domain: sdv sde

text:
Sánchez Martín

J M Gómez

; Cheptou

& Rubio De Casas

Differences in seed dormancy and germination in amphicarpic legumes: manifold bet-hedging in space and time

Keywords: Dormancy/colonization trade-off, Environmental heterogeneity, Geocarpy, Germination, Mediterranean environments, Physical dormancy, Lathyrus amphicarpos L., Lathyrus cicera L., Vicia amphicarpa L., Vicia sativa L

Background and Aims

Dispersal through space or time via dormancy is one of the primary processes whereby organisms can influence the environment they experience. In plants, strong evolutionary correlations are expected between the two kinds of dispersal because both are performed by the seeds. In this paper, we investigate these evolutionary correlations using amphicarpic plants, which produce simultaneously aerial seeds with high spatial dispersal propensity and subterranean seeds that do not disperse.

Methods

We investigated the variation in dormancy and germination in aerial and subterranean seeds of two amphicarpic legumes (Vicia amphicarpa and Lathyrus amphicarpos) and in two closely related homocarpic taxa (V. sativa and L. cicera).

Key results

Our results showed complex interactions between spatial and temporal dispersal. Right after dispersal, aerial seeds are more dormant than their subterranean counterparts, but this trend reverses with afterripening. Seeds of homocarpic plants germinate at higher percentages than any morph of their amphicarpic congeners and lose dormancy homogeneously with afterripening.

Conclusions

The varied dormancy strategies of amphicarpic seeds are expected to increase variation in emergence timing, providing multiple levels of diversifying bet-hedging. This strategy might be adaptive under highly unpredictable conditions by enabling plants to rely on historically favorable sites in good seasons without preventing spatial and temporal migration.

Introduction

Dispersal is one of the primary processes whereby organisms can influence the environment they experience. An individual's fitness depends on its capacity to reach an appropriate habitat for development and reproduction. Dispersal is usually regarded as the spatial displacement of the progeny from the parental breeding site. However, dispersal also has a temporal dimension, as progeny emerging at different times experience different conditions and different population compositions from their parents. Temporal dispersal is often mediated by dormancy of the seeds or the eggs [START_REF] Levin | The Problem of Pattern and Scale in Ecology: The Robert H. MacArthur Award Lecture[END_REF]. The distribution of progeny across heterogeneous environments will ultimately be determined by the interaction between spatial and temporal dispersal, i.e., by when and where offspring emerge.

In plants, dispersal results from the interaction of environmental factors (including dispersal vectors) and a propagule (the fruit or seed) whose phenotype is largely determined by the maternal plant. These interactions can result in a variety of outcomes, both in terms of distance to the maternal plant and the relative quality of the progeny patch. Dispersal of plants in time is determined to a large extent by seed dormancy [START_REF] Donohue | Germination, Postgermination Adaptation, and Species Ecological Ranges[END_REF]. Dormant seeds do not germinate under conditions generally favorable for germination [START_REF] Finch-Savage | Seed dormancy and the control of germination[END_REF]. This state is often innate (i.e., seeds become "primary dormant" before detachment form the maternal plant) and lost with time in a process called "afterripening". However, seeds can also go through cycles oscillating between a germination ready state and one of "secondary" dormancy [START_REF] Auge | Secondary dormancy dynamics depends on primary dormancy status in Arabidopsis thaliana[END_REF]. For instance, physical dormancy or hardseededness (PY; prevention of germination by impermeable seed coats) is only attained under specific humidity and temperature conditions. In such instances, seeds might disperse in a non-dormant state and acquire PY if exposed to drought [START_REF] Jaganathan | Influence of maternal environment in developing different levels of physical dormancy and its ecological significance[END_REF]. Consequently, whether a seed is dormant will ultimately depend on its endogenous characteristics as well as on the environment that it encounters.

In some cases, plants can produce different types of seeds (Figure 1) that differ drastically in terms of their spatial dispersal propensity and/or dormancy level. This type of strategy termed seed heteromorphism, or heterospermy [START_REF] Imbert | Ecological consequences and ontogeny of seed heteromorphism[END_REF] is generally regarded as a bet-hedging mechanism, as the different morphs can disperse to different sites or germinate under different conditions increasing the likelihood that at least some of the progeny will be recruited at any given time [START_REF] Venable | The Evolutionary Ecology of Seed Heteromorphism[END_REF]. Amphicarpy constitutes one of the most striking cases of heterospermy. Amphicarpic plants produce both aerial and subterranean (geocarpic) flowers and fruits [START_REF] Darwin | The Power of Movement in Plants[END_REF][START_REF] Cheplick | The ecology of amphicarpic plants[END_REF]. In amphicarpic species, a single plant simultaneously produces two sets of dispersal structures that differ in practically all their characteristics. Seeds produced by the aerial flowers disperse without difficulty, while the subterranean seeds are inevitably limited in their dispersal capacity [START_REF] Cheplick | Life History Evolution in Amphicarpic Plants[END_REF][START_REF] Zhang | Effect of seed morph and light level on growth and reproduction of the amphicarpic plant Amphicarpaea edgeworthii (Fabaceae)[END_REF]. In addition, both types of seeds usually show differences in morphology as well as in dormancy (Baskin and Baskin 2014a). Generally, aerial parts produce a large number of seeds that are relatively small and dormant, while the subterranean fruits produce a smaller number of large, non-dormant seeds [START_REF] Cheplick | The ecology of amphicarpic plants[END_REF][START_REF] Kaul | The subterranean flower[END_REF]. Seed morph proportions are genetically controlled but also generally somewhat plastic and can vary depending on the maternal environment [START_REF] Cheplick | Life History Evolution in Amphicarpic Plants[END_REF][START_REF] Clavijo | The Influence of Achene Type and Plant Density on Growth and Biomass Allocation in the Heterocarpic Annual Catanache lutea (Asteraceae)[END_REF][START_REF] Sadeh | Plastic bet-hedging in an amphicarpic annual: an integrated strategy under variable conditions[END_REF].

Amphicarpy is assumed to derive from "homocarpy" (i.e., only aerial flowers and seeds) as a result of adaptation to highly unpredictable or frequently disturbed environments [START_REF] Tardío | Anficarpia: un curioso fenómeno del reino vegetal: plantas que emiten frutos tanto aéreos como subterráneos[END_REF][START_REF] Lev-Yadun | Archaeology. The cradle of agriculture[END_REF][START_REF] Gutterman | Survival Strategies of Annual Desert Plants[END_REF]). The existence of the two different morphs might ensure reproduction and the possibility of leaving descendants even under recurring unfavorable conditions such as predation or fire [START_REF] Darwin | The Power of Movement in Plants[END_REF][START_REF] Kaul | The subterranean flower[END_REF].

