 M -1 .cm - 1 ). All measurements were performed in triplicate.

Protocol with (±)-4-hydroxyhexan-3-one 6. One unit of TK gst activity was defined as the amount of enzyme that catalyzes the formation of 1 µmol of-hydroxy ketoneproduct per minute at 25 °C in HEPES buffer (50 mM, pH 8). TK gst enzymatic assay was carried out with (4S,4R)-hydroxyhexan-3-one 6 and D-erythrose as acceptor leading to the corresponding ketose and propanal. ADH was used to reduce the released propanal to propanol with NADH. 110 µL (4S,4R)-hydroxyhexan-3-one 6 (250 mM), 50 µL D-erythrose (250 mM), 10 µL ThDP (2.5 mM), 10 µL MgCl 2 (25 mM), 10 µL NADH (10 mg . mL -1 ), 10 µL of ADH (25 U . mL -1 ) and 50 µL TK gst solution (5 mg . mL -1 ) were added in a well of a 96-well multiplate and adjusted to 250 µL with HEPES buffer (pH 8, 50 mM). The disappearance of NADH was followed by spectrophotometry at 340 nm (value of ε NADH at 340 nm is 6220 M -1 .cm -1 ). All measurements were performed in triplicate. GC assay method for enantiomeric excess determination. Samples of compounds 6 and 6′

General

were trimethylsilylated before injection using N,O-bis(trimethylsilyl)trifluoroacetamide (BSTFA) with 1% of trimethylchlorosilane (TMCS). First, 250 µL of reaction mixture was filtered using Centricon (10 min, 14,500 rpm, cutoff 30,000 g.mol -1 ) to remove the enzyme, and the filtrate was then mixed with 250 µL of chloroform to extract 6 and 6′ from the aqueous phase. Finally, the organic layer was mixed with 75 µL of the BSTFA mixture and with 1% of TMCS. Samples were stirred for 30 s and incubated at 60 °C for 15 min. After cooling, samples were directly injected. The carrier gas was helium (1 mL . min -1 ). The injector temperature was

Scheme 3 .

 3 Scheme 3. Reaction catalyzed by TK for the synthesis of -hydroxy ketones from -ketoacid as

Figure 1 .

 1 Figure 1. Model of wild-type TK gst active site based on the X-ray crystal structure of TK ban

Scheme 4 .

 4 Scheme 4. Screening assay to determine TK gst activity toward cleavage of -hydroxy ketones 1-

  length showed lower activities than 1 with wild type and variants except with F435I and 3. In this last case, (3S)-1,3-dihydroxyhexan-2-one 3 gave the best TK gst improvement compared to wild type (18 fold) owing to the presence of Ile in place of Phe, allowing an interaction with the C6 of 3 (Figure2).

Figure 2 . 3 .

 23 Figure 2. Model of F435I TK gst with (3S)-1,3-dihydroxyhexan-2-one 3 based on the X-ray

Figure 3 .

 3 Figure 3. Model of H102L/H474S/L118I TK gst with (±)-4-hydroxyketone 6 based on the X-ray

  4 U.mg -1 of total protein (SI). SDS-PAGE with Precision PlusProteinTM All Blue Standards#161-0373(10-250 kDa, BioRad) as standard was used to determine the purity and molecular mass of the samples revealed with Coomassie Blue G-250 (SI).Determination of WT TK gst specific activity. One unit of TK gst activity was defined as the amount of enzyme that catalyzes the formation of 1 µmol of -hydroxy ketone product per minute at 25 °C in HEPES buffer (50 mM, pH 8). TK gst enzymatic assay was carried out with Lerythrulose and D-ribose-5-phosphate (DR5P) leading to D-sedoheptulose-7-phosphate (DS7P) and glycolaldehyde. The yeast alcohol dehydrogenase (ADH) was used to reduce the released glycolaldehyde to ethylene glycol with nicotine adenine dinucleotide reduced form (NADH). 50 µL L-erythrulose (250 mM), 50 µL D-R5P (12.5 mM), 10 µL ThDP (2.5 mM), 10 µL MgCl 2 (25 mM), 10 µL NADH (10 mg. mL -1 ), 10 µL of ADH (25 U . mL -1 ) and 10 µL TK gst solution(1 mg

  procedure of TK gst (wild type and variants)-catalyzed cleavage of -hydroxy ketones. In a 2 mL Eppendorf tube were mixed 800 µL of (di)-hydroxy ketones 2, 3 and 6 (500 mM) with 80 µL of glycolaldehyde (1.25 M), 20 µL MgCl 2 (100 mM), 20 µL ThDP(10 mM) and 680 µL of distilled water. pH was adjusted to 7.5 and 400 µL of TK gst WT or variants (25 mg . mL -1 ) were added to start the reaction. Reactions were monitored by TLC (solvent system: ethyl acetate/cyclohexane 9/1, stain : p-anisaldehyde) and 1 H NMR. The concentration of substrates and products were determined by in situ 1 H NMR.

