Ruth Vanderschueren

Jasmien Doevenspeck

Florence Helsen

Sandra Mounicou

Jakob Santner

Jan Delcour

Eduardo Chavez

Erik Smolders

Jan A Delcour D

Cadmium migration from nib to testa during cacao fermentation is driven by nib acidification

Keywords:

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

INTRODUCTION

In 2019, the European Commission enforced a regulation which sets the maximum allowed cadmium (Cd) concentrations in cacao-derived consumer products (European Commission 2014). Similar limits were also approved by the Codex Alimentarius (Codex Alimentarius Commission 2018). These regulations have a large impact on the cacao industry, especially in Latin America where soils naturally contain elevated Cd concentrations. Larger Cd concentrations have been reported in cacao beans and chocolates from Latin America, compared to other geographical origins such as Africa (Bertoldi et al.2016;[START_REF] Vanderschueren | The elemental composition of chocolates is related to cacao content and origin: A multi-element fingerprinting analysis of single origin chocolates[END_REF]. A recent extensive meta-analysis [START_REF] Vanderschueren | Mitigating the level of cadmium in cacao products: Reviewing the transfer of cadmium from soil to chocolate bar[END_REF]) demonstrated the impact of the Cd regulation on cacao farmers in Latin America. More than 50% of compiled bean Cd data exceeded the unofficial industry threshold for EU export (0.60 mg Cd kg -1 ). The regulation has prompted researchers to study mitigation strategies to lower the Cd concentrations in cacaoderived products. While research thus far focused mostly on the use of soil amendments [START_REF] Ramtahal | The effectiveness of soil amendments, biochar and lime, in mitigating cadmium bioaccumulation in Theobroma cacao L[END_REF]) and selection of cultivars for reduced Cd uptake and translocation [START_REF] Lewis | Genetic variation in bioaccumulation and partitioning of cadmium in Theobroma cacao L[END_REF][START_REF] Engbersen | (EC) No 1881/2006 as regards maximum levels of cadmium in foodstuffs[END_REF], the potential of postharvest mitigation has also been pointed out [START_REF] Meter | Cadmium in cacao from Latin America and the Caribbean -A review of research and Potential Mitigation Solutions[END_REF].

The effect of fermentation on bean Cd concentrations was recently demonstrated by [START_REF] Vanderschueren | The impact of fermentation on the distribution of cadmium in cacao beans[END_REF], their results indicated that Cd migrates from the nib to the testa and the mucilage during fermentation. This migration decreased the nib Cd concentration by up to a factor 1.3, but only occurred with sufficient nib acidification, i.e. nib pH <5. This migration is counterintuitive since Cd concentrations in the testa are typically higher than in the nib Cd [START_REF] Ramtahal | Relationships between cadmium in tissues of cacao trees and soils in plantations of Trinidad and Tobago[END_REF][START_REF] Lewis | Genetic variation in bioaccumulation and partitioning of cadmium in Theobroma cacao L[END_REF][START_REF] Vanderschueren | The impact of fermentation on the distribution of cadmium in cacao beans[END_REF], and migration thus occurs against the total Cd concentration gradient. The authors speculated that the concentration gradient of mobile Cd is opposite to the total concentration gradient, especially in more acid, fermented beans.

The objective of this study is to better understand the causes of Cd migration during cacao fermentation. More specifically, this study was set up (i) to reveal whether nib acidification is the driving force behind Cd migration during cacao fermentation; and (ii) to identify whether artificial acidification during fermentation using organic acids typically formed during fermentation (acetic and lactic acid) can be used to lower the Cd concentration in the final product. The work consisted of two lab-scale fermentation experiments with different organic acid treatments, and one micro-fermentation experiment established inside a full-scale commercial fermentation unit. The idea behind the lab-scale fermentation experiments is that the effects of acidity on mobile and total nib Cd concentrations can be disentangled from the other effects in the fermentation process, such as heat-induced cell breakdown or the effects of ethanol penetration. A novel method was adopted to map Cd mobility at sub mm resolution. An imprinting technique was used followed by Laser Ablation Inductively Coupled Plasma Mass Spectrometry (LA-ICP-MS) analysis of the imprint. This visualisation method allows to test the hypothesis that the mobile Cd concentration gradient in cacao beans is opposite to the total Cd concentration gradient.

MATERIALS AND METHODS

Lab-scale fermentations with organic acid treatments

Experimental setup and sampling

Two separate lab-scale (5 kg) fermentation experiments were executed in a greenhouse in Guayaquil (Ecuador) with administration of lactic and acetic acid before, during, or at the end of fermentation (Table 1). Lab-scale fermentations were executed in planter pots with a flat cone shape (small diameter 17.5 cm, large diameter 23.5 cm, height 22.5 cm). Three to five holes in the bottom of each planter pot ensured drainage of fermentation sweatings. Cacao beans (Nacional cultivar) were provided by local farmers in Manabí and Guayas provinces (Ecuador).

Prior to setting up each fermentation experiment, 60 kg fresh beans were deposited on a plastic sheet and homogenised manually before dividing them among the planter pots (5 kg each). The planter pots were placed inside Styrofoam boxes to retain heat. The fermentation lasted 5 (labscale fermentation A) or 6 days (lab-scale fermentation B). The cacao mass was mixed manually on day 2 in experiment A, and on days 2 and 5 in experiment B. Treatment doses, treatment time and total fermentation time for both experiments are shown in Table 1. To apply the treatments, the cacao beans were spread out in plastic trays and treatment solutions were applied using spray bottles. The application of acids before fermentation and on the second fermentation day logically disturbed the fermentation process and associated temperature increase. Deionised water treatments were included on days 0 and 2 to assess the effect of liquid application on the overall course of fermentation. Those water treatments allowed to disentangle the effect of liquid treatment from the acidification effect. The treated cacao beans were mixed manually to ensure an even distribution of the treatment components and were redeposited in the plastic planter pots.

Samples containing 500 g beans were taken after mixing, at the start of fermentation, at the end of fermentation, and before treatment. Samples of the first experiment (A) thus included day 0, day 2, day 4 for the cacao treated that day, and day 5; samples of the second experiment (B) included day 0, day 5 and day 6. The mucilage was removed manually using paper towels, after which the intact beans were oven-dried for 72 hours at 65 °C. After drying, beans were peeled to separate nib and testa, and both fractions were ground using an electric coffee grinder.

Water-extractable Cd

The Cd mobility was assessed by measuring Cd concentrations in deionised water extracts of the nibs. Ground nib samples were sieved to obtain a more homogenous particle size distribution (800 µm test sieve, VWR International). Sieved material (0.50 g) was incubated with 4.0 mL deionised water in an end-over-end shaker at 20 °C for 5 days. After incubation, samples were centrifuged (15 minutes at 2000 g, Heraeus Multifuge X3R, Thermo Scientific, Waltham, MA, USA). The supernatant was filtered through a 0.45 µm syringe filter and filtered samples were diluted 200 times prior to Inductively Coupled Plasma Mass Spectrometry (ICP-MS) analysis (described below).

Additionally, the effect of artificial acidification on Cd mobility was determined using waterextractable Cd as a proxy. To this end, ground nib and testa material from the blank treatments of experiment A (unfermented samples at day 0 and fermented samples at day 5) were used.

Duplicate ground materials (0.50 g for nibs and 0.25 g for testa) were incubated in 12.0 mL deionised water, either or not acidified to pH 4.5 with hydrochloric acid (HCl, 37% w/w). The pH of the contact solutions was adjusted prior to incubation and remained unchanged after incubation (<0.1 pH unit difference). Samples were incubated in an end-over-end shaker, centrifuged, and their supernatant was filtered and diluted prior to ICP-MS analysis, as outlined below.

Determining the elemental composition

Subsamples containing 100 mg dry nib or testa material were acid digested in an open digestion block for 8 h in 3.0 mL Normatom ® nitric acid (HNO 3 67-69% w/w, VWR International, Radnor, PA, USA), reaching a maximum temperature of 130 °C. Digests were brought to a volume of 10.0 mL and diluted ten times with Milli-Q water (18.2 MΩ cm -1 ) prior to elemental analysis with ICP-MS (Agilent 7700x, Agilent Technologies, Santa Clara, CA, USA). Cadmium concentrations were measured by monitoring the 111 Cd isotope in helium (He) collision cell mode, using 103 Rh as an online internal standard. The limit of quantification (LOQ) for the analysis was 0.001 mg Cd kg -1 dw. Duplicate samples were included in the digestions at intervals of ten samples and the coefficient of variation (CV) of these duplicates ranged from 0.4 to 13% (average CV 3.5%). Blank samples (in quadruplicate) and certified reference NIST 2384 baking chocolate (in triplicate, certified concentration 0.073 ± 0.008 mg Cd kg -1 ) were included in all digestions and treated identical to the cacao samples. Recoveries of the certified reference material ranged between 87 and 100%. In addition to Cd, several other elements were also analysed (Al, P, K, Ca, Cr, Mn, Co, Ni, Cu, Zn, As, Mo and Pb). Specific parameters regarding the ICP-MS analysis for those elements and a discussion on the quality control can be found in the Supplementary Information (SI).