While dormant subterranean seeds can ensure recruitment within the maternal site, aerial seeds can colonize new patches (Ruiz de [START_REF] De Clavijo | The Ecological Significance of Fruit Heteromorphism in the Amphicarpic Species Catananche lutea (Asteraceae)[END_REF]. However, in spite of this potential adaptive value, it is a rather unusual reproductive strategy. There are very few amphicarpic species, spread across phylogenetically distant families, such as Poaceae (e.g., Amphicarpum), Cyperaceae (Eleocharis), Brassicaeae (Cardamine) or Fabaceae (Amphicarpa, Macrotyloma, Lathyrus or Vicia), where it seems to be particularly frequent [START_REF] Barker | A review and survey of amphicarpy, basicarpy and geocarpy in the African and Madagascan flora[END_REF]. The rarity of the strategy and the fact that evolutionarily distant genera include amphicarpic species can be posited as evidence for adaptive convergence under particular environmental conditions. It is commonly assumed that amphicarpy can only be favored in highly unpredictable environments [START_REF] Sadeh | Plastic bet-hedging in an amphicarpic annual: an integrated strategy under variable conditions[END_REF][START_REF] Choo | Advantages of amphicarpy of Persicaria thunbergii in the early life history[END_REF]. Theoretical models have shown that this strategy provides an extreme case of diversifying bet-hedging, compromising short-term success to ensure high geometric mean fitness [START_REF] Hidalgo | Environmental unpredictability and inbreeding depression select for mixed dispersal syndromes[END_REF]. When conditions fluctuate drastically in both space and time, selection can favor simultaneously phenotypes that diversify risk through migration and those able to ensure the occupation of a previously favorable patch [START_REF] Buoro | Life-history syndromes: Integrating dispersal through space and time (J Clobert, Ed.)[END_REF][START_REF] Rubio De Casas | Gene-flow through space and time: dispersal, dormancy and adaptation to changing environments[END_REF]. In seeds, this can lead to the differentiation of two specialized morphs, a "High dispersal/High dormancy" seed (which migrates in space and time) and a "Low dispersal/Low dormancy" seed (which ensures occupation of the parental patch; [START_REF] Arshad | Dispersal biophysics and adaptive significance of dimorphic diaspores in the annual Aethionema arabicum (Brassicaceae)[END_REF]. Amphicarpic plants might conform to this model, as subterranean seeds are generally described as less dormant than aerial ones [START_REF] Cheplick | Life History Evolution in Amphicarpic Plants[END_REF]. The former would disperse neither in space nor in time, and migration would be a function exclusively of the latter.

However, a trade-off between spatial and temporal dispersal is also commonly associated with heterospermy. In these cases, the morph that disperses more readily through space has limited dormancy [START_REF] Imbert | Ecological consequences and ontogeny of seed heteromorphism[END_REF]Baskin and Baskin 2014). For amphicarpic plants, this sort of evolutionary trade-off would have led to reduced dormancy in the aerial seeds (which would be specialized to disperse in space) relative to both their homocarpic ancestors and the subterranean morph. Simultaneously subterranean seeds should have developed relatively deep dormancy to provide dispersal through time. Therefore, comparing the behavior of homocarpic seeds (which have to ensure both spatial and temporal dispersal) to that of amphicarpic seeds might be useful to illustrate the potential existence of evolutionary tradeoffs between seed functions and whether bet-hedging causes specialization to disperse simultaneously in space and time.

In the present paper, we investigate evolutionary correlations and trade-offs in seed dispersal and dormancy by estimating differences in dormancy between seed morphs of two amphicarpic legumes, and between these and those of close homocarpic relatives with almost identical ecology and phenology. We aimed to establish if amphicarpy entails the specialization of seeds either as migrant (High dispersal/High dormancy) vs. persistent (Low dispersal/low dormancy) or as spatial (High dispersal/Low dormancy) vs. time (Low dispersal/High dormancy) capsules. To test these different scenarios, we investigated:

1. Whether different seed morphs (aerial vs. subterranean) of the Mediterranean amphicarpic legumes Lathyrus amphicarpos and Vicia amphicarpa differ in their germination and dormancy.

2. If seed dormancy is constant in time, i.e., if fresh and afterripened seeds exhibit similar levels of dormancy and whether these potential temporal differences vary between seed morphs.

3. If seeds from amphicarpic species, particularly aerial ones, differ in dormancy and germination from those of their closest homocarpic species, L. cicera and V. sativa, which share similar ecology and geographical distribution.

Materials and Methods

Plant species and dormancy classes

We studied two amphicarpic species (Lathyrus amphicarpos and Vicia amphicarpa) and their closest non-amphicarpic relatives (L. cicera and V. sativa). All four of these annual species belong to the tribe Fabeae of the Fabaceae, and are common, often co-occurring, in Mediterranean herbaceous communities in Southern Iberia. Besides their ecological requirements, the plants share many morphological synapomorphies within genera. The only trait differentiating V. amphicarpa and V. sativa is the production of subterranean reproductive structures in the former. In the case of the two Lathyrus, besides the subterranean flowers and fruits, morphological differences are limited to the abaxial shape of the fruit pods and the ramification of the tendrils [START_REF] Gallego | Flora Vascular de Andalucía Oriental[END_REF]. These ecological and morphological similarities are probably a consequence of the close evolutionary relationships (in fact, V. amphicarpa is often considered as a subspecies of V. sativa; [START_REF] Kenicer | Systematics and biogeography of Lathyrus (Leguminosae) based on internal transcribed spacer and cpDNA sequence data[END_REF].

The aerial fruits of these homocarpic and amphicarpic legumes are also similar and disperse their seeds ballistically, shooting them violently when the valves dehydrate in early summer (dehiscence; [START_REF] Garrison | Ballistic seed projection in two herbaceous species[END_REF], while subterranean fruits remain attached to the maternal plant.

Many legumes have seeds with hard and impermeable coats, which prevent water uptake (i.e., physical dormancy, PY). Under natural conditions, this type of dormancy is usually lost when seeds undergo temperature fluctuations that open the coats, which in the laboratory can be replaced by a mechanical scarification (Baskin et al. 2000;Baskin and Baskin 2000;[START_REF] Zalamea | Do soil microbes and abrasion by soil particles influence persistence and loss of physical dormancy in seeds of tropical pioneers?[END_REF]. Also, the seeds of certain Fabaceae can exhibit another class of dormancy, known as physiological dormancy (PD), which is caused by conditions within the embryo that prevent germination until inhibiting chemicals are broken down or are no longer produced by the seed. In nature, this deactivation is often caused by periods of storage in the seed bank under dry conditions (i.e., afterripening) followed by specific temperature combinations that ensure that the seeds germinate at the optimum moment for the development of the seedlings.