Table 1 .

 1 Selected positions in the TK gst active site for the improvement of TK gst activity toward

	-hydroxyketone 1-7 cleavage.

Table 2 .

 2 Results of TK gst wild type (WT) and variant-catalyzed cleavage of -hydroxy ketones 3 and 6 in the presence of glycolaldehyde (GoA) .

  TK gst L382F, TK gst D470I, TK gst F435I, TK gst L191I, TK gst H102L/H474S, TK gst

	200 rpm. Isopropyl β-D-1-thiogalactopyranoside (IPTG) at 0.5 mM was added when the OD
	600 nm range reached 0.6-0.8 A.U. Cells were then grown at 30 °C, 200 rpm overnight. After
	centrifugation (8000 rpm, 4 °C for 15 min), the recovered cells were washed twice with
	phosphate buffer NaH 2 PO 4 .2H 2 O (50 mM), NaCl (300 mM) at pH 8.0 and finally harvested
	(4,000 rpm, 4 °C for 15 min). The recovered bacterial pellets were stored at -25 °C (≈ 5 g of wet
	bacterial pellet . L -1 of biomass).
	Expression of TK gst libraries. Each clone was transferred with a replicator from glycerol
	stock to individual wells in 96-well plates containing 200 µl per well of LB growth media and 30
	H102L/H474S/L382F, TK gst H102L/H474S/D470I, TK gst H102L/H474S/F435I or TK gst
	H102L/H474S/L191I. These strains were stored at -80 °C in glycerol 60% (10% final). Apre-
	culture (20 ml), containing kanamycin (30 µg.mL -1 ) grown at 27 °C, 200 rpm for 12 h, was used
	to inoculate 1 liter of culture medium containing kanamycin (30 µg.mL -1 ) grown at 37 °C,
	20

atoms to release the corresponding aldehydes. SYNOPSIS

set at 200 °C. 2 µL aliquots were injected with a split ratio of 1:30. Initial column temperature was 80 °C, held for 10 min. The temperature was then ramped to 90 °C at 1 °C. min -1 , to 230 °C at 5 °C. min -1 , and then held for 5 min. Franck Charmantray : 0000-0001-8638-8349 Nicolas Duguet : 0000-0002-9425-7450 Wolf-Dieter Fessner : 0000-0002-9787-0752 Present Addresses † Univ Lyon, Université Claude-Bernard Lyon 1, CNRS, INSA-Lyon, CPE-Lyon, Institut de Chimie et Biochimie Moléculaires et Supramoléculaires (ICBMS), F-69100 Villeurbanne (France) ‡ Institut für Organische Chemie und Biochemie, Technische Universität Darmstadt, Alarich-Weiss-Strasse 4, 64287 Darmstadt (Germany) ACKNOWLEDGMENTS This work was funded by the Auvergne Rhône-Alpes Region through the "Pack Ambition Recherche" VALCOUPENZ -ID 1701105201-61617 (grant to N.D and L.H) and by ERA CoBioTech TRALAMINOL -ID: 64 (grant to W.D.F and L.H.). Computations were performed on the supercomputer facilities of the Mésocentre Clermont Auvergne Université.

AUTHOR INFORMATION

Corresponding Author *Laurence.hecquet@uca.fr *Franck.chamantray@uca.fr ORCID Laurence Hecquet : 0000-0003-2971-5686

Author Contributions

All the authors jointly wrote the article. All the authors have given their approval to the final version of the manuscript.