Temperature and pH

The temperature (T) in the centre of the fermentation boxes was measured daily using a digital thermometer (VWR International). The pH was determined only at sampling times to avoid excessive disturbance of the fermentation process. The pH of the mucilage, not including the testa, was determined in suspension after shaking 25 g beans with mucilage in 100 mL deionised water on an orbital shaker (KS 130 orbital shaker, IKA laboratory equipment, Staufen, Germany)

for 5 minutes at 320 rpm. The nib pH was determined by shaking 5.0 g dried ground material in 20 mL deionised water on an orbital shaker for 5 min and measuring the pH in the supernatant after centrifugation for 25 minutes at 1800 g (MTL-5MS Tabletop Low Speed Centrifuge, Micronlab, Shandong, China).

2.1.5.

Micro-fermentation

Cacao material and setup

The micro-fermentation experiment was carried out at a plantation in Guayas province (Ecuador), using Nacional cultivar cacao. Fifteen ripe cacao pods were opened using a machete.

Half of the beans from each pod were immediately sampled and processed without fermentation.

The other half of the beans from each pod were collected in mesh bags with drawstring closing and placed in the centre of a wooden fermentation box (82 x 65 x 65 cm) filled with Nacional cacao from the same field (225 kg). Fermentation lasted four days and the mass was mixed manually on the second day. The mesh sample bags were kept in the centre of the box throughout the process and were sampled when fermentation was complete (fermented samples) (SI, Figure S 1).

Unfermented and fermented beans were cleaned with paper towels to remove the mucilage, oven-dried for 72h at 65 °C, and peeled to separate nib and testa. Due to the small sample size, testa and nibs of three individual beans per sample were digested separately and their elemental composition was analysed with ICP-MS. The temperature in the centre of the fermentation box was measured daily using a digital thermometer (Custom, CT-422WR, Hitchcock, TX, USA).

Nib and mucilage pH were determined before and after fermentation in the bulk cacao surrounding the micro-fermentation samples, as described above.

Visualisation of Cd mobility using LA-ICP-MS

A novel visualisation method was developed to map mobile Cd in cacao beans at mm resolution, using the principles of the Diffusive Gradient in Thin film technique (DGT) developed by [START_REF] Santner | Two decades of chemical imaging of solutes in sediments and soils -a review[END_REF]. In brief, a transversely cut bean was exposed to a gel that contains a metal ion selective binding agent. The mobile metal ions migrate from the source to the gel. After a given deployment time, the gel is removed, dried, and its surface composition is scanned with LA-ICP-MS.

Metal binding gels containing micro-milled Chelex-100 resin were prepared as described by [START_REF] Zhou | Comparison of Chelex based resins in diffusive gradients in thin-film for high resolution assessment of metals[END_REF], with a resin-to-gel ratio of 4.5% (w/w). Due to the small size of the resin beads (<10 µm), this gel can be used for chemical analysis with high spatial resolution. Mobile Cd was visualised for 5 of the 15 micro-fermentation pods and included two unfermented and two fermented beans from each pod. Immediately after sample collection (either before or after fermentation, not dried), beans were transversely cut in half using a clean razor blade and placed on the gel overnight. A Nuclepore ® membrane (0.2 µm pore size, Whatman Maidstone, UK) was placed in between the beans and the gel as a barrier to avoid particulate transfer. The micro Chelex gel acts as a zero sink for cationic trace metals. It binds mobile Cd ions present in the contact surface with the bean and thereby creates a print of the mobile Cd in the beans. Imprinted Chelex gels were stored in plastic bags with several drops of ultrapure water to keep the gels hydrated. Gels were dried on 0.45 µm Millipore membrane with Whatman chromatography paper as a support (3 mm CHR), in a slab gel dryer (DrygelSr, Hoefer, Holliston, MA, USA) for at least 48 hours. During drying, gels were covered with clean polyethylene plastic to protect the reactive layer. Dried gels were fixed on glass microscopy slides using double sided tape and submitted to LA-ICP-MS analysis.

For elemental analysis, a quadrupole 7700cs ICP-MS (Agilent Technologies) mounted with platinum cones was used. The sensitivity and operational conditions of the ICP-MS (stability, background and mass calibration) were initially checked using a 1.0 µg L -1 Y, Tl, Li, Ba and Ce tuning solution. The ICP-MS was then coupled to a 213-nm laser ablation system equipped with a TV2 cell (NWR213, ESI, Fremont, CA, USA) and coupling was optimised using a NIST 612 glass by monitoring 238 U and 232 Th for maximal sensitivity, and a U to Th ratio as close as possible to the unit (NIST 612 certified concentrations 37.79 ± 0.08 mg Th kg -1 and 37.38 ± 0.08 mg U kg -1 ). Visualisation of Ca, Cd, K, Ni and Zn fixed on micro Chelex gel was performed by running several ablation line scans with a 20 Hz laser shot repetition rate, fluency maintained between 4.1 and 4.5 J cm -2 , a laser beam of 50 µm² and a scan speed of 50 µm s -1 . Ablated material was transported with 800 mL min -1 He and mixed with Ar gas before the ICP torch inlet. The ICP-MS was used in He mode, allowing monitoring of 111 Cd (0.2 s), 114 Cd (0.2 s), 39 K (0.005 s), 44 Ca (0.15 s), 60 Ni (0.1 s), and 64 Zn (0.1 s as integration time).

Several sets of two or three ablation lines (distance between lines 100 µm) were run on each sample (SI Figure S 6). Signal intensities were averaged for each set of ablation lines. To account for the differences in elemental composition between samples and elements, signal intensities measured for each element (counts per second, cps) were divided by the elemental concentration determined in each sample by acid digestion and ICP-MS analysis as described above. Several gels were lost prior to LA-ICP-MS imaging as they were either compromised during storage and transportation or broken during gel drying. In addition, gels that displayed disproportionally low K intensities, i.e. factor 10 lower than the other samples, were excluded from the discussion below. Even though the affinity of the Chelex resin is lower for K compared to the other elements evaluated here (Cd, Ca, Ni and Zn), K concentrations in the nibs are high and K is readily water-extractable (see results below). Therefore, low K intensities were considered as an indication for poor contact between gel and bean, or as an indication that the gel was erratically used upside down, as these gels are typically asymmetric because Chelex resin beads settle to the bottom during gel casting. 2]. Water treatment on day 0 increased the mucilage pH from 4.0 before fermentation to 6.9 on day 5, indicating suboptimal fermentation. Water treatment prior to fermentation likely delayed and/or impeded the outgrowth of yeasts in the first fermentation phase, which reduced the ethanol production. Ethanol is the main substrate for the acetic acid bacteria in the third and final fermentation phase. During this phase, the mucilage pH decreases as these bacteria convert ethanol to acetic acid, and the fermentation temperature increases to >45 °C because this process is strongly exothermic. Thus, reduced production of ethanol may explain the higher final mucilage pH and lower maximum temperature observed in the pre-fermentation water treatment.

At the end of experiment A, the nib pH was significantly lower in the control, and in the acetic acid day 2 and day 4 treatments compared to acetic acid day 0 and water treatments (Tukey's HSD test, P-value < 0.05, Table 2). The acetic acid treatments in experiment A thus did not lower the nib pH below that of the control, and results of the water treatments indicated that liquid application hindered nib acidification with fermentation. This again suggests reduced availability of ethanol substrate for the acetic acid bacteria, as nib acidification mostly results from penetration of acetic acid into the nibs. Acetic acid treatment before fermentation (day 0) immediately decreased the nib pH, but this effect disappeared with time [significant decrease from pH 6.5 ± 0.1 (standard deviation, stdev) to 5.9 ± 0.1 within one hour after treatment, Student's t-test, P-value < 0.05, results not shown). Despite this immediate penetration of the acetic acid into the nibs, pre-fermentation treatments yielded higher final nib pH values compared to control treatments, which was likely related to the effects of the liquid application on the overall fermentation (i.e. less acetic acid production from fermentation activity).

Much larger acid doses were applied in experiment B compared to A (35 and 53 g kg -1 cacao compared to 13 and 15 g kg -1 cacao, Table 1). These treatments were applied only after fermentation and resulted in significantly lower final nib pH values (on average 4.4 for lactic acid and 4.5 for acetic acid treatment) compared to control or water treatments (average nib pH 5.2, Tukey's HSD test, P-value < 0.05, Table 2). A detailed overview of the changes in nib pH with fermentation time can be found in SI Figure S 3. Within the first hour after application, the average nib pH decreased significantly from 5.3 ± 0.2 (stdev) to 4.6 ± 0.1 with lactic acid and from 5.3 ± 0.1 to 4.7 ± 0.1 with acetic acid application (Students t-test, P-value <0.05).