For Mediterranean annuals that disperse their seeds at the onset of the dry season in early summer, the optimal germination season occurs in autumn, once temperatures have cooled down and water availability for seedling establishment can be reliably expected (i.e., winter annual life cycle). Therefore, PD is usually lost following a period of several weeks or months of dry afterripening and/or a short spell of cold temperatures (~4ºC) under moist conditions [START_REF] Finch-Savage | Seed dormancy and the control of germination[END_REF]Baskin and Baskin 2014). In the laboratory, this is usually mimicked by dry storage at room temperature for several months (aprox. four) and short cold stratification [START_REF] Varela | Latencia y germinación de semillas[END_REF]. However, when using dry storage, it is essential to keep in mind that in non-dormant seeds, post-dispersal drying might induce secondary dormancy, and thus dry afterripening might cause dormancy rather than alleviate it, specially if temperatures right after dispersal are high (Finch-Savage and Leubner-Metzger 2006). Furthermore, since PY is exogenous to the embryo and PD endogenous, the two dormancy classes might be present simultaneously in what is called combinational dormancy (PY +PD), which is relatively frequent in temperate legumes (Van Assche and Vandelook 2010).

Experimental design

Seeds were obtained from different populations across S. Spain, incorporating wide genetic variability (Figure S1). We collected three to twenty seeds from a total often to twelve individuals from 17 populations (three L. amphicarpos, four L. cicera and five of each of the Vicia species) and grew 10 random individuals from each of these populations under homogeneous conditions in a common greenhouse (IFAPA, Granada, Spain) allowing for open pollination within the greenhouse. After seed collection, some stock was saved so as to cultivate three descendants of each of the initial samples every year. Following the natural cycle of the plants, sowing took place at the beginning of October and mature seeds were collected in early June. This process was replicated several years and the seeds collected were used for the different germination experiments. Specifically, F1 seeds were used for experiment 1, F2 seeds for experiment 2 and F3 for experiment 3. F1 & F2 plants were cultivated in the IFAPA greenhouses, and F3 in the 'Plateforme des Terrains d'Experience du LabExCeMEB' (Montpellier, France). In all germination experiments, seeds were pooled at the population level. The seeds of each species and seed morph were assigned to the same "type", i.e., amphicarpic plants had two seed types and homocarpic species had only one; as a result, all experiments included six seed types.

Experiment 1 was aimed at quantifying PY. In this experiment, we used seeds afterripened for eight months at room temperature to minimize potential PD effects on germination. Seeds underwent: an untreated control (treatment 0 hereafter) or mechanical scarification with nail pliers (treatment S hereafter). We used 10 replicates per treatment and seed type, each replicate consisting of 6 seeds in a Petri dish (5 cm diameter) filled with 0.5% agar solution (n = 60 seeds/treatment x 6 types = 360 seeds). Once seeds were sowed, we sealed the dishes with parafilm, analyzing germination percentage after, one, two, five, 11 and 14 days.

In Experiment 2, we used seeds that had undergone just four months of afterripening and were therefore expected to retain some PD. This experiment was similar to Exp. 1 but we added two new treatments: a cold dark stratification (3ºC for 3 days, treatment C) to check physiological dormancy and a combined treatment (C+S) to control for combinational dormancy (PY + PD).

To further clarify the effect of PD and estimate whether seeds at the time of dispersal lacked hardseededness and if secondary PY was a consequence of dry afterripening, a third experiment (Experiment 3) was carried out. For this experiment, seeds were harvested when plants had finished their cycle but were not yet fully dry (at least 75% of pods on each plant were yellow). They were then exposed to either control or cold stratification conditions (as described above).

In Experiments 1 & 2, seeds were disinfected in NaClO 5% for 30 seconds prior to being sowed and cleaned with distilled water. In Experiment 3, we did not use any disinfection treatment to avoid any accidental opening of a water gap that may facilitate imbibition, which also caused a decrease in the number of final viable seeds (many seeds had to be discarded because of the presence of fungi, particularly in the case of the subterranean morphs). To ensure imbibition, 10µL of water were added to each seed after plating. Germination conditions were set at 21ºC constant temperature and 12h light cycles. Seeds were incubated under these conditions for 14 days in the first experiment and 11 days in the other two experiments. Germination, identified as a visible protrusion of the radicle, was checked every 48h during incubation. At the end of each experiment, petri dishes were opened and seed viability assessed by firmness to touch, and non-viable seeds were discarded from all analyses.

Data analysis

Data was analyzed in R.3.3.1 (Team 2016) using generalized linear models with germination percentage as the response variable and seed type and treatment as fixed predictors. In every case, we selected the model with the best fit to the data based on AIC values. According to this criterion, linear models exhibited the best fit to the data of the two first experiments (Table S2. Supplementary materials) although our data did not conform to the assumptions of parametric models. We controlled for potential statistical artifacts by repeating all analyses using the model with the second lowest AIC value (Tweedie log; Table S2. Supplementary materials). Since the results of both models were essentially identical, we kept the results of the linear models. For the results of Experiment 3, the model selected was a negative binomial generalized model. In all models, we used the aerial amphicarpic morph of each genus as the reference level. That way, we can quickly test if these differ from either the subterranean seeds of the same species or the homocarpic seeds of close congeners.

Results

Dormancy and germination of afterripened seeds

Experiments 1 & 2 demonstrated the presence of PY in all seed types except those of Vicia sativa, which did not appear to have a significant degree of dormancy (Table 1; Fig. 2).

Physical dormancy was particularly notable in amphicarpic species, especially in the subterranean seeds. Conversely, physiological dormancy and combinational dormancy did not influence germination of these species (Fig. 2). Cold stratification had only a significant effect on L. cicera and the subterranean seeds of L. amphicarpos, and in both cases its effect was much smaller than that of physical scarification, while neither V. amphicarpa morph nor the aerial seeds of L. amphicarpos responded to PD alleviation by cold stratification (Table 1; Fig. 2). In the absence of any dormancy breaking treatment, homocarpic species germinated to higher percentages than any amphicarpic seeds, while aerial seeds germinated to higher percentages than subterranean ones, although this last difference was only significant in V. amphicarpa (Fig. 2, Table 1).

Dormancy and germination of fresh seeds

The results of Experiment 3 demonstrated the presence of relatively deep primary dormancy in amphicarpic species. In this experiment, homocarpic species geminated better than amphicarpic ones. Aerial seeds of both amphicarpic species appeared to be particularly dormant. This dormancy was probably physiological to some degree because although cold stratification did not alleviate it, it was much diminished in afterripened seeds (Expp. 1 & 2; Fig. 2). The seeds of L. cicera also seemed to have a relatively high level of dormancy, with %germination < 40% and a complete absence of response to cold stratification. The subterranean seeds of L. amphicarpos had a similar degree of dormancy, but in this case, it was significantly reduced after cold stratification, which together with the results obtained with AR seeds can be taken as indication of the presence of PD in these seeds. The seeds of V. sativa also responded to cool stratification, although in this case, even control seeds had %germination > 60% (Table 1, Fig. 2).

Detailed model coefficients are available in supplementary material tablesS3-S8.

Discussion

Our results showed that amphicarpy entails variability in dispersal in time as well as in space.

When compared to homocarpic seeds, amphicarpic seeds have greater dormancy overall.