Sébastien Bougleux
email: bougleux@ceremade.dauphine.fr

Gabriel Peyré
email: peyre@ceremade.dauphine.fr

Laurent Cohen
email: cohen@ceremade.dauphine.fr

Anisotropic Geodesics for Perceptual Grouping and Domain Meshing

This paper shows how computational Riemannian manifold can be used to solve several problems in computer vision and graphics. Indeed, Voronoi segmentations and Delaunay graphs computed with geodesic distances are shaped according to the anisotropy of the metric. A careful design of a Riemannian manifold can thus help to solve some important difficulties in computer vision and graphics. The first contribution of this paper is thus a detailed exposition of Riemannian metrics as a tool for computer vision and graphics. The second contribution of this paper is the use of this new framework to solve two important problems in computer vision and graphics. The first problem studied is perceptual grouping which is a curve reconstruction problem where one should complete in a meaningful way a sparse set of curves. Our anisotropic grouping algorithm works over a Riemannian metric that propagates the direction of a sparse set of noisy incomplete curves over the whole domain. The proposed method prunes the Delaunay graph in order to correctly link together salient features in the image. The second problem studied is planar domain meshing, where one should build a good quality triangulation of a given domain. Our anisotropic meshing algorithm is a geodesic Delaunay refinement method that exploits a Riemannian metric in order to locally impose the orientation and aspect ratio of the created triangles.

Geodesic Distance over Riemannian Manifold

Various important problems in computer graphics and computer vision require the integration of a local anisotropy over a complex planar domain. This local anisotropy is dictated by directional features such as curves or textures that should be exploited to perform sampling, segmentation, grouping or meshing. In this paper, we focus on two representative problems in these fields: perceptual grouping of salient features and meshing of a complicated planar domain.

The proposed approach encodes the local anisotropy within a tensor field that corresponds to a Riemannian metric. This local information is integrated into global constraints thanks to the geodesic distance over this Riemannian domain. Figure 1 shows some examples of Riemannian metrics frequently encountered in computer vision and graphics. This section reviews basis facts about Riemannian manifolds and explains how geodesic distances can be computed efficiently with Fast Marching methods.

Riemannian Metric and Geodesic Distance

This paper considers 2D Riemannian manifolds that are defined over a compact planar domain Ω ⊂ R 2 . At each point x ∈ Ω, one has a tensor H(x) ∈ R 2×2 which is a positive symmetric matrix. This tensor field defines a local metric that allows to measure the length of a piecewise smooth curve γ : [0, 1] → Ω as follows

L(γ) def. = 1 0 γ ′ (t) T H(γ(t))γ ′ (t)dt.
In image processing and computer vision, the manifold is often the image domain Ω = [0, 1] 2 equipped with a metric H derived from the image f to process. In computer graphics and numerical analysis, one often deals with complicated planar domains Ω with holes and corners. Figure 1 shows some examples of Riemannian metrics. An important issue is thus to design, for a specific application, a tensor field H(x) that encodes the important information about the problem to solve. Sections 3 and 4 show computer vision and graphics applications where the tensor field is derived either from a background image or from some user input.

At each location x ∈ Ω, the Riemannian tensor can be diagonalized

H(x) = λ 1 (x)e 1 (x)e 1 (x) T + λ 2 (x)e 2 (x)e 2 (x) T with 0 < λ 1 λ 2 , (1)
and e 1 , e 2 are two orthogonal (un-oriented) eigenvector fields. A curve γ passing at location γ(t) = x with speed γ ′ (t) has a shorter local length if γ ′ (t) is collinear to e 1 (x) rather than to another direction. Hence shortest paths (to be defined next) tends to be tangent to the direction field e 1 .

The anisotropy of a metric H(x) is defined as

α(x) = λ 1 -λ 2 λ 1 + λ 2 = 2 √ ab -c 2 a + b ∈ [0, 1], for H(x) = a c c b . (2)
A metric with α(x) close to 1 is highly directional near x, whereas a metric with α(x) = 1 is locally isotropic near x.

Geodesic Distances and Shortest Paths

The geodesic distance between two points is defined as

∀ (x, y) ∈ Ω 2 , d(x, y) def. = min γ∈P(x,y) L(γ) (3)
where P(x, y) denotes the set of piecewise smooth curves joining x and y P(

= {γ \ γ(0) = x and γ(1) = y} .

The distance d turns the domain Ω into a metric space. By carefully designing a local Riemannian metric, one can create global interactions within the domain that are crucial to solve computer vision and graphics problems such as perceptual grouping or planar domain meshing.

A shortest path γ between two points (x, y) ∈ Ω 2 according to the Riemannian metric is called a geodesic. It satisfies L(γ) = d(x, y). If the metric H is well chosen, then geodesic curves can be used to follow salient features in images.