Penetration of both acids into the nib thus occurs quickly and does not require fermentation, as equally fast acid penetration was observed in the pre-fermentation treatment of experiment A. It should be noted that, while both lactic and acetic acid treatments in experiment B resulted in a lower nib pH compared to the blank and water treatments, final nib pH values were still within the range for fermented cacao reported in literature, i.e. pH 4.0-5.5 [START_REF] Schwan | The microbiology of cocoa fermentation and its role in chocolate quality[END_REF][START_REF] Belitz | Coffee, tea, cocoa[END_REF][START_REF] Papalexandratou | Spontaneous organic cocoa bean box fermentations in Brazil are characterized by a restricted species diversity of lactic acid bacteria and acetic acid bacteria[END_REF][START_REF] Vuyst | The cocoa bean fermentation process: from ecosystem analysis to starter culture development[END_REF]. The differences in final nib pH between acid treated and untreated cacao (control or water) in experiment B thus may have been related to less extensive fermentation in these small planter pots.

Effect on tissue Cd concentrations

A detailed overview of nib and testa Cd concentrations as a function of fermentation time can be found in SI Figures S 3 and S 4, a summary is given in Table 2. Nib Cd concentrations were not significantly affected by fermentation (Tukey's HSD test, P-value ≥ 0.05), except for the replicates treated with acetic acid on day 2 in experiment A, where nib Cd decreased from 2.3 ± 0.3 mg Cd kg -1 before fermentation to 2.0 ± 0.1 mg kg -1 after fermentation (day 5) (SI, Figures S 3 and S 4). In both fermentation experiments, no significant differences could be observed in final nib Cd concentrations among the different treatments (Tukey's HSD test, P-value ≥ 0.05, Table 2). However, final testa Cd concentrations did differ significantly among treatments in both experiments. Because the testa represents a much smaller weight fraction than the nib, the total Cd stock in the testa is also smaller. For example, considering the weight fractions of nib and testa from which mucilage had been removed [average weight fractions nib 0.93 and testa 0.07 [START_REF] Vanderschueren | The impact of fermentation on the distribution of cadmium in cacao beans[END_REF]], a cacao bean from the control in experiment A with a total weight of 1 g (nib and testa) would have a total stock of 1.8 µg Cd in the nib but only 0.26 µg Cd in the testa. This explains why outward Cd migration from nib to testa can result in a significant increase in the testa Cd concentration but not in a detectable decrease in the nib Cd concentration. However, patterns were not consistent as testa Cd concentrations increased in some treatments but decreased in others (SI,Figures S 3 and S 4). The inconsistency in the effect of fermentation on testa Cd concentrations may have been related to further migration of Cd toward the mucilage. This hypothesis could not be verified as the mucilage was not collected for elemental analysis in this study.

While both lactic and acetic acid treatments in experiment B resulted in a similar final nib pH of 4.5, testa Cd increased more extensively with acetic acid compared to lactic acid treatment (Table 2). Both acids thus penetrated and acidified the cacao nib, but testa Cd results suggested that only acetic acid treatment resulted in relevant Cd mobilisation. However, this hypothesis was not confirmed by the water extractions, as the water-extractable Cd fractions in the nibs of both lactic and acetic acid treated cacao were similar at the end of fermentation and did not differ from those in blank or water treatments. In experiment A, the total nib and testa Cd concentrations decreased and increased significantly with decreasing nib pH (Pearson correlation test, P-value < 0.05, Figure 2). In experiment B, no significant correlations could be found. Still, a trend of increasing testa Cd concentrations with decreasing nib pH could be discerned, with the acetic acid treated samples standing out in the low pH/high testa Cd area of the graph (Figure 2.B, top left). This relation between tissue Cd and nib pH suggests a pH dependent mobilisation of Cd in the nib tissue. Indeed, the water-extractable nib Cd fraction increased significantly with decreasing nib pH in both experiments (Pearson correlation test, P-value < 0.05). The effects of different fermentation treatments on tissue elemental concentrations and water-extractable nib fractions of Ca, Cu, K, Mn, Ni, P and Zn are reported in SI Table S 1 andFigure S 6. The water extraction of unfermented bean tissues, that were artificially acidified to pH 4.5, indicated an increase of water-extractable nib Cd at lower pH (Figure 3), again suggesting the importance of acidification for Cd migration. Similar effects of acidification were observed for e.g. Mn and Zn (SI, Figure S 7). Water-extractable Cd fractions were much lower in the testa compared to the nib. The increased water-extractability of Cd in the fermented testa may represent the Cd that migrated from nib to testa during fermentation, while the Cd present in this tissue prior to fermentation was less water-extractable (Figure 3).

Micro-fermentation

Effect on tissue Cd concentrations

The temperature in the centre of the box increased from ambient temperature (26 °C) to a maximum of 50 °C on the third day, after which it decreased to 46 °C by day 4. The mucilage pH increased from 3.9 (day 0) to 4.4 (day 4), while the nib pH decreased from 6.3 (day 0) to 4.8 (day 4). Even though all pods were collected from a single field, both nib and testa Cd concentrations varied by more than a factor 3 among the pods (CV 37% for nibs and 31% for testa), while variations among beans within a pod were relatively small (average CV 12% for nibs and 10% for testa) (Figure 4). Similar large variations in bean Cd within a single field were found previously by [START_REF] Argüello | Soil properties and agronomic factors affecting cadmium concentrations in cacao beans: A nationwide survey in Ecuador[END_REF]. They reported an average within-field CV of 39% in a nationwide study in Ecuador. Pairwise analysis of single pod micro-fermentations revealed an effect of fermentation on both nib and testa Cd concentrations. A pairwise t-test indicated a significant decrease in the nib Cd concentration with fermentation by -0.30 mg Cd kg -1 , and an increase in the testa Cd concentration by +0.75 mg Cd kg -1 (P-value <0.05). In contrast, unpaired analysis of the ensemble of data did not detect these changes (SI Table S 2). The nib Cd concentration decreased on average by a factor 1.25, which is similar to the factor 1.3 reported previously [START_REF] Vanderschueren | The impact of fermentation on the distribution of cadmium in cacao beans[END_REF]. Significant decreases in nib elemental concentrations were also observed for Cu (factor 1.3), K (factor 1.3), Ni (factor 1.3) and P (factor 1.1), while testa elemental concentrations increased for Cu (factor 2.9), K (factor 2.5), Mn (factor 2.6), Ni (factor 2.2), P (factor 5.5) and Zn (factor 2.1) (SI Table S 2).

Considering the weight fractions of nib and testa from which mucilage had been removed

[average weight fractions nib 0.93 and testa 0.07 [START_REF] Vanderschueren | The impact of fermentation on the distribution of cadmium in cacao beans[END_REF]], the average mass of Cd lost from the nibs (0.28 ± 0.20 mg Cd kg -1 bean ) was significantly larger than the mass of Cd gained in the testa (0.05 ± 0.03 mg Cd kg -1 bean ). Similar discrepancies were found in the mass balances of Cu, K, Ni and P, i.e. the decrease in mass of these elements in the nib was significantly larger than the mass gained in the testa. This discrepancy in mass balances is likely related to leaching of elements into the mucilage (the liquid phase surrounding the beans).

Indeed, Cd, Cu, K, Ni and P concentrations in the mucilage (among other elements) have been reported to increase with fermentation by a factor 2-8 depending on the element and on the fermentation conditions. It was also reported that fermentation reduced the total bean Cd level by 15%, likely due to leaching of Cd in the mucilage sweatings [START_REF] Vanderschueren | The impact of fermentation on the distribution of cadmium in cacao beans[END_REF].

Visualisation of elemental mobility

While duplicate mobile elemental prints were originally generated before and after fermentation for five micro-fermentation pods, only eight samples are discussed here because several samples had to be excluded, as explained above. The contact area between the beans and the Chelex resin gels could be clearly discerned on dried gels, as the samples stained the gel (Figure 5 and SI Table S 2 andFigure S 6). The maps of the mobile element prints obtained here (signal larger for nib than for testa) were markedly different from the LA-ICP-MS images of total element composition in unfermented cacao beans obtained previously, where the total Cd signal was larger for the testa than for the nib [START_REF] Vanderschueren | The impact of fermentation on the distribution of cadmium in cacao beans[END_REF]. This suggests that fermentation induces an increase in Cd mobility in the nibs but not in the testa, resulting in a mobile concentration gradient opposite to the total Cd concentration gradient. The lack of element signal peaks at the edges of the Chelex imprints was in accordance with the lower water-extractable Cd fraction found in the testa compared to the nib. Imaging with LA-ICP-MS using imprints of cacao beans on a metal chelating gel was used to successfully visualise element mobility in unfermented and fermented cacao beans. As such, the visualisation method was used here to better understand the underlying mechanisms of Cd migration during cacao fermentation, while bulk chemical analysis using acid digestion and ICP-MS (see above) allows to quantify the extent of migration occurring. This novel imaging technique can be employed in the future for the visualisation of element mobility in a variety of sample matrices and may be strengthened by including analysis of chelating gel standards with predetermined Cd concentrations which would allow (semi-)quantitative analysis of the images.