However, the two seed morphs produced by amphicarpic taxa differ significantly in their dormancy. Moreover, spatio-temporal dispersal patterns appear to be more diverse than expected, and no straightforward correlation between temporal and spatial dispersal could be established. Seed dormancy also seemed to be determined by the degree of after-ripening, but this effect differed among morphs, and as a result the behavior of each morph varied through time independently. The combination of multiple dormancy and dispersal levels within the seeds of individual maternal plants indicate that amphicarpy is a multifarious bet-hedging strategy that enables plants to spread risk spatially and temporally.

Dormancy in aerial and subterranean seeds of amphicarpic plants

Our results did not support the existence of a clear dormancy/colonization trade-off in amphicarpic plants or the differentiation of "High dispersal/High dormancy" and "Low dispersal/Low dormancy" strategies [START_REF] Rees | Evolutionary ecology of seed dormancy and seed size[END_REF][START_REF] Arshad | Dispersal biophysics and adaptive significance of dimorphic diaspores in the annual Aethionema arabicum (Brassicaceae)[END_REF], but rather indicated the selection of more complex dormancy and dispersal strategies. The seeds of amphicarpic species exhibited in every case greater dormancy than the seeds of their closest congeners.

The relative gain in physical dormancy was particularly prominent. However, physiological and combinational dormancy might also be ecologically relevant, especially in the case of subterranean seeds: initially they present a superficial physiological dormancy similar to aerial seeds of the other species (Exp. 3), but when dehydrated they acquire physical dormancy (Exp. 1 & 2). This suggests that in these seeds, afterripening modifies dormancy in a sophisticated fashion, increasing PY while simultaneously decreasing PD [START_REF] Finch-Savage | Seed dormancy release in Arabidopsis Cvi by dry after-ripening, low temperature, nitrate and light shows common quantitative patterns of gene expression directed by environmentally specific sensing[END_REF][START_REF] Van Assche | Combinational dormancy in winter annual Fabaceae[END_REF]Baskin and Baskin 2014b).

Hardseededness appeared to be potentially deeper in subterranean seeds, which in both Lathyrus and Vicia had lower %germination and responded more dramatically to scarification treatments than aerial seeds (Expp. 1 & 2). This difference among afterripened seeds of the two morphs appeared to be more significant in V. amphicarpa than in L. amphicarpos, although the trend was similar in both cases. However, these differences could be the result of secondary dormancy, likely induced by dehydration. Physical dormancy depends on the water content of the seed, the seed coat only becoming impermeable after partial dehydration [START_REF] Van Assche | Seasonal cycles in the germination capacity of buried seeds of some Leguminosae (Fabaceae)[END_REF][START_REF] Jaganathan | Influence of maternal environment in developing different levels of physical dormancy and its ecological significance[END_REF]. It would appear that the coats of subterranean seeds of amphicarpic species are initially water-permeable, as these seeds germinated better than fresh aerial seeds. This shallow dormancy of subterranean seeds might allow them to germinate immediately after "dispersal", i.e., as soon as the seeds are mature, if the appropriate environmental conditions are met. These legumes have the typical Mediterranean annual life cycle, and mature their seeds at the end of the spring or early summer, concurrent with the onset of the dry summer period. Therefore, it is unlikely that subterranean seeds will encounter conditions adequate for germination, although given the stochasticity of the Mediterranean climate [START_REF] Blondel | Biology and wildlife of the Mediterranean region[END_REF], this might occur in exceptionally wet and cold years. In such instances, subterranean seeds might enable the re-occupation of the maternal spot before the regular germination season in the fall. However, they will most commonly dry during the summer, persist in the soil bank, and emerge scattered through time within the maternal patch [START_REF] Zeide | Reproductive Behavior of Plants in Time[END_REF][START_REF] Tardío | Anficarpia: un curioso fenómeno del reino vegetal: plantas que emiten frutos tanto aéreos como subterráneos[END_REF]. Thus, under realistic ecological conditions, hardseededness in underground seeds ensures temporal dispersal, but it can also provide temporal bet-hedging, adding to the differentiation of banking strategies associated to amphicarpy.

Differences in germination and dormancy between amphicarpic and homocarpic relatives

Our phylogenetically paired comparisons indicated that amphicarpy represents in every case an increase in seed dormancy. In both genera, amphicarpic seeds, either aerial or subterranean, displayed deeper dormancy that those of their homocarpic congeners. Dormancy was particularly shallow in the case of V. sativa, the seeds of which seemed to be almost entirely non-dormant. In homocarpic species, seeds appeared to be ready to germinate following four-months of dry afterripening. Under natural conditions, this would entail emergence in the fall of all the propagules dispersed earlier in the year and the absence of an inter-annual seedbank.

Given the morphological similarities in the aerial fruits within genera, there is no reason to expect drastic differences in the spatial dispersal between taxa. Lathyrus and Vicia disperse their aerial seeds ballistically. However, amphicarpic species appear to have increased their level of dormancy compared to their homocarpic relatives, which display a more evident colonizing behavior. At this point, it is unclear if that is because selection favors an overall increase in dormancy in all amphicarpic seeds or if the diversification of specialized dormancy types is somehow constrained in these species. Maybe the differentiation of a "dormant morph" limits the colonizing potential of the other morph, i.e., selection for increased dormancy might ultimately affect all of the seeds produced by a plant.

Alternatively, it is possible that selection might have favored complex dormancy patterns in both seed types to ensure simultaneous seed banks in different locations. Such an outcome might be expected when environmental conditions are highly unpredictable in both space and time, which is the only scenario in which amphicarpy is expected to be adaptive [START_REF] Hidalgo | Environmental unpredictability and inbreeding depression select for mixed dispersal syndromes[END_REF].

In summary, our results prove that the correlation between spatial and temporal seed dispersal can have in multiple different outcomes. There are no simple trade-offs between dispersal and dormancy and their joint evolution leads to the emergence of bet-hedging strategies that spread risk at multiple scales in both space and time. Tables Table1. 1. Statistical summary of the three germination experiments. In every test, %germination of the aerial seeds of the amphicarpic species under the control (0) treatment was used as the reference level. For example, the values shown for Experiment 1 correspond to the difference between S and 0 aerial amphicarpic seeds, between subterranean amphicarpic seeds in both treatments (i.e., 0 and S) and aerial control seeds and between homocarpic seeds in both treatments and aerial control seeds. Treatment codes: 0 -control; C -cold dark stratification (3 days at 3ºC-4ºC); S -physical scarification; S+C: physical scarification followed by cold stratification. Only p-values are displayed, for other coefficients, please refer to Tables S3-S8. Values represent the significance of the comparison between each seed type and the reference for any given treatment. *Significant at p < 0.05

A. Lathyrus

Morph

Acknowledgements

This research was supported by the Spanish Ministries of Education and Professional Training (FPU grant FPU17/00629 awarded to RSM) and of Economy and Competitiveness (grant CGL2016-79950-R) including FEDER funds as well as by the Junta de Andalucía/FP7 Talentia program. We also thank the staff at the IFAPA Camino de Purchil and CEFE-CNRS greenhouses for their help with plant cultivation and Dr. D. Lawrence Venable for his helpful comments.