In order to perform the numerical computation of geodesic curves and distances, we fix a set of starting points S = (x k ) k ⊂ Ω and define the distance map to this set

∀ x ∈ Ω, U S (x) def. = min k d(x, x k ).
An important theoretical result is that if x → H(x) is continuous, U S is the unique viscosity solution of the following Hamilton-Jacobi non-linear PDE

||∇ x U S || H(x) -1 = 1 with ∀ k, U S (x k ) = 0, (4)
where ||v|| A = √ v T Av. This classical result allows to replace the optimization problem (3) that defines geodesic distances by a partial differential equation.

Once the distance map U S has been computed by solving the Eikonal equation (4), one can extract a geodesic γ between a point x and its closest point x k ∈ S by the following gradient descent

γ ′ (t) = - H(γ(t)) -1 ∇ γ(t) U S ||H(γ(t)) -1 ∇ γ(t) U S || with γ(0) = x. (5)
The geodesic curve γ extracted using this gradient descent is parameterized with unit speed since ||γ ′ || = 1. In particular if S = {x 1 } then one can compute the geodesic curve between x 1 and any point in Ω.

Figure 2 shows examples of geodesic curves computed from a single starting point S = {x 1 } in the center of the image Ω = [0, 1] 2 and a set of points on the boundary of Ω. The geodesics are computed for a metric H(x) whose anisotropy α(x) (defined in equation ( 2)) is decreasing, thus making the Riemannian space progressively closer to the Euclidean space.

Image f α = 1 α = .5 α = .1 α = 0 Fig. 2.
Examples of geodesics for a tensor metric with an increasing anisotropy α (see equation [START_REF] Sethian | Level Sets Methods and Fast Marching Methods[END_REF] for a definition of this parameter). The tensor field H(x) is computed from the structure tensor of f as defined in equation [START_REF] Cohen | Multiple Contour Finding and Perceptual Grouping Using Minimal Paths[END_REF], its eigenvalues fields λi(x) are then modified to impose the anisotropy α.

For the particular case of an isotropic metric H(x) = W (x) 2 Id x , the geodesic distance and the shortest path satisfy

||∇ x U S || = W (x) and γ ′ (t) = - ∇ x U S ||∇ x U S || . ( 6
)
This corresponds to the Eikonal equation, that has been used to compute minimal paths weighted by W [START_REF] Cohen | Global Minimum for Active Contour models: A Minimal Path Approach[END_REF].

Numerical Computations of Geodesic Distances

In order to make all the previous definitions effective in practical situations, one needs a fast algorithm to compute the geodesic distance map U S . The Fast Marching algorithm, introduced by Sethian [2] is a numerical procedure to efficiently solve in O(n log(n)) operations the discretization of equation ( 6) in the isotropic case. Several extensions of the Fast Marching have been proposed in order to solve equation ( 4) for a generic metric, see for instance Kimmel and Sethian [START_REF] Kimmel | Computing Geodesic Paths on Manifolds[END_REF] for triangulated meshes and Spira and Kimel [START_REF] Spira | An efficient solution to the eikonal equation on parametric manifolds[END_REF], Bronstein et al. [START_REF] Bronstein | Weighted distance maps computation on parametric three-dimensional manifolds[END_REF] for parametric manifolds.

We use the Fast Marching method developed by Prados et al. [START_REF] Prados | Control theory and fast marching methods for brain connectivity mapping[END_REF], which is a numerical scheme to compute the geodesic distance over a generic parameteric Riemannian manifold in 2D and 3D in O(n log(n)) operations. As any Fast Marching method, it computes the distance U S by progressively propagating a front, starting from the initial points in S. Figure 3 shows an example of Fast Marching computation with an anisotropic metric. The front propagates faster along the direction of the texture. This is because the Riemannian tensor is computed following equation ( 8) in order for the principal direction e 1 to align with the texture patterns.

Fig. 3. Examples of anisotropic front propagation (from 9 starting points). The colormap indicates the values of the distance functions at a given iteration of the algorithm.

The metric is computed using the structure tensor, equation [START_REF] Cohen | Multiple Contour Finding and Perceptual Grouping Using Minimal Paths[END_REF], of the texture f shown in the background.

Computational Geometry with Riemannian Metric

This section, shows how several tools from computational geometry extend to the setting of a Riemannian metric.