Impact of the experimental setup

The results of the micro-fermentation experiment revealed a factor 1.25 decrease in nib Cd (Figure 4), but no such effect was found in the lab-scale experiments (Table 2). The lack of effect in the lab-scale experiments was likely not related to insufficient acidification. The final nib pH for the control in lab-scale fermentation A (pH 4.6) was even lower than that reached in the micro-fermentation (pH 4.8). The difference in effects between setups may be related to temperatures typically reported in literature [START_REF] Lima | Theobroma cacao , "the food of the gods": Quality determinants of commercial cocoa beans, with particular reference to the impact of fermentation[END_REF]) and lower than the temperature reached in the micro-fermentation (50 °C). The lab-scale temperature profiles showed a delay in initial temperature rise, especially in experiment B where the temperature only started to increase after two days (Figure 1). [START_REF] Jespersen | Occurrence and diversity of yeasts involved in fermentation of West African cocoa beans[END_REF] reported that yeast outgrowth in the outer layers of the fermenting cacao mass is delayed in comparison to that in the centre of the mass. They found that the maximum yeast cell count was reached after 24 hours in the centre, while it was only reached after 72 h in the outer layers. The delay in fermentation activity observed in the labscale fermentations, indicated by the lag phase in the temperature profile, may be related to the small size and large surface-to-volume ratio of the lab-scale vessels. Fermentation temperature may thus explain the lack of decrease in nib Cd concentrations in the lab-scale experiments.

Sufficient heat may be required to allow acidification-driven mobilisation of Cd in cacao nibs during fermentation. Additional research under controlled temperature conditions is required to verify or reject this hypothesis.

The differences in detected effects between both setups may also be related to the large variation in bean Cd. Results of the micro-fermentation experiment demonstrated how large inter-pod variability (up to a factor 3) can mask the effect of fermentation on bean Cd concentrations (Figure 4). The coefficient of variation of the nib Cd concentration between all samples prior to treatment in lab-scale fermentation B was large (CV 13%). A power test was performed to assess the number of replicates in the lab-scale experiments, using the online tool developed by Rollin Brant [University of British Columbia (https://www.stat.ubc.ca/~rollin/stats/ssize/n2.html)]. To account for differences in the initial bean Cd concentrations, the CV was used instead of the standard deviation. Based on that analysis, at least four replicates are required to detect a factor 1 effect with statistical significance (C = 1 %, α = % and power = 8 %) However, the power test also indicated that two replicates per treatment were sufficient to detect the effect in lab-scale fermentation A (CV between all nib samples prior to treatment was only 6%).

Organic acid fermentation treatments as a mitigation strategy for Cd in cacao

The micro-fermentation experiment indicated that nib acidification is the driver for nib Cd mobilisation, and thus also for reducing the final nib Cd concentration. However, organic acid application in lab-scale fermentations did not reduce the nib Cd concentration, probably because of the lower temperatures compared to commercial fermentations. Organic acid treatments reduced the nib pH, but only when applying large doses, and the final nib pH was still within the range of commercial values (pH 4-5). Thus, nib Cd does decrease with decreasing nib pH but organic acid treatments during fermentation may not be the best option to acidify the nibs.

Optimisation of fermentation conditions can likely yield final nib pH values similar to those resulting from organic acid treatments. For example, acetic acid concentrations in mucilage are reported to be higher when turning is performed (Camu et al. 2008a). The impact of such fermentation practices on the flavour should be taken into account, as excessive acetic acid has been related to less pronounced flavour in the final product (Camu et al. 2008b).

CONCLUSION

The

  1. Effect on fermentation parameters Both water and organic acid treatments affected the fermentation temperature. It dropped to ambient temperature (i.e., 30 °C or below) within 24 hours after treatment in both experiments (Figure 1 and SI, Figure S 2). Pre-fermentation treatment (day 0) delayed heat production in the fermenting mass. In fermentation A, this effect was stronger for the acid compared to the corresponding water treatment (SI, Figure S 2). The temperature profiles of fermentation Asuggest that the effect of treatment on temperature was smallest when applying the acids after fermentation, as was performed in fermentation B.The mucilage pH was strongly affected by the treatments as this tissue was in direct contact with the solutions. The final mucilage pH was significantly lower in acid treated cacao compared to control and water treatments in both experiments [Tukey's Honestly Significant Difference (HSD) test, P-value < 0.05, Table

Figure S 8

 8 Figure S 8). This imprint allowed targeting the contact area during analysis. Visualisation of the

  differences in temperature. While small lab-scale fermentations of 5 kg cacao have practical advantages for experimental work, the small fermentation volume offers less heat retention compared to micro-fermentation experiments performed in a box of >200 kg cacao. The temperature in the centre of the lab-scale vessels reached 45 °C in the controls of both experiments (A and B, Figure 1 and SI Figure S 2), which was on the lower end of maximum

  mobility of Cd in cacao nibs increases during fermentation, as indicated by enhanced water extractions and enhanced diffusible Cd identified by LA-ICP-MS imaging. Water-extractable nib Cd was negatively correlated to nib pH and increased through artificial acidification, indicating nib pH as the driving force behind nib Cd mobilisation. Migration of Cd from the nibs to the testa during fermentation is likely related to increased Cd mobility in the nibs due to acidification. Organic acid amendments before or during fermentation were ineffective in reducing the nib pH due to their negative impact on fermentation (lower fermentation temperatures and excessively high mucilage pH). Application of acetic or lactic acid as a postfermentation treatment lowered the nib pH compared to blank and water treatments, but only when applying large doses. Although no significant reductions in nib Cd concentrations could be observed with acid treatment, the reduced nib pH did increase testa Cd concentrations and nib Cd mobility. Micro-fermentations with single pod-derived beans inside a commercial fermentation box indicated a factor 1.25 reduction in nib Cd. Optimisation of fermentation parameters for acetic acid production can likely offer similar nib acidification compared to acid treatments, without the added cost. However, a trade-off needs to be made between nib Cd reduction and flavour quality as excessive nib acidity is considered an off-flavour.

ACKNOWLEDGEMENTS

RV is the recipient of an FWO-SB fellowship (FWO Vlaanderen). JAD receives Methusalem research excellence financing at KU Leuven. Aquitaine Region and Feder are acknowledged for the funding of the laser ablation system from the Aquitrace Project (convention number 20131206001-13010973). The authors thank the Ecuadorian farmers who supplied material and allowed us to use their infrastructure. We also thank Dr. David Arguëllo and Jesse Dekeyrel for assistance with the lab work.


Stéphane G Roux
email: stephane.roux@ens-lyon.fr

Janka Lengyel
email: janka.lengyel@ens-lyon.fr

Patrice Abry
email: patrice.abry@ens-lyon.fr

François Sémécurbe
email: francoissemecurbe@gmail.com

Stéphane Jaffard
email: jaffard@u-pec.fr

Local multifractal analysis of marked spatial point processes

Nous proposons une méthode d'estimation locale des propriétés multifractales. Cette methode, qui repose sur une moyenne d'incrément à une dimension, est applicable à tout type de processus ponctuel marqué pour lesquels l'information n'est pas accessible partout. Nous montrons la robustesse d'estimation sur trois types de supports différents.

Introduction

Scaling or multifractal analysis [1,2,3] is now a standard procedure for estimating local regularities in empirical data. Scale invariance is indeed a property that has been extensively observed in numerous phenomena of very different nature [START_REF]Proc. Symposia in Pure Mathematics[END_REF][START_REF] Sémécurbe | [END_REF]. In general, multifractal analysis consists in measuring scaling exponents, whose values are then commonly involved in various detection, identification or classification tasks. Despite its increasing popularity, existing methods for multifractal estimation are only able to deal with signals or images that are defined everywhere on a full support or where non-existent values are replaced by 0. Even though the latter approach is reliable if the density of missing values is very small, it can not be used for datasets where access to information is irregular, e.g. in the context of geospatial point patterns. Even though an original methodology to derive (multi-) fractal properties of such heterogeneous geographical distributions was recently developed [START_REF] Sémécurbe | [END_REF] it focused exclusively on unmarked points so far. Against this background, we propose a procedure based on an average of one-dimensional increments allowing for the targeted extension of these methods for marked 2D point processes with arbitrary underlying spatial patterns. Building upon a brief overview of the classical multifractal procedure, we introduce the two main contributions of this work : the multiscale quantity specifically tailored for irregular 2D distributions and the local multifractal analysis. We conclude by providing a realworld example using geographical data.

Classical Multifractal Analysis

Multifractal analysis aims at characterizing the local regularity of a process V . Different pointwise regularity exponents can be used, most commonly the Hölder exponent h(x 0 ) > 0. It is defined as the largest α so that there exists, in a neighborhood of x 0 , a constant C > 0 and a polynomial P (of order smaller than α) satisfying |V (x 0 -r) -P (x 0 -r)| < C|r| α . The multifractal spectrum D(h) is defined as the collection of Hausdorff dimensions (dim H ) of the sets of points x, where h(x) takes the values h. D(h) = dim H {x such that h(x) = h}. The shape and the width of D(h) provide information on the intensity of fluctuations in local regularity.