Dr Wided Hagui

Kiruthika Periasamy

Jean-François Soulé
email: jean-francois.soule@univ-rennes1.fr

Dr Mrs K Periasamy

J.-F Soulé

Synthesis of 2,2'-Bipyridines through Catalytic C-C Bond Formations from C-H Bonds

Keywords: C-H bond Activation, Homogeneous Catalysis, Ligand Design, N Ligands, Synthetic Methods Accepted

The present review describes the homogenous catalytic methodologies for the synthesis of 2,2'-bipyridines involving C-C bond formation from C-H bond. There are three main approaches to build well-decorated 2,2'-bipyridines from C-H bond cleavage: i) the formation of C2-C2' bond via the dimerization of (activated) pyridines, ii) the C-H bond heteroarylation of activated pyridine, or iii) the late-stage functionalization of (activated) 2,2'-bipyridines including photoredox processes. Besides the scope and limitation of each methodologies, the relevant mechanisms of reaction are also discussed.

Introduction

2,2'-Bipyridines are among the most fundamental and privileged bidentate ligands in organometallic and inorganic chemistry for both catalyst and material designs. [1] For example, 4,4'-di-tert-butyl-2,2'bipyridyl (dtbpy) is employed for the preparation of heteroleptic Ir(III) complexes such as (Ir[dF(CF3)ppy]2(dtbpy))PF6 a well-know photoredox catalyst (Figure 1). [2] 2,2'-Bipyridines is also a key scaffold for preparing cyclometalated complexes, such as [PtCl(Phbpy)], displaying emission features useful for the fabrication of optical devices. [3] Moreover, chiral 2,2'-bipyridines, such as Bolm ligand, are very useful in asymmetric catalysis. [4] Since the first synthesis in 1888 from picolinic acid, [5] chemists have continually innovated to prepare well-decorated 2,2'-bipyridines, including asymmetrically substituted ones. [1a] One of the most classical approaches for the synthesis of symmetrically 2,2'-bipyridines is based on reductive homocoupling of 2-halopyridines in the presence of Cu, Ni, or Pd catalysts. [6] Asymmetrically substituted 2,2'-bipyridines can be prepared using cross-coupling reactions, i.e., Stille, Negishi, Suzuki-Miyaura, Kumada, and Hiyama coupling reactions. Another possibility of the synthesis of 2,2'bipyridines is based on the cyclization methods, namely Kröhnke cyclization [7] and [2+2+2] cocyclotrimerization. [8] Although these methods enable the formation of the desired symmetrically and asymmetrically substituted 2,2'-bipyridines, they suffer from a limited access of starting materials (multi-steps synthesis) and the low stability and poor reactivity of 2-halopyridines and 2-pyridyl organometallics. [9] Since these last decades, C-H bond functionalization for the formation of C-C bond has revolutionized the synthesis of organic molecules, including 2,2'-bipyridines. [10] Our review focus on the recent progress in catalytic cross-coupling reactions through C-H bond functionalization for the flexible access to well-decorated 2,2'-bipyridines. Among the different approaches to prepare (functionalized) 2,2'-bipyridines, there are three strategies which involves C-H bond cleavage a key step: i) the formation of C2-C2' bond via dimerization of (activated) pyridines, ii) the C-H bond heteroarylation of activated pyridines, or iii) late-stage functionalization of (activated) 2,2'-bipyridines (Figure 2). In catalytic reactions with transition metals, the coordination ability of the 2,2'-bipyridine products remains a "problem" because it can affect, positively or negatively, the catalytic properties of the metal centers. Several strategies based on catalysts, conditions, or substrate design have been discovered to avoid such deactivation.

Dimerization of Pyridine Derivatives

The merest method of preparing 2,2'-bipyridine is the homocoupling of pyridine, initially reported with heterogeneous catalysts such as Raney's Nickel [11] or palladium on charcoal. [12] However, this approach is limited to the synthesis of symmetrical molecules, and the use of heterogeneous catalysts is often associated with a low functional group tolerance. Several homogeneous systems have been developed to improve the scope of this transformation, and the dimerization of protected pyridines such as pyridine N-oxides was allowed reported.

Dimerization of Pyridines

In 2007, Suzuki and co-workers reported an early example of dehydrogenative coupling of 4-substituted pyridines catalyzed by homogeneous diruthenium complexes: Cp*Ru(μ-H)4RuCp* (Ru1; Cp*=η 5 -C5Me5) and (Cp*Ru)2(μ-PMe2)(μ-H)(μ-η 2 :η 2 -C6H6) (Ru2) (Scheme 1). [13] The phosphine-free Ru1 complex catalyzed the dimerization of several 4-substituted pyridines and selectively produced the corresponding 4,4'-difunctionalized-2,2'-bipyridines, without the formation of by-products such as terpyridine. The µ-phosphido Ru2 complex displayed some times better activity than Ru1 depending on the nature of the para-substituent of the pyridine, et vis versa. The renationalization of the reactivity and the choice of catalysts is difficult to explain. It is noted that the dehydrogenative coupling of ethyl iso nicotinate is more challenging due to its lower basicity than that of other substituted pyridines.

Scheme 1. Ru-Catalyzed Dehydrogenative Coupling of 4-Substituted Pyridines.

The same group has continued their efforts toward the design and synthesis of polynuclear organometallic compounds to dimerize several kinds of 4-substituted pyridines without using a sacrificial oxidant or hydrogen acceptor (Scheme 2, Top). They showed that (Cp*Ru)3(μ-H)3(μ 3 -H)2 (Ru3) (Cp* = η 5 -C5Me5) catalyzed the dimerization of 4-methylpyridine, 4-ethylpyridine, and N,Ndimethylpyridin-4-amine in low yield. The pyridines bearing an electron-withdrawing group are not tolerated (Scheme 2, top). [14] In 2016, Takao and co-workers showed that the trinuclear complex composed of ruthenium and cobalt (Ru4) catalyzed more efficiently the dimerization of 4methylpyridine, 4-tertbutylpyridine, N,N-dimethylpyridin-4-amine, but again fail with electron-poor pyridine or unsubstituted pyridine (Scheme 2, bottom). [15] Accepted Manuscript In 2019 Murakami, Itami and co-workers have developed a general and practical method for synthesizing 2,2'-bipyridines in high yields using Pd-catalyzed dehydrogenative coupling pyridine (Scheme 3). [16] The reaction was performed using 5 mol% Pd(OAc)2 in the presence of AgOPiv as base/oxidant in CPME (cyclopentyl methyl ether) at 140 o C. Interestingly, under these conditions, the dimerization of pyridine occurred in 80% yield. Moreover, both para-substituted pyridine bearing an electron-withdrawing group (e.g., trifluoromethyl, ester) or an electron-donating group (e.g., methyl, phenyl) underwent dimerization in high yields. Ortho-substituted pyridine such as α-picoline displayed a lower reactivity as the 2,2'-bipyridyl product was isolated in only 31% yield. When the reaction is performed with meta-substituted pyridines, a mixture of two regioisomers is observed. Interestingly, the sterically hindered 3-chloropyridines reacted preferentially at C2-positions to give the corresponding 2,2'-bipyridines with 33 to 61% yield. This regioselectivity might be explained by the high acidity of the C-H bond at the ortho-position of the chlorine atom; a feature communally exploited in Pd-catalyzed C-H bond arylation of polyfluororenzene [17] or polychlorobenzene derivatives. [18] The unique regioselectivity afforded a convertible platform to access various 3,3'-disubstituted-2,2'bipyridines, a new class of axially twisted 2,2'-bipyridyl ligands. The reaction is not limited to pyridine rings, and quinolines or pyrazines are also successfully dimerized.