Starting from a set of points S = {x i } m i=1 , one can define graphs and triangulations that reflect the geometry of the Riemannian manifold. These points and the corresponding graphs are the basic building blocks of the algorithms for perceptual grouping and planar domain meshing.

In the following, the boundary ∂Ω of Ω is assumed to be a set of closed smooth curves. At least one point of S is located on each curve, and these boundary points segment ∂Ω as a set of sub-curves

∂Ω = i,j θ i,j with θ i,j ∈ P(x i , x j ) θ i,j ∩ S = {x i , x j }.
(one can have x i = x j if there is only one point on a curve).

Ω Ω c x 1 x 2 x 3 x 4

Delaunay and Voronoi Graphs

The segmentation of the domain Ω in Riemannian Voronoi cells is

Ω = C 0 xi∈S C i where C i = {x ∈ Ω \ ∀ j = i, d(x i , x) d(x j , x)} . (7)
The outer Voronoi cell is defined as C 0 = Closure(Ω c ). The Delaunay graph D S of S is a graph where two points are connected if their respective Voronoi cells are adjacent

(x i , x j ) ∈ D S ⇐⇒ C i ∩ C j = ∅.
To each Delaunay edge (x i , x j ) ∈ D S corresponds a double point w i,j , which is the closest point to x i and x j on the common Voronoi cell boundary

w i,j = argmin x∈Ci∩Cj d(x, x i ).
The Delaunay graph D S is a planar graph, where each Delaunay edge (x i , x j ) ∈ D S has a geometric realization that is defined as if one of x i or x j is not on ∂Ω, the curve (x i , x j ) is the union of the two geodesics joining the double point w i,j to x i and x j , if x i ∈ ∂Ω and x j ∈ ∂Ω, the curve (x i , x j ) is the portion of the boundary θ i,j ⊂ ∂Ω. If the sampling S of Ω is dense enough, then one can prove that D S is a valid planar triangulation, see [START_REF] Leibon | Delaunay triangulations and Voronoi diagrams for Riemannian manifolds[END_REF].

A triple point s i,j,k is located at the intersection of three adjacent Voronoi cells

s i,j,k ∈ C i ∩ C j ∩ C k .
The set of triple points of S is denoted as Σ S . For a generic set of points S, these triple points are isolated. Saddle points s i,j,k where one of the index i, j or k is 0 are located on the boundary ∂Ω of the domain.

Numerical Computation

The Voronoi segmentation [START_REF] Leibon | Delaunay triangulations and Voronoi diagrams for Riemannian manifolds[END_REF] can be computed in parallel to the computation of the geodesic distance map U S . This requires to track the index of the closest point in S of any point in the front during the Fast Marching propagation. This can be done with any algorithm mentioned in section 1.3.

During this propagation, one can keep track of the double points w i,j for each Delaunay edge (x i , x j ) ∈ D S . Such a point corresponds to the first meeting location of the two fronts emanating from both x i and x j . After the propagation is finished, the Delaunay edge curve (x i , x j ) is extracted by solving two gradient descents, equation [START_REF] Bronstein | Weighted distance maps computation on parametric three-dimensional manifolds[END_REF], to compute the two geodesics joining w i,j to x i and x j .

All the geometric tools (Delaunay graph, Voronoi cells, double and triple points) can thus be extracted using a single front propagation, which requires O(n log(n)) operations. Furthermore, when they have been computed for a set of points S, they can be extended to S ∪ {x m+1 } by a local propagation restricted to the cell C m+1 , thus requiring on average O(n log(n)/m) operations.

Figure 4 shows examples of Voronoi segmentation for Riemannian metrics with a decreasing anisotropy. One can see how the Voronoi cells C i are stretched along the main tensor direction e 1 for highly anisotropic metrics.

Application to Perceptual Groupping

Perceptual grouping is a curve reconstruction problem where one wants to extract a curve from an image containing a sparse set of curves embedded in noise. This problem is relevant both to model good continuation perception laws and to develop efficient edge detection methods. In this paper we restrict ourselves to the detection of a set of non-intersecting open or closed curves, although other kinds of topological or regularity constraints could be enforced.

Our algorithm extends the isotropic geodesic grouping method of Cohen [8] by designing a Riemannian metric that propagates the anisotropy of the sparse curves to the whole domain. This metric helps to disambiguates difficult situations where some curves are close from one to each other. This allows a better reconstruction with less user intervention.