Practical estimation of the multifractal spectrum requires the use of multifractal formalism, a procedure originally inspired by thermodynamics (cf., e.g., [2] ). It is based on estimating the different moments of certain multiscale quantities and observing their evolution across scales. Historically, multiscale coefficients were defined as the increments of the process, however these were later generalized to wavelet coefficients (defined as the high pass filtering of the process [6]) or to some non-linear transformations of these [7,8].

Let T (x, r) denote the multiscale coefficient at point x and scale r. A process V defined on R d is said to possess scaleinvariance or scaling properties if -for some statistical orders q -the time averages of |T (x, r)| q in a fixed scale display power law behaviors with respect to radii r S q (r) = E{|T (x, r)| q } = lim

N →+∞ 1 N N x=1 |T (x, r)| q ∼ F q |r| ζq ,
where N is the total number of points x. The scaling exponents ζ q are related by a Legendre transform to its multifractal spectrum D(h) = min q (d + qH -ζ q ).

If we suppose that ζ q is smooth at 0, we can use a Taylor expansion of ζ q = p≤1 c p q p p! and show that the log-cumulant of multiscale coefficients behaves linearly with respect to scales r. For the two first orders one obtains :

C 1 (r) = E{log |T (x, r)|} = c 1 log(|r|) + d 1 (1) C 2 (r) = E{log |T (x, r)| 2 } -C 2 1 (r) = c 2 log(|r|) + d 2 (2)
which give a quadratic approximation of the singularity spectrum

D(h) ∼ 1 + (h-c1) 2 2c2 .

Local Multifractal Analysis

Here, we propose a novel methodology that is able to measure local scaling properties of random fields. It involves two essential steps : the definition of multiscale quantities and their local weighting functions.

Multiscale coefficient. We start by introducing a multiscale coefficient, defined as the average of all one-dimensional increments originating from x, i.e.

T (x, r) = 1 Nx(r) x ′ , ||x ′ -x||<r (V (x ′ ) -V (x)) (3)
where N x (r) is the number of points in radius r that is centered on x. In a two-dimensional space, this coefficient can be seen as a convolution of the field V with the high-pass filter ψ r (x) = πr 2 δ 0 -1 B0(r) which verifies the admissibility condition R 2 ψ r (x)dx = 0 for all r. This filter can be regarded as a wavelet and the scaling in equations 1, 1 and 2 are still valid as long as 0 < H < 1 [6].

Local weighting function. To carry out the local multifractal analysis, we first introduce a local environment L, that is larger than the radius (or radii) r. Second, we introduce a regular grid of arbitrary size with points x g , which will serve as the focal points for local estimations. Moving on, we compute a weighted average based on the distance between a chosen estimation site x g and the locations of the original points x : where weights are defined as w xg,x = f (||x g -x||/L). We use a quartic (biweight) kernel defined as f (x) = (1 -x 2 ) 2 if ||x|| < 1 and 0 otherwise. For better estimation quality, we restrict our analysis to estimation sites where N xg ( √ 2l) ̸ = 0, l standing for the distance between points x g . Locally we recover the scaling

S q (x g , r) = x w xg,x |T (x, r)| q C 1 (x g , r) = x w xg,x log |T (x, r)| C 2 (x g , r) = x w xg,x (log |T (x, r)| -C 1 (x g , r)) 2 (4)
log S q (x g , r) ∼ ζ q (x g ) log(|r|) + F q (x g ) C 1 (x g , r) ∼ c 1 (x g ) log(|r|) + d 1 (x g ) C 2 (x g , r) ∼ c 2 (x g ) log(|r|) + d 2 (x g ) (5)
Finally, global estimation can be defined by choosing a collection of estimation sites N (x g ) and derive the average over this set of measurement points :

log S q (r) = 1 N xg xg log S q (x g , r) ∼ ζ q log(|r|) + F q C 1 (r) = 1 N xg xg C 1 (x g , r) ∼ c 1 log(|r|) + d 1 C 2 (r) = 1 N xg xg C 2 (x g , r) ∼ c 2 log(|r|) + d 2 (6)

Numerical simulations

In this section, first, the procedure for generating the analyzed 2D marked point processes is described ; second, they are used to assess the performance of the local multifractal analysis introduced in Section 3. Multifractal random walk. The multifractal random walk (MRW) [9,10], is a popular and representative member of the group of multiplicative cascade processes. It constitutes one of the most prominently used class of multifractal models for applied research. More precisely, MRW is a non-stationary, non-Gaussian process with stationary increments. Its multifractal properties are those of the well-known multiplicative lognormal cascade of Mandelbrot. In one dimension, It is defined as k1) , where n is an integer, ω H (k) are the increments of a fractional Brownian

V (0) = 0 and V (n) = n k1=1 ω H (k 1 )e ω(
ζ q = c 1 q -c2 2 q 2 and D(h) = 1 + (1-c1) 2 2c2 with c 1 = H + c 2 .
The two-dimensional isotropic MRW field is obtained using the generalization of the 1D case proposed in [11,12]. For the purposes of this study, we generate a full image of an MRW process of size M = 4096 with parameters H and c 2 . In the followings, this image will be construed as a marked point process with spatial positions at [0, M -1] 2 . Mono-and multifractal support. In order to emphasize general applicability that is independent of underlying point patterns, we construct two Sierpinski carpets using the IFS fractal generator "GenFrac" [START_REF] Genfrac | Fractal Generator[END_REF]. We deploy one (d 0 = 1.76) and then four different (d 0 = 1.56) reduction factors to obtain a monoand multifractal structure, with different capacity dimensions d 0 = -log(N r )/ log(r). Since our methodology may be of interest to research on geospatial systems, it is compelling to observe such rather different point patterns characterized by distinct variations in their point densities.

For each centroid point of the 2D carpet elements, we assign the value of the MRW at the rounded point position. As a result, we obtain three marked point process Γ i = x i , y i , V i , i = 1, 2 or 3. Fig. 1 shows the support structure of the three processes and their assigned MRW values that were generated with two different H exponents (0.7 and 0.3) and a single shared c 2 = 0.04. We note that restricting the fully defined image of the multifractal random field to a fractal set of points may be mathematically ill-defined. However, this does not constitute a problem here as the methodology from Section 3 operates exclusively on a finite collection of scales. Performance assessment. In accordance with (3), we compute the multiscale coefficients at each location x g (see Fig 1). For the purposes of this analysis, we apply ten scales logarithmically spaced between r = 2 2 and 2 7 . A non-weighted linear regression is then carried out at each point x g (spaced with a resolution of 25 pixels), making use of all available scales to obtain ζ 2 (x g ), c 1 (x g ) and c 2 (x g ). Final results form an image of size 162X162 : In Fig. 2, we show the resulting slopes obtained for the cumulant of order one, c 1 (x g ), computed on the processes shown in Fig. 1, whilst Fig. 3 displays the associated cumulant of order two c 2 (x g ).

In general, obtained values do fluctuate locally from point to point but the fluctuations are homogeneous for the three supports. Except from some very sparse areas in the center of the multifractal Sierpinski carpet, the estimation does not significantly depend on its support (as can be clearly seen in Fig. 2). Note that this situation is an additional motivation for computing c 1 (x g ) and c 2 (x g ), i.e. exponents derived from values of D(h) close to the maximum of the spectrum. Indeed, sets of small dimensions might become "invisible" when their trace on a fractal set is considered. Moving on to consider local estimation results, Fig. 2 demonstrates that the range of locally obtained H exponents is evidently distinct for the two generated multifractal random fields : Darker blue and green colors are dominating in the bottom row (H = 0.3) whilst bright green and yellow colors prevail in the top (predefined H = 0.7). In contrast, the same color range is observable for all cases in Fig. 3, as they all share one common intermittency coefficient (c 2 = 0.04). The quality of the regression is very good with a coefficient of determination R 2 for ζ 2 (x g ) above 0.95 for all three point processes : 0.997%, 0.984%, 0.795% for the image, the self-similar and the multifractal carpet respectively. For c 1 (x g ), we get the same quality for the image and the selfsimilar carpet (with 0.993% and 0.917%) but a poorer estimation for the multifractal support with only 0.52% of the points above R 2 > 0.96. For c 2 the quality of the fit degrades strongly.

Finally, the top row in Fig. 4 shows the statistics for the three local slopes, ζ 2 (x g ), c 1 (x g ) and c 2 (x g ), and the bottom line the corresponding global scaling for S 2 (r), C 1 (r) and C 2 (r) as an average over the entire 2D field x g . What can be clearly seen is the strong correspondence between the predefined theoreti-

Application to geographical data

Due to its local nature and broad range of applicability, the methodology may find a number of important applications in settings where information is commonly rather sparse, e.g in the domains of human or environmental geography. In Fig. 5, we deployed the methodology to analyze housing prices ("demande de valeurs foncières" [START_REF]Demande de Valeurs Foncières[END_REF]) in and around the city of Paris, France. Small-scale multifractality of land price is of great potential urban planning and policy relevance and it has already been associated with more classical measures of socioeconomic segregation and inequalities [13,14]. We must emphasize that data distribution is highly irregular, it marks the centroids of cadastral areas where purchases were being made in our observed time window between 2014 and 2020. Notwithstanding, we were able to derive both local parameters, c 1 (x g ) and c 2 (x g ) with a sufficient regression quality : the average R 2 for the entire observed 2D field depicted in Fig. 5 lies at 0.84 for the cumulant of order one c 1 (x g ) and at 0.93 for the cumulant of order two c 2 (x g ).