Accepted Manuscript

Dimerization of Pyridine N-Oxides

Pyridine N-oxides are readily available and bench-stable substrates, including in transition-metal crosscouplings. They often displayed higher reactivity in C-H bond functionalizations. In 2013, during their investigations on Pd-catalyzed oxidative C-H/C-H direct coupling of heterocyclic N-oxides, Kuang and co-workers reported two examples of homocoupling of pyridines N-oxides (Scheme 4). [19] The treatment of 2-methylpyridine or 3-methylpyridine N-oxides by 5 mol% of Pd(OAc)2 in the presence of 2 equivalents of Ag2CO3 and 1 equivalent of pyridine in 1,4-dioxane at 120 o C afforded the corresponding dimerized products in 58% and 46% yield, respectively. In the case of 3-methylpyridine

Accepted Manuscript

N-oxide, only one regioisomer is obtained from the activation at the less sterically hindered C6-H bond.

It should be mentioned that no homocoupling product was observed under the above conditions when pyridine, 4-methylpyridine, 4-nitropyridine, 2-phenylpyridine, quinoline, and isoquinoline N-oxides were used as substrates.

Scheme 4 Pd-catalyzed Homocoupling of Pyridine N-Oxides

In 2016, during their investigation on Pd-catalyzed oxidative homocoupling of 1,2,3-triazole N-oxides to construct bis-triazole N-oxides, Zhu, Liu and co-workers explored the possibility of extending the reaction to the more challenging pyridine N-oxide (Scheme 5). [20] In the presence of 5 mol% Pd(OAc)2 associated with 1,10-phenanthroline as a ligand, 1 equivalent of t-BuOK as a base, the pyridine N-oxide underwent dimerization to afford the 2,2'-bipyridine N-oxide product in 43% yield. Unfortunately, the authors have shown only one example in the pyridine series. The dimerization of pyridine N-oxides is not only catalyzed by palladium. In 2017, Jain and co-workers reported the synthesis of 2,2′-homocoupled azine N,N′-dioxides, or azine N-oxides via Cu-catalyzed homocoupling reaction. The best reaction conditions for the homocoupling of 2-phenylpyridine Noxide were 3.0 equivalents of t-BuOLi in toluene at 120 °C for 72 h in the presence of 10 mol% Cu(OAc)2 (Scheme 6). [21] They only studied the substitution on the aryl ring and not on the pyridine ring. The reaction with ortho-, meta-and para-aryl-substituted 2-arylpyridine N-oxides gave the corresponding N,N′-dioxide products.

Scheme 6: Cu-Catalyzed Homocoupling of 2-Phenylpyridine N-oxide Derivatives

Accepted Manuscript

Interestingly, they showed that when the reaction is carried out without catalytic copper, 2,2'-bipyridine N-monoxides were obtained instead of dioxide (Scheme 7). [21] Similar conditions for base-promoted bipyridine N-monoxides synthesis from pyridine N-oxides were previously reported. [22] Scheme 7: Homocoupling of 2-phenyl pyridine N-oxide derivatives in the absence of copper catalysts

The authors proposed two different reaction mechanisms depending on the presence or the absence of Cu catalysts (Figure 4). First, the reaction is initiated by deprotonation of the C2-H by t-BuOLi to generate the lithium salt of pyridine N-oxide (I). In 2016, inspired by the work of Natsume and Itai, [23] Larionov have investigated t-BuOK-mediated deoxygenative homocoupling of N-pyridine oxides using t-BuOK under metal-free conditions (Scheme 8, top). [24] It should be pointed out that the reaction directedly delivers the 2,2'-bipyridine products, not the N-oxide or N,N'-dioxide products. The reactions were carried out using azoisobutyronitrile (AIBN) as a catalyst. The reaction was conducted with a broad scope and tolerates several N-heterocyclic systems and various substituents such as halogen substituents (F, Cl, Br) and benzylic positions. The authors have shown the applicability of this deoxygenative in gram-scale synthesis. More interestingly, the authors also succeeded the cross-coupling of two different heterocyclic N-oxides (Scheme 8, bottom). [24] Although substantial amounts of homocoupling products were also formed, the reaction produced the desired asymmetrically 2,2'-bipyridine derivatives in moderate yields. This condition is

Accepted Manuscript

C-H bond Heteroarylation of Pyridine Derivatives

Pyridine remain a challenging substrate in C-H bond arylation with transition metal catalysis. Moreover, the reaction took place preferentially at C3 or C4 position, [25] due to the electronic repulsion

Accepted Manuscript

between the nitrogen lone pair and polarized C-M bond. Therefore, "protected" or "activated" pyridines shall be employed to switch the regioselectivity at the C2 position and enable the synthesis of 2,2'-bipyridines after a final deprotection step.

In 2008, Charette and co-workers reported Pd-catalyzed direct C-H bond arylation of Niminopyridinium ylides with bromo(hetero)arenes at the C2-position (Scheme 9). [26] The ortho-C-H bond of N-iminopyridinium ylides often displays higher reactivity than the ortho-C-H bond of pyridines avoiding chemoselectivity issues. The arylation of N-iminopyridinium ylides was performed using 5 mol% of Pd(OAc)2 , 15 mol% P(t-Bu)3, in the presence of 3 equivalents of K2CO3 as a base in toluene at 125 °C. The reaction was very sensitive to water. Therefore, molecular sieves have to be employed.

Only one example of heteroarylation with 2-bromopyridine has been reported affording the corresponding 2,2'-bipyridine precursor in 50% yield. The authors have demonstrated that N-N bond can be cleave under reductive conditions (e.g., Zn dust or Pt/C combined with H2) to prepare 2arylpyridines, but they did not report the deprotection for the synthesis of 2,2'-bipyridine derivatives.