The idea of using anisotropic information to perform perceptual grouping is introduced in [START_REF] Guy | Inferring global perceptual contours from local features[END_REF] where the completed contours are local minimizers of a saliency field. Many variational definitions of perceptual contours have been proposed using local regularity assumption, for instance with the ellastica model of Mumford [START_REF] Mumford | Elastica and computer vision[END_REF]. In contrast, our completed contours are anisotropic shortest paths that connect Riemannian Voronoi cells, thus being the global minimum of a length criterion.

Design of an Anisotropic Tensor Field

The Riemannian metric H(x) needs to be computed from the noisy input image f . In order to compute robustly the local direction of the features, we use a local pooling of the gradient information that constitutes a sparse set of tensors with a confidence measure. This sparse tensor field is then integrated by diffusion into a dense field.

Structure Tensor. The local orientation of a feature around a pixel x is given by the vector orthogonal to the gradient v(x) = (∇ x f ) ⊥ , which is computed numerically with finite differences (using maybe some little smoothing to cancel noise). This local direction information can be stored in a rank-1 tensor T 0 (x) = v(x)v(x)

T . In order to evaluate the local anisotropy of the image, one needs to average this tensor

T (x) = T 0 * G σ (x) (8)
where the 4 entries of the tensor are smoothed against a gaussian kernel G σ of width σ > 0. The metric H corresponds to the so-called structure tensor, see for instance [START_REF] Kothe | Edge and junction detection with an improved structure tensor[END_REF]. This local tensor T is able to extract both the local direction of edges and the local direction of textural patterns (see figure 4, left). Another option, that we do not pursue here, is to use the square of the Hessian matrix of f instead of the structure tensor.

Estimation of a dense tensor field. The structure tensor field T (x) defined in [START_REF] Cohen | Multiple Contour Finding and Perceptual Grouping Using Minimal Paths[END_REF] gives a robust estimation of the local anisotropy only close to image features, where the gradient is large. In homogenous areas (typically outside the salient features), the tensor is nearly isotropic with small eigenvalues. In order to have a dense anisotropic field, one needs to extend the anisotropy over the whole domain using some kind of interpolation. This notion of interpolation of local orientation is similar to the computation of good continuation field, as studied for instance in stochastic completion fields [START_REF] Williams | Stochastic completion fields: a neural model of illusory contour shape and salience[END_REF] or tensor voting [START_REF] Medioni | A Computational Framework for Segmentation and Grouping[END_REF].

In this paper, we propose a simple interpolation method that computes a dense tensor field with a linear diffusion outside a region of high confidence. The region of high saliency is computed by thresholding the anisotropy map α of T (x) defined in equation ( 2)

Ω α = {x ∈ Ω \ α(x) > 1 -ε} ,
where ε is a small constant, in the numerical examples we choose ε = .05. The orientation of the tensors in Ω α are computed with a high confidence, and in order to compute a dense tensor field, we compute the following steady state of a heat diffusion

∀ x / ∈ Ω α , ∆T d i,j (x) = 0 and ∀ x ∈ Ω α , T d i,j (x) = T i,j (x),
where ∆ is the Laplacian with Neumann reflecting conditions on the boundary ∂Ω and where (T i,j ) i,j=1,2 are the components of the structure tensor.

In order to turn the dense tensor field T d into a Riemannian metric, we apply a non-linear mapping to its eigenvalues,

T d (x) = µ 1 e 1 e 1 T + µ 2 e 2 e 2 T =⇒ H(x) = ψ 1 (µ 1 )e 1 e 2 T + ψ 2 (µ 2 )e 2 e 2 T . (9)
where ψ i is a decreasing function. In this paper, we use ψ i (x) = (η + |x|) -1 for a small value of η.

Figure 5 shows examples of computation of dense tensor fields.

Anisotropic Perceptual Grouping

Our anisotropic grouping algorithm proceeds by computing a perceptual graph DS of a set of points S provided by the user. This perceptual graph is a sub-graph of the Delaunay graph DS ⊂ D S .

The use of the anisotropic metric H(x) defined in ( 9) helps to reduce the user intervention to a minimum by grouping together only curves that obey a good continuation property with respect to the anisotropic tensor field. This is performed by ordering the edges (x i , x j ) ∈ D S with respect to their respective geodesic distance d(x i , x j ). The edges are progressively inserted as at the corresponding curves are closed or open but non-intersecting. This topological constraint is enforced by monitoring the current degree δ i of each point, which should be smaller or equal to 2. The algorithm is detailed in Table 1. Table 1: Anisotropic perceptual grouping algorithm.