Conclusions. To conclude, this paper set out to illustrate how the introduced methodology is suitable for the local multifractal analysis of marked spatial point processes whose underlying support structure may be strongly non-uniform. More precisely, it could recover -with sufficient regression qualitycertain predefined characteristic parameters of multifractal random fields, namely the Hurst exponent H and the intermittency coefficient c 2 . It has to be stressed that the latter analysis was possible both on the global and local levels and that estimations were also statistically satisfactory on real-world geospatial data.

FIGURE 1 -

 1 FIGURE 1 -Images showing the same MRW process with a full support (right), a self-similar Sierpinski carpet (middle) and a multifractal Sierpinski carpet. The top row displays a realization of a MRW with H = 0.7 and c 2 = 0.04, the bottom row with H = 0.3 and c 2 = 0.04.

FIGURE 2 -

 2 FIGURE 2 -Local analysis. Cumulant of order 1 (eq. 5) obtained on an image (left), a self-similar (middle) and a multifractal Sierpinski carpet (right). The parameters used are H = 0.7 and c 2 = 0.04 (top) and H = 0.3 and c 2 = 0.04 (bottom).

FIGURE 3 -

 3 FIGURE 3 -Local analysis. Cumulant of order 2 (eq. 5) obtained on an image (left), a self-similar (middle) and a multifractal Sierpinski carpet (right). The parameters used are H = 0.7 and c 2 = 0.04 (top row) and H = 0.3 and c 2 = 0.04 (bottom row).

4 FIGURE 4 -FIGURE 5 -

 445 FIGURE 4 -The first row shows the histogram of the local scaling ζ 2 (x g ) (left), c 1 (x g ) (middle) and c 2 (x g ) obtained from the data on the full support in Fig. 1 left. The second row displays the global average of the scaling, see (6), for the latter three exponents. The parameters are H = 0.7 and c 2 = 0.04 (black symbols) and H = 0.3 and c 2 = 0.04 (blue symbols). The estimation is done for the image (+), the self-similar Sierpinski carpet (•) and the multifractal Sierpinski carpet (⋆).


Dr Yannick Geiger

Dr Stéphane Bellemin-Laponnaz

Non-Linear Effects in Asymmetric Catalysis: Impact of Catalyst Precipitation

Keywords: asymmetric catalysis, non-linear effect, chiral amplification

Non-linear effects between the enantiomeric excesses of both the ligand and the product are ubiquitous phenomena in asymmetric catalysis, allowing asymmetric amplification (or depletion) and are widely used tools for mechanistic investigations. Non-linear effects are caused by catalyst aggregation; however, the effect of catalyst precipitation on NLEs has not been systematically investigated to date, except in special cases such as ternary phase systems. In this article, we show through simulations and with several literature cases at hand how precipitation affects shape and amplitude of NLE curves. The limit of solubility of the homo-or heterochiral dimeric species causes broken-shaped NLE curves or very pronounced NLEs even though the equilibria between the different species in solution are not favourable, at first sight. Peculiar features such as horizontal segments, inverse S-shaped curves and a strong effect of total catalyst concentration are also observed. Overall, this study allows to get a better understanding of chiral catalytic systems and gives an outlook at other types of phase separation leading to NLEs.

Introduction

Asymmetric synthesis is of paramount importance for obtaining enantiopure molecules, in particular in pharmaceutical industry where chiral drugs are almost exclusively applied as single enantiomers. [START_REF] Collins | Chirality in Industry II: Developments in the Commercial Manufacture and Applications of Optically Active Compounds[END_REF] In such catalytic -or stoichiometric -asymmetric syntheses a chiral auxiliary (often a chiral ligand attached to a metal) is used, usually in its enantiomerically pure form in order to maximise the enantiomeric excess of the product (eeP). However, high eeP can also be obtained from non-enantiopure compounds if the ligand ee (eeL) does not scale linearly with eeP as is usually expected (Figure 1a). Deviations from eeP vs eeL-linearity are called non-linear effects (NLE): these can allow high eeP with a scalemic ligand (even higher than the ligand ee itself) if the NLE is positive (Figure 1b, (+)-NLE); in special cases eeP may even increase upon eeL reduction, so that the maximum eeP is not obtained with the enantiopure but with a non-enantiopure ligand (Figure 1c, hyperpositive NLE). In contrast, a negative non-linear effect (Figure 1d, (-)-NLE) leads to a depleted eeP. NLEs, [START_REF] Girard | [END_REF][3][4][5] first rationalized by Kagan and co-workers, [6,7] are caused by aggregation of the chiral complexes. This work follows studies on the influence of a compound's enantiomeric purity on its physical properties by Horeau and Guetté [8] and on its chemical reactivity in stoichiometric reactions by Wynberg and Feringa in the mid-70's. [9] It should also be noted that already in 1936 Langenbeck and Triem reported a NLE in the double addition of menthol to oxalyl chloride, along with the conceptual basics for asymmetric amplification and an outlook to catalytic reactions. [10,11] Probing for NLEs has become a tool frequently used for mechanistic investigations in asymmetric catalysis, especially if other analytical tools failed. [12] In addition, the ability of (+)-NLEs to amplify the ee of the chiral source is believed to play a key role in the emergence of biological homochirality and therefore feeds the debate on the origin of life. [13] To date, models for NLEs consider mostly aggregation levels of 2, as in the Noyori model [14,15] (catalytically active monomers in equilibrium with inactive dimers), Kagan's ML2 model [7] (active dimers and inexistent monomers) and our recent monomer-dimer competition model [16][17][18][19] (both monomers and dimers are present and catalytically active). The source of nonlinearity relies in the relative values of the dimerization constants coming into play, which may lead to non-statistical distributions of the different species in solution, as well as in the relative kinetic activities of the different catalysts (except for the Noyori model where only one type of catalyst exists). Another factor likely to influence that distribution is the selective precipitation of one or several of these aggregates. This is much less discussed, although there are numerous reported experimental cases for catalyst precipitation causing non-linear effects -the first by Narasaka and collaborators [20] in an asymmetric Diels-Alder reaction, followed by many more in transition metal catalysis [16,[21][22][23][24][25][26][27][28][29][30][31][32][33][34][35] and organocatalysis [36][37][38][39][40][41] -but to date, none of the known models accounts for that possibility. An exception is Kagan's "reservoir effect" model which, however, lacks the mathematical expressions needed to simulate NLE curves properly. [7] More systematic studies were made by Klussmann et al., but with a focus on the special case of ternary phase systems, i. e. simultaneous hetero-and homochiral precipitation, which lead to a horizontal segment in the eeL vs eePplot ("buffered eeP"). [36][37][38] Altogether, it was recognized that the interpretation of NLEs could be complicated if catalyst precipitation was not properly taken into account.

Our study aims at filling this gap by expanding previous NLE models to allow for hetero-or homochiral dimer precipitation and thus to study the precipitation's effect on simulated eeL vs eePcurves. We will take some literature cases as a starting point to explore various scenarios and to analyse peculiar features in NLEs (broken-shaped curves, sign switch of an NLE) found both in simulations and in the literature examples. Furthermore, the models allow us to predict other unusual features in NLEs, such as inverse S-shaped curves and horizontal segments. Finally, an outlook concerning other types of phase transition (i. e. liquidliquid phase separation) will be given.

Results and Discussion

Among the numerous examples of NLEs occurring with precipitation, two stand out through peculiar features: a cyclopropanation reaction reported by Estevan et al. [23] (Figure 2a, black dots) and an alkene hydrogenation reported by Terrade et al. (Figure 2b), [34] both catalysed by chiral ligand-supported Rh catalysts. The first is remarkable through its broken shape: at more than 80% ligand enantiomeric excess (eeL) linearity and a homogeneous reaction mixture are observed. Below that value, a (+)-NLE occurs, along with the precipitation of racemic catalyst. Such a linear-to-NLE transition was already observed in some homogeneous systems, [START_REF] Girard | [END_REF]6,42] however not in the sharp way the data in Figure 2a suggest (making them compatible with existing models, such as Kagan's ML4 model). In the second example a strong (-)-NLE is observed when catalyst and substrate are added rapidly (Figure 2b, hollow squares), however it changes to a strong (+)-NLE if the catalyst is let to incubate for 1 h before addition of the substrate, during which a racemic precipitate appears (filled squares; cf. also Klussmann's study on delayed equilibration of proline and TADDOL precipitates). [38] This is a remarkable feature since it allows to study the NLE on one and the same system both with and without catalyst precipitation.

Finally, decreasing the concentration in Rh atoms from 25 to 2 mM leads to an almost linear plot, regardless of the protocol used, and to the absence of any precipitate (hollow and filled dots).