Scheme 9 Pd-catalyzed C2-H Bond Heteroarlation of N-Iminopyridinium Ylides

Since the pioneer report by Fagnou on Pd-catalyzed C-H bond arylation of pyridine N-oxides at C2 position with aryl halides, [27] an increasing number of methods for the direct arylation of arenes and heteroarenes based on transition-metal-catalyzed C-H cleavage has been developed. In 2011, Tzschucke and co-workers described Pd-catalyzed direct coupling between pyridine N-oxides and 2halopyridines as the critical step of an efficient and versatile preparation of substituted 2,2'-bipyridines from readily available starting materials. [28] This method significantly reduces the number of steps in comparison to conventional cross-coupling reactions. [6] They demonstrated that pyridine N-oxides should bear an electron-withdrawing group to be reactive. Indeed reaction of equimolar amounts of 2bromopyridine and 4-alkylpyridine N-oxides in the presence of 5 mol% Pd(OAc)2 associated with 5 mol % P(t-Bu)3 and 2 equivalents of K2CO3 in toluene failed to give detectable amounts of coupling product. In contrast, the reaction between ethyl isonicotinate N-oxide with 2-bromopyridine in 1.5:1 ratio affords the desired coupling product bipyridine N-oxide (bipy) in 60% yield among with 21% yield of terpyridine N-oxide (tpy) -resulting from double arylation (Scheme 10). They also evaluated the reactivity of some 2-bromopyridine-surrogate (i.e., 2-chloropyridine, 2-iodopyridine, 2-pyridyl triflate, 2-pyridyl nonaflate, 2-pyridyl tosylate), but none of them outperformed 2-bromopyridine. The bipy / tpy selectivity is slightly dependent of the stoichiometry: an excess amount of ethyl isonicotinate N-oxide (4 equivalents) give bipy in 70%, while using 2.5 equivalents of 2-bromopyridine tpy.

Scheme 10: Effect of the Substrate Ratio and Pseudohalide Pyridines on the Mono/Diaryaltion

Accepted Manuscript

They next explored the substrate scope of pyridine N-oxides and 2-bromopyridines (Scheme 11, top). [28] Pyridine N-oxides bearing a para-electron-withdrawing substituent such as ester, nitro, cyano, trifluoromethyl, or chloro gave moderate to good yields in favor of 2,2'-bipyridine N-oxide products.

In comparison, unsubstituted pyridine N-oxide gave only a 23% yield of the corresponding 2,2'bipyridine N-oxide. In most cases, the authors observed the formation of diarylation products (terpyridine N-oxide), and with electron-rich pyridine N-oxides, they also observed the formation of N-(pyridine-2-yl)-2-pyridone. The mono arylation of meta-substituted pyridine N-oxides is more challenging as two ortho C-H bonds can be arylated. From 3-fluoropyridine N-oxide, the reaction takes place at the most acidic C-2 position (i.e., ortho-position of fluorine atom). [17] From 3-cyano or 3-ester pyridine N-oxides, the arylation occurred at the less hindered C-6 position. Ortho-substituted pyridine N-oxides were also arylated in good yields. The nature and the position of substituent on the 2bromopyridine seem not to have a critical effect on the reactivity. In most cases, the authors also demonstrate that the reduction of 2,2'-bipyridine N-oxides could be carried with hydrogen or sodium borohydride using palladium on charcoal as a catalyst in nearly quantitative (Scheme 11, bottom). [28] Scheme 11 top) Synthesis of Asymmetrically Substituted Bipyridine by Pd Catalyzed C-H Bond Heteroarylation of Pyridine N-Oxide. bottom) Reduction of 2,2'-Bipyridine N-oxides

In 2013, Tzschucke and co-workers reinvestigated this reaction to prepare asymmetrically substituted terpyridines using iterative Pd-catalyzed direct arylation of pyridine N-oxides (Scheme 12, top). [29] The reaction conditions are similar to those previously reported, [28] albeit K3PO4 as the base resulted in better yields than K2CO3. In a first step, pyridine N-oxides substituted by an electron-deficient group at the para-position are arylated with diversely substituted 2-bromopyridines. For the second heteroarylation, 1.2 equivalent of another 2-bromopyridine derivative is employed. The second heteroarylation often seems easier to proceed to afford the asymmetrically substituted terpyridines terpyridine N-oxide in moderate to good yields. They also reported the synthesis of symmetrical terpyridine N-oxides using an excess amount of 2-bromopyridines. Finally, the authors reduced several terpyridine N-oxides via hydrogenolysis using Pd/C as a catalyst in the presence of hydrogen (Scheme 12, bottom). [28] Scheme 12. top) Preparation of Asymmetrically Substituted Terpyridines N-Oxides via Pd-catalyzed C-H Bond Heteroarylation of Pyridine N-Oxides. bottom) Reduction of terpyridine N-oxides They also conducted competition experiments and kinetic studies, which agree with Hartwig and coworkers' cooperative catalysis mechanism (Figure 6). [30] Dimeric cyclometalated complex A is in equilibrium with an undetected monomeric species A', which induces C-H cleavage of the pyridine Noxides to deliver intermediate B. Meanwhile, a second aryl palladium complex D is generated by oxidative addition of the aryl bromide to Pd(0) complex C. Then, transmetalation between these two aryl palladium complexes (B and D) afford the bisaryl palladium complex E, which undergoes fast reductive elimination to form the C2-arylated pyridine N-oxide and regenerate the Pd(0) catalyst C.

Figure 6. The Cooperative Catalysis Mechanism for the Direct Arylation of Pyridine N-Oxides

Proposed by Hartwig. [30]

Late-Stage Functionalization of 2,2'-Bipyridine Derivatives

An alternative approach to build up well-decorated 2,2'-bipyridines, instead of generating the C2-C2'bond, involves the C-H bond functionalization of the 2,2'-bipyridine scaffolds. This late-stage functionalization simplifies the synthetic pathway and can access complexes 2,2'-bipyridines that would be time-consuming to access from simple pyridines. [31] However, the main challenge is the strong coordination ability of the 2,2'-bipyridines. There are two pathways to achieve regioselective functionalization of 2,2'-bipyridyl C-H bond: 1) the rollover cyclometation. Which often afford the modification of C3 and/or C3'-position(s) or 2) radical pathways to functionalize the C6 and/or C6'position(s).

Transition Metal-Catalyzed C-H Bond Functionalizations of Bipyridines via "Rollover" Cyclometalation

Although bipyridines generated stable chelated with transition metal (bipy)MLn, Miura and co-workers demonstrated in 2009 that bipyridines could be difunctionalized via cyclometalation (Scheme 13). [32] The reaction of 2,2'-bipyridyl or 5,5'-dimethyl-2,2'-bipyridine with ethynyltriisopropylsilane in the presence of [RhCl(cod)]2 and PPh3 smoothly proceeded to give the corresponding 3,3'-dialkenylated products in good to excellent yields.