Figure 6 compares the results of perceptual grouping using an isotropic metric (which is equivalent to the algorithm developed in [START_REF] Cohen | Multiple Contour Finding and Perceptual Grouping Using Minimal Paths[END_REF]) to our algorithm that uses the dense tensor field T d . The isotropic method fails because closed curves are connected regardless of their relative orientation. In contrast, our anisotropic metric enables a correct grouping of curves that obey a good continuation property. varying sizes are highly desirable because of their capability to represent efficiently functions with directional singularities that one encounters for instance in parabolic PDE's near shocks. This section proposes a new anisotropic meshing algorithm that conforms the shape and size of the triangles to the tensor field of a Riemannian metric. Classical planar domain meshing algorithms are based on Euclidean Delaunay triangulation. They proceed by progressively inserting triple points, which are circumcenters of Delaunay triangles, see for instance [START_REF] Ruppert | A Delaunay refinement algorithm for quality 2-dimensional mesh generation[END_REF][START_REF] Chew | Guaranteed-quality mesh generation for curved surfaces[END_REF]. These points are inserted in order to split triangles that are poorly shaped, and also to ensure a minimum size of the triangles.

Application to Meshing of Planar Domains

Anisotropic meshes can be built using a local modification of the metric [START_REF] Borouchaki | Delaunay mesh generation governed by metric specifications[END_REF] or anisotropic elastic forces [START_REF] Bossen | A pliant method for anisotropic mesh generation[END_REF] and bubble packing [START_REF] Yamakawa | High quality anisotropic tetrahedral mesh generation via ellipsoidal bubble packing[END_REF]. In order to incorporate global constraints within a provably correct Delaunay refinement scheme, Labelle and Shewchuk propose in [START_REF] Labelle | Anisotropic voronoi diagrams and guaranteed-quality anisotropic mesh generation[END_REF] to use the anisotropic distance d(x, y) 2 = (xy)

T H(x)(xy). This is a simplified anisotropic metric not based on shortest paths. Several issues arise with such a definition. It is not symmetric and the Voronoi diagram might not be connected. However it can be used in conjunction with Ruppert's Delaunay refinement algorithm to provide anisotropic triangulation with guaranties on the aspect ratio of the triangles. This algorithm is extended in 3D by [START_REF] Boissonnat | Anisotropic diagrams: Labelle shewchuk approach revisited[END_REF] and to domains with curves by [START_REF] Yokosuka | Guaranteed-quality anisotropic mesh generation for domains with curves[END_REF]. The simplified distance d has been applied to image sampling [START_REF] Feng | Anisotropic noise samples[END_REF], optimal samples placement with centroidal tesselations [START_REF] Du | Anisotropic centroidal Voronoi tessellations and their applications[END_REF] and surface remeshing [START_REF] Valette | Generic remeshing of 3d triangular meshes with metric-dependent discrete Voronoi diagrams[END_REF].

Anisotropic geodesic meshing. We propose a new Delaunay refinement algorithm that extends the algorithm of Ruppert [START_REF] Ruppert | A Delaunay refinement algorithm for quality 2-dimensional mesh generation[END_REF] with Riemannian metrics. It extends the isotropic farthest point seeding strategy of [START_REF] Peyré | Geodesic remeshing using front propagation[END_REF] with anisotropic metrics and domains with arbitrary boundaries.

Our anisotropic meshing algorithm proceeds by iteratively inserting a triple point s i,j,k ∈ Σ S to an already computed set of points S. In order to compute an anisotropic mesh with triangles of high quality with respect to the local metric, one inserts s i,j,k for a Delaunay triangle (x i , x j , x k ) with the smallest circumradius to shortest edge ratio

ρ(s i,j,k ) = d(s i,j,k , x i ) min(d(x i , x j ), d(x j , x k ), d(x k , x i ))
,

which is a quantity computed for each triple point in parallel to the Fast Marching propagation. In the Euclidean domain, a triangle (x i , x j , x k ) with a low value of ρ(s i,j,k ) is badly shaped since its smallest angle is close to 0 . As explained in [START_REF] Labelle | Anisotropic voronoi diagrams and guaranteed-quality anisotropic mesh generation[END_REF], this property extends to an anisotropic metric H(x) if angles are measured using the inner product defined by H(x).