Figure 2. Non-linear effects observed for (a) the enantioselective cyclopropanation of styrene catalysed by a chiral dirhodium(II) complex (black dots) [23] with a computer-drawn curve calculated from eq. (1-4) (blue line; k2 = k2' = 0, ee1 = 0.91, KHomo = 100, KHetero = 195, [Cattot] = 0.004, Ks = 1.07•10 -4 , k1 and ee2 not relevant); b) the enantioselective hydrogenation of dimethylitaconate in presence of a chiral Rh(I) complex at 25 mM (squares) or 2 mM (dots) in Rh catalyst, with immediate addition of the substrate (hollow squares/dots) or after a 1 h equilibration time for the catalyst (filled squares/dots), leading to racemate precipitation at 25 mM (filled squares). [34] Graph b) was adapted with permission from reference 28. Copyright Wiley-VCH GmbH.

Both examples suggest that precipitation of a racemate does not only lead to a (+)-NLE, but can also override the system's inherent tendency to linearity (Figure 2a) or to asymmetric depletion (Figure 2b). Moreover, Figure 2a shows that precipitation can lead to eeP vs eeL-curves with a peculiar, broken-shaped aspect. In the following, we will verify whether this can be reproduced in simulations, which account for heterochiral precipitation, by using an extended version of the monomer-dimer competition model we introduced recently.

Model studies: heterochiral dimer precipitation

The monomer-dimer competition model is a two-component catalysis where both monomeric and dimeric complexes catalyse the reaction with different enantioselectivities (Figure 3). [16][17][18][19] Depending on the parameters chosen it can give rise to classic (+)-or (-)-NLEs, as encountered in the Noyori [14,15] and ML2 [7] models (which can both be considered as special cases within the monomer-dimer competition models), as well as to more exotic ones such as hyperpositive [16] (cf. Figure 1c) or enantiodivergent NLEs, [18] for which we also discovered experimental examples. The overall shape of the NLE curve depends on dimerization constants (KHomo and KHetero), kinetic constants (k1, k2 and k2'), enantioselectivities (ee1 and ee2) as well as on the total catalyst concentration [Cattot]. All these parameters intervene in equations (1-3) which allow to simulate NLE curves. Here, we introduce an additional solubilization equilibrium for the RS-dimer, whose concentration is now governed by the solubility constant KS. The mathematical expressions for this model are the same as for the standard model (i. e. all-soluble) one except for β, which becomes a constant depending only on KHetero and KS (eq. ( 4), cf. Supporting Information for the mathematical development). The precipitated RS(s) is not catalytically active and thus influences the overall catalytic outcome by pulling on the various equilibria, as we will show in several computed examples. It is assumed that upon start of the catalytic reaction all equilibria have reached a steady state. The catalytically active species are issued from the irreversible reaction of a metal salt (M) and a chiral ligand (LR and LS) with enantiomeric excess eeL and concentration [Cattot]. k1, k2 and k2' are rate constants, ee1 and ee2 enantioselectivities, KHomo and KHetero dimerization constants, KS a solubilisation constant, α and β are working variables. The formed chiral product has an overall enantiomeric excess eeP. Equations (1-3) allow to simulate eeP vs eeLcurves following the all-soluble system; eq. ( 4) replaces eq. ( 3) if RSprecipitation occurs.

Figure 4 shows simulations following classic scenarios, depicting a Noyori-type [15] (inactive dimers, i. e. k2 = k2' = 0, Figure 4a) and a ML2-type [7] system (inactive and inexistent monomers, i. e. k1 = 0 and very high values of KHomo and KHetero, Figure 4b). The parameters were chosen to generate either a (+)-NLE (Figure 4a) or no NLE at all (Figure 4b). Upon integration of a KS-value sufficiently low to induce RS-precipitation, the curves become broken-shaped as in Estevan's example: at low eeL (left part of the graphs) precipitation occurs while at high eeL (right part) [RS] stays below its solubility limit. The point at which both parts join, the transition point, is defined by [RS] = KS in the all-soluble model (cf. Supporting information for details). The lower KS, the more the left part of the broken-shaped curve dominates; at very high KS the soluble part literally vanishes from the graph. RS-precipitation induces a "local" (+)-NLE which enforces a pre-existing (+)-NLE (Figure 4a) or even induces one in the first place (Figure 4b), which concords nicely with the observations made by Estevan in Figure 2a. Precipitation acts thus in a similar way as an increase of KHetero [15,18] and allows (+)-NLEs even in special cases where an NLE would be forbidden otherwise (e. g. k2' = k2 in a ML2scenario, [7] as in Figure 4b, or alternatively when KHetero = 2KHomo [15,19] ). Moreover, we could determine a set of parameters that fits well with Estevan's system (continuous line in Figure 2a), reproducing both the linear and the (+)-NLE domains as well as the transition point. The latter can be missed easily if it is far to one side of the graph, or if an insufficient number of datapoints was collected. For example, in the hyperpositive NLE of the NBE-catalysed addition of ZnMe2 to benzaldehyde, where precipitation occurs at very high eeL, both all-soluble (with a high KHetero) and heterochiral precipitation models fit equally well (cf. Supplementary Figure 1).

(+)-NLE induction through racemic precipitation works also with an underlying (-)-NLE, as seen in Figure 5. With a sufficiently low KS, the precipitation-induced (+)-NLE can completely replace the underlying (-)-NLE; one can thus observe asymmetric amplification despite e. g. KHetero < 2KHomo. [15,18] This concords with Terrade's case in Figure 2b, where the NLE switches from (-) to (+) when the system is given sufficient time to let the racemic catalyst precipitate (although their system is much more complex than our model, with several diastereoisomers present when the system is not left to self-sort). However, it is worth looking also at intermediate KS-values (e. g. Ks = 0.002 and 0.001, blue and purple highlighted curves): these lead to inverse S-shaped curves with a (+)-NLE on the left and a (-)-NLE on the right part of the graph. So far, S-shaped NLE curves were modelled only with Kagan's ML4 model, which invokes aggregates as high as tetramers. [7] Here, an aggregation degree of two combined with RS-precipitation is sufficient to induce such a shape. In principle, the transition point between the all-soluble and RS-precipitation regimes makes it visually distinguishable from an ML4-type S-shaped curve (which is perfectly continuous) but in practice this might not be the case, especially if no or few experimental data points were taken around the transition point. (1-4)] with Ks as varying parameter. The dashed line simulates a linear system based on ee1 alone, which intersects the NLE curves (visible on the blue, purple and orange curves) which thus become inverse S-shaped NLEs, with a (+)-NLE at low and a (-)-NLE at high eeL, Fixed parameters: KHomo = 150, KHetero = 33, ee1 = 100, [Cattot] = 0.11 (k2 = k2' = 0, k1 and ee2 not relevant, Noyori-type system).

The second peculiar feature in Terrade's case (Figure 2b) is the switch from a homogeneous system to a system with heterochiral precipitation by increasing the total concentration of catalyst [Cattot]. This is interesting since [Cattot] is a parameter easily modifiable for the experimentalist to study NLEs induced by precipitation. Simulations with variation of [Cattot] in a Noyori-type system (Figure 6) show that within only one order of magnitude of [Cattot], the catalytic system can be changed from a homogeneous system with linearity to a strong (+)-NLE with RS-precipitation. This concords well with Terrade's case (Figure 2b) where an increase from 2 to 25 mM in Rh catalyst is sufficient to induce a strong NLE. In addition, scenarios in which both monomers and dimers are present and catalytically active, with opposite enantioselectivities, may be affected by RS-precipitation in a similar way as seen in Figure 4 and Figure 5 (cf. Supplementary Figures 2 and3 in the Supporting Information): eeP increases especially at low eeL, enhancing an already present hyperpositive NLE or turning an inverse (-)-NLE (where catalysis is dominated by the RR dimer, yielding negative eeP values) into a hyperpositive and enantiodivergent one. A (-)-NLE can be turned into an inverse Sshaped curve much like in Figure 5 and finally into a hyperpositive NLE.