The major obstacle to the removing of badly shaped triangles is that the boundary ∂Ω should always be part of the Delaunay graph. This can be broken if a point x k is located too close to the boundary. A boundary sub-curve θ i,j is said to be encroached by x k ∈ S if it exists a triple point w i,k,0 ∈ θ i,j , see figure 7. Such an encroached edge cannot be part of the Delaunay triangulation, and is automatically split by the algorithm by inserting a mid point. Similarly, triple points are not added if they encroach any boundary sub-curve (the sub-curve is subdivided instead).

Another difficulty is that the Delaunay graph D S of S is not necessarily a valid triangulation if the sampling S is not dense enough, see [START_REF] Leibon | Delaunay triangulations and Voronoi diagrams for Riemannian manifolds[END_REF]. This is because of some isolated point, that is connected to only one other point of S in D S . The algorithm automatically add points on the Voronoi cell boundary of such a point. Table 2 details this algorithm. A bound ρ ⋆ on ρ enforces the refinement to reach some quality criterion, while a bound U ⋆ enforces a uniform refinement to match some desired triangle density. -If in S ∪ {s ⋆ }, s ⋆ encroaches some θi,j ⊂ ∂Ω, subdivide: S ← S ∪ argmax x∈θ i,j US (x).

-Otherwise, add it: S ← S ∪ s ⋆ . Update US with a local Fast Marching. 5. Stop: while ρ(s ⋆ ) > ρ ⋆ or US (s ⋆ ) > U ⋆ , go back to 2.

Conclusion

This paper has detailed how several notions from computational geometry extend seamlessly to the geodesic setting. This allows to incorporate some important knowledges about the directionality of the features to solve more efficiently problems in computer vision and graphics. We explore two particular applications, perceptual grouping and domain meshing, where this anisotropy allows to overcome several limitations of previous approaches.

Fig. 1 .

 1 Fig. 1. Examples of Riemannian metrics (top row) and geodesic distances and curves (bottom row). The blue/red colormap indicates the geodesic distance to the starting point. From left to right: euclidean (H(x) = Id2 restricted to Ω = [0, 1] 2 ), planar domain (H(x) = Id2 restricted to Ω = [0, 1] 2 ), isotropic (H(x) = W (x)Id2 with W computed from the image), Riemannian manifold metric (H(x) is the structure tensor of the image, see equation (8)) and 3D surface (H(x) corresponds to the first fundamental form).

1 Fig. 4 .

 14 Fig. 4. Examples of anisotropic distances (top row) and Voronoi diagrams (bottom row) with an decreasing anisotropy α. The metric tensor is computed using the structure tensor, equation (8).

Fig. 5 .

 5 Fig. 5. Computation of a dense tensor field T d (x) from a sparse structure tensor T (x).

1 . 2 .

 12 Initialization: set DS ← ∅, and Π ← DS . Select edge: set (xi, xj) ←argmin (x,y)∈Π d(x, y). Remove it: Π ← Π -{(xi, xj)}.

3 .

 3 Check topology: if δi < 2 and δj < 2, then update DS ← DS ∪ {(xi, xj)} and set δj ← δj + 1 and δi ← δj + 1. 4. Stop: while Π = ∅, go back to 2.

Fig. 6 .

 6 Fig. 6. Comparison of isotropic and anisotropic Riemannian metric for perceptual grouping.

3 Fig. 7 .

 37 Fig. 7. Left: the vertex x3 encroaches the boundary curve θ1,2. Right: the vertex x3 does not encroach anymore because (x1, x2) is a Delaunay edge.

Figure 8

 8 Figure 8 show an example of anisotropic meshing, where the user controls the shape of the triangle by designing the tensor field.

Fig. 8 .

 8 Fig. 8. Example of anisotropic meshing with a synthetic tensor field.

1 .

 1 Initialization: set S with at least one point on each curve of Ω, compute US with a Fast Marching. 2. Boundary enforcement: while it exists θi,j ⊂ ∂Ω encroached by some x k ∈ S, subdivide: S ← S ∪ argmax x∈θ i,j US (x). Update US with a local Fast Marching. 3. Triangulation enforcement: while it exists (xi, xj) ∈ DS with xi or xj isolated, insert w * = argmax

	d(xi, w).
	w∈C i ∩C j
	4. Select point: s ⋆ ← argmin	ρ(s).
	s∈Σ S -∂Ω

Table 2 :

 2 Anisotropic planar domain meshing algorithm.