Further model studies: homochiral dimer precipitation

In a similar way as shown before, one can imagine the precipitation of homochiral instead of heterochiral dimers. Thus, the monomer-dimer competition model can also be extended for homochiral dimer precipitation, by introducing again a solubility constant KS (Figure 7). Although conglomerate precipitation (i. e. RR and SS dimers crystallise independently of each other, in opposition to racemate crystals where both crystallise within the same lattice in a 1:1 ratio) is less common [43,[START_REF] Jacques | Enantiomers, Racemates, and Resolutions[END_REF] we will also discuss this model as it exhibits interesting properties (cf. Supporting Information for the mathematical treatment and supplementary discussion). The model represents a scenario opposite to the RS-precipitation discussed before: here the racemate remains in solution and the major homochiral dimer enantiomer (in our case RR) precipitates, generating a (-)-NLE at the upper end of the eeL-scale as shown in simulation graphs (Supplementary Figure 5). It thus acts similar to an increase in KHomo and can induce a (-)-NLE where no NLE (Supplementary Figure 5a) or a (+)-NLE (Supplementary Figure 5b) would be visible otherwise. In the latter case, inverse S-shaped NLEs at intermediate KS-values are possible, much like those shown for the heterochiral precipitation model in Figure 5. Transition points between RR-precipitation and all-soluble models are also observed. Variation of [Cattot] (Supplementary Figure 6) has a similar impact as in Figure 6: an increase of an order of magnitude is sufficient to change an all-soluble system with a weak (+)-NLE to a (-)-NLE with homochiral precipitation. A peculiarity of this model is that it can lead to NLEs with a horizontal segment (i.e. constant product ee when varying ligand ee). This is typically achieved whenever KS is sufficiently low to also make the minor enantiomer dimer (in our case SS) precipitate, in addition to the major enantiomer (e. g. yellow lines with KS = 0.00001 in Figure 8 and KS = 0.001 in Supplementary Figure 5). eeP is then equal to 0 between eeL = 0 and the transition point (which here marks the eeL value where SS has reached saturation); RR precipitates over the whole NLE range and no allsoluble regime is observed. However, RR-precipitation alone can also induce horizontal segments if the RS-dimer is catalytically inactive and highly favoured over the homochiral dimers within the monomer-dimer competition model (Figure 8, green and blue lines at KS = 0.01 and 0.001). Under these conditions, as long as RR precipitates, the R and RR complexes in solution are in a constant ratio and the only competent catalysts present, therefore eeP stays constant over varying eeL. These features are remarkable since horizontal segments have been observed before only in amino acid-catalysis where simultaneous precipitation of both enantiopure and racemate crystals lead to a "buffered" eeP. [36,37] Figure 8. Monomer-dimer competition model with homochiral dimer precipitation. Simulation of the relationship between eeP and eeL. Fixed parameters: k1 = 10, k2 = 100, k2' = 0, KHomo = 3, KHetero = 5000, ee1 = 100%, ee2 = -100%, [Cattot] = 0.11. The broken shape of the curves originate from a transition between all-soluble (low eeL) and RR-precipitation (high eeL), except for the yellow curve (KS = 0.00001) where also the solubility limit of the SS dimer has been reached, leading to RR/SS-coprecipitation instead of an all-soluble regime at low eeL.

Finally, it should be mentioned that homo-and heterochiral precipitation can occur simultaneously in ternary phase systems. This has already been investigated by Klussmann et al. [36][37][38] and won't be further developed here.

Outlook: liquid-liquid phase separation

In this context, it is worth looking at another study reported by Šulce and co-workers. [START_REF] Šulce | [END_REF] Here, amino acid-functionalized platinum nanoparticles (NP) adsorbed on Al2O3 catalyse the hydrogenation of ketones. Proline and tert-leucine as chiral auxiliaries show no NLE or very weak (+)-NLEs, while phenylglycine gives rise to plots with a linear-to-(+)-NLE transition (Figure 9) similar to Estevan's case in Figure 2a. However, since the system is heterogeneous anyway it is unclear where that transition comes from. A possible explanation is that it originates not from the catalytic process but from the catalyst preparation, which can also impact NLEs. [34,46] The amino acid (in excess) is adsorbed onto the Pt NPs in a biphasic cyclohexanone-water mixture. Interestingly, phenylglycine is structurally close to phenylalanine which is prone to liquid-liquid phase separation (i.e. formation of pseudophases in order to shield hydrophobic residues from water) [47,48] and may act as a surfactant in the biphasic mixture, which has also been shown to enable the emergence of NLEs. [49] Thus, one can imagine that stability or reactivity differences between pseudophases (or soft interfaces) of different enantiomeric composition lead to a biased ee of the adsorbed phenylglycine. In principle, one can expect transition points in NLEs to be caused by any type of phase transition, not only solid-liquid. Thus, the transition point seen here might point at a chirally-biased phase separation which would have gone unnoticed otherwise.

Conclusion

This study shows how selective precipitation of dimeric species, either homo-or heterochiral, can generate influence non-linear effects in asymmetric catalysis. Catalyst precipitation amplifies existing NLEs or induces NLEs in the first place, even though the underlying thermodynamic equilibria in solution would not allow for a NLE or generate a NLE of opposite sign. This also allows for the emergence of new phenomena such as broken-shaped, partly horizontal or inverse S-shaped NLE curves, which are possible otherwise only with models involving higher-order aggregation. We have identified a transition point, at which the all-soluble model switches into either one of the precipitation models, as a characteristic feature of catalyst precipitation -which in practice may, however, be difficult to spot in certain cases, especially if the experimental curve is not sufficiently resolved around the transition point. Obtaining a large and detailed dataset is good practice and should be done anyway; a change of reaction parameters (catalyst concentration, temperature) does also help by shifting the point into an easier-to-spot part of the NLE curve. Moreover, these observations concord with similar features found in literature examples. It is interesting to note that reported precipitation-induced NLEs in the literature include only heterochiral precipitation, mostly leading to a (+)-NLE. This highlights the propensity of chiral molecules to form racemate crystals rather than conglomerate crystals. [43][44][50] Beyond precipitation, other phase separations such as liquid-liquid and soft interfaces may also influence NLEs, which then get an increased utility: characteristic features such as a transition point can give a hint for a phase transition coming into play, even if it is invisible to the eye of the experimentalist -as e. g. in liquid-liquid phase separation. This may expand considerably the utility of NLEs as probes for catalytic reaction mechanisms.

Figure 1 .

 1 Figure 1. Product ee vs ligand ee-graphs of catalytic asymmetric reactions showing examples for a) no NLE, b) a positive NLE [(+)-NLE)], c) a hyperpositive NLE and d) a negative NLE [(-)-NLE)].

Figure 3 .

 3 Figure 3. General scheme for the monomer-dimer competition model including a precipitation of the heterochiral dimer RS (with KS as the solubility constant).The catalytically active species are issued from the irreversible reaction of a metal salt (M) and a chiral ligand (LR and LS) with enantiomeric excess eeL and concentration[Cattot]. k1, k2 and k2' are rate constants, ee1 and ee2 enantioselectivities, KHomo and KHetero dimerization constants, KS a solubilisation constant, α and β are working variables. The formed chiral product has an overall enantiomeric excess eeP. Equations (1-3) allow to simulate eeP vs eeLcurves following the all-soluble system; eq. (4) replaces eq. (3) if RSprecipitation occurs.

Figure 4 .

 4 Figure 4. EeP vs. eeL-simulation according to the RS-precipitation model [eq. (1-4)] with Ks as varying parameter. Fixed parameters: (a) KHomo = 33, KHetero = 150, ee1 = 100%, [Cattot] = 0.11 (k2 = k2' = 0, k1 and ee2 not relevant, Noyoritype system), (b) k2 = k2' = 1, KHomo = 3.3•10 8 , KHetero = 1.5•10 9 , ee2 = 100%, [Cattot] = 0.11 (k1 = 0, ee1 not relevant; the high dimerization constants ensure that monomer concentrations are negligible, as required for a ML2-type system). The dashed line in a) simulates a linear system based on ee1 alone.

Figure 5 .

 5 Figure 5. EeP vs. eeL-simulations according to the RS-precipitation model [eq.(1-4)] with Ks as varying parameter. The dashed line simulates a linear system based on ee1 alone, which intersects the NLE curves (visible on the blue, purple and orange curves) which thus become inverse S-shaped NLEs, with a (+)-NLE at low and a (-)-NLE at high eeL, Fixed parameters: KHomo = 150, KHetero = 33, ee1 = 100, [Cattot] = 0.11 (k2 = k2' = 0, k1 and ee2 not relevant, Noyori-type system).

Figure 6 .

 6 Figure 6. Impact of varying the total catalyst concentration [Cattot]with heterochiral precipitation (eq. (1-4)). Fixed parameters: KS = 0.01, KHomo = 120, KHetero = 250, ee1 = 100%, k2 = k2' = 0 (k1 and ee2 not relevant), Noyori-type system.

Figure 7 .

 7 Figure 7. Scheme showing the monomer-dimer competition model with homochiral dimer precipitation. Either only the major enantiomer dimer (here, RR) precipitates with the solubility constant KS, or both RR and SS precipitate. The equations describing the model and further discussion found the Supporting Information.

Figure 9 .

 9 Figure 9. Enantioselective hydrogenation using phenylglycine-functionalized platinum nanoparticles: eeP vs eeL-plot with methyl 4,4-dimethyl-3-oxovalerate (R = tBu, R' = Me) as substrate (a) or ethyl 3-oxo-3-phenylpropanoate (R = Ph, R' = Et) as substrate (b). The two figures were reprinted with permission from reference 45. Copyright Wiley-VCH GmbH.

Acknowledgements

We thank Valentin Bellemin-Laponnaz (ISAE-SUPAERO, Toulouse) for helpful discussion in particular with MATLAB software. This work was supported by the Interdisciplinary Thematic Institute ITI-CSC via the IdEx Unistra (ANR-10-IDEX-0002) within the program Investissement d'Avenir.

Experimental Section

Full description of the models and equations, as well as simulation data, are provided in the supporting information section.