Scheme 13. Rhodium-Catalyzed Alkenylation of Bipyridine C-H Bonds with Terminal Silylacetylenes

In 2012, Chang and co-workers investigated Rh-catalyzed dialkylation of the 2,2′-bipyridyl C-H bond at C5 and C5'-positions (Scheme 14). [33] They found that N-heterocyclic carbene (NHC, IMes) as ligand play a significant role exhibiting a strong trans-effect in (bipy)Rh(NHC) chelate structure, thus weakening a rhodium-pyridyl bond, which is trans to the bound NHC. This feature favors the rollover cyclometalation [34] pathway enabling selective bis-functionalization of 2,2′-bipyridines and 2,2′bisquinolines. A set of 2,2′-bipyridines were bisalkylated using 0.03 equivalent of Rh(acac)3/2IMes . Cl ligand and 0.3 equivalent of t-BuONa base at 130 ºC for 2 h. A wide range of aliphatic terminal alkenes, styrenes, and terminal alkenes bearing functional groups such as removable hydroxy-protecting groups were successfully employed. Notably, the hydroarylation was highly selective at the terminal olefins in the presence of an internal double bond, albeit partial isomerimerization was also observed. Alkenes containing strained internal double bonds such as norbornene smoothly reacted. Moreover, the reaction can be carried out using ethylene (5 atm) as alkylation reagents. The authors also extended this transformation to the hydroarylation of alkynes.

Accepted Manuscript Based on computational calculations and experimental data, the authors proposed a mechanistic pathway of bishydroarylation depicted in Figure 7. They postulated that a Rh(I) complex ligated to tertbutoxide and NHC is the catalytically active species. Firstly, a monodentate rhodium complex B from the stable complex A (R' = H) via endothermic bond rupture of the stable chelate Rh-bipyridyl bond. This "rollover" decomplexation is promoted by the strong σ-donating NHC ligand (IMes) character, which weakening the trans Rh-N bond relative to NHC. The oxidative addition of the C-H bond is afforded Rh-hydride species C. Olefin insertion into Rh-H bond followed reductive elimination of a monoalkylated bipyridine E (R' = CH2CH2R) give the first functionalization. Finally, E re-enters into the second catalytic cycle by starting with a bidentate complex F (R′ = CH2CH2R) to follow the same pathway as in the first catalytic cycle, affording the bishydroarylated product.

Accepted Manuscript In 2016, Chang used the same rollover-decomplexation strategy for the bisalkylation of terpyridine derivatives (Scheme 15). [35] They employed this time a well-defined Rh(NHC) catalyst [RhCl(COD)(IMes)] associated with t-BuONa. The alkylation took place at the C3 and C5-positions of the central pyridine unit. The reaction smoothly proceeded, affording a library of functionalized tridentate compounds in high yields over a broad range of alkenes, including aliphatic alkenes, styrenes, congested alkenes, and also ethylene gas. A similar mechanism to the Rh-catalyzed C-H bond bisalkylation of bipyridyl C-H bonds is proposed (Figure 7). ). [36] The 2,2-bipyridine-6-carboxamides were regioselectively arylated at the C3'-position using 10 mol% Pd(OAc)2 associated with Cs2CO3 (4 equivalents) in mesitylene (2 mL) at 160 ºC. Notably, the primary amide group at the C6-position is required, as reaction from substrates bearing secondary amide carboxylic acid, carboxylic ester, or 2,2-bipyridine itself failed to afford the corresponding arylated bipyridine products. Both electron-rich and electron-poor of aryl iodides were efficiently coupled, and the reaction displayed a broad functional group tolerance such as halogen (Br, I), trifluoromethoxy, carboxylic acid ester, aldehyde, ketone, cyano, nitro, and trifluoromethyl groups.

Accepted Manuscript

Accepted Manuscript In 2019, the same group employed a similar strategy (i.e., rollover cyclometalation pathway of 2,2bipyridine-6-carboxamides) in Rh-catalyzed C-H acylmethylation with sulfoxonium ylides (Scheme 17). [37] Under the optimized conditions, namely [Cp*RhCl2]2 (3.0 mol %) as a catalyst, NaOAc (1 equiv) as a base, and AgSbF6 as an additive in acetonitrile at 120 ºC under air for 24 h, the C3 alkylated 2,2-bipyridine-6-carboxamides were isolated in good to excellent yields. In contrast with the C-H bond arylation using Pd catalysis, the functionalization took place on the pyridine unit bearing the amide group, and 2º and 3º amides could be employed. However, no reaction occurred without an amide group at the C6 position. Aroyl sulfoxonium ylides holding electron-donating and electronwithdrawing groups as well as heteroaroyl sulfoxonium ylides and alkanoyl sulfoxonium ylides could be employed as acylmethylation with high yields.

Scheme 17. Rh-Catalyzed Regioselective C3-H Acylmethylation of [2,2′-Bipyridine]-6carboxamides with Sulfoxonium Ylides

Accepted Manuscript

The authors proposed a plausible reaction pathway (Figure 8). In 2020, Chen, Liu, Zhu, and co-workers developed a novel approach for the C-H functionalization of 2,2'-bipyridine derivatives through a rollover cyclometalation pathway (Scheme 18). [38] The substituent at the C6 position of 2,2'-bipyridine weakened the adjacent N-Rh bond promoting the rollover cyclometalation, C-H activation, and functionalization sequence. The pyridine ring of 2,2'-bipyridine holding the C6 substituent was annulated via an oxidative double C-H bond alkenylation using 5 mol% [Cp*RhCl2]2, in the presence of 2.5 equivalents of AgOAc and 5.0 equivalents of NaOAc in 1,2dichloroethane (DCE) at 110 ºC. The author shows that a substituent at the 6-position of 2,2'-bipyridine is mandatory and can be Br, Cl, F, Me, and MeO. Diarylalkynes diversely substituted by electrondonating or -withdrawing groups at the para-or meta-position of the aryl ring nicely reacted, while no reaction occurred with ortho-substituted diarylalkynes. Reactions with di(2-naphthyl)acetylenes, di(2thienyl)acetylenes or hex-3-yne were also successful. Moreover, reaction with unsymmetrical alkyne (e.g., prop-1-yn-1-ylbenzene) led to a single regioisomer.

C-H Bond Functionalization of 2,2'-Bipyridines using Photoredox Catalysis

Recently, photoredox-catalyzed Minisci-type reaction has become a preferred method for the sustainable functionalization of pyridines. [2c, 39] The reaction involves the formation of a radical, which adds to the C2 or C4 position of the pyridine ring (Figure 10). In the cases of 2,2'-bipyridines, the In 2017, during their investigations on C-H bond alkylation of pyridine derivatives using redox-active esters [N-(acyloxy)phthalimide] in the presence of photoredox catalysis, Shang, Fu, and co-workers described one example of alkylation of 2,2'-bipyridine (Scheme 19, top). [40] From 4,4'-di-tert-butyl-2,2'-bipyridine and isobutyric acid NHPI ester (5 equivalents) in the presence of 2mol% Ir[dF(CF3)ppy]2(dtbbpy)PF6, 4 equivalents of trifluoroacetic acid in DMA under irradiation at room temperature, a mixture of mono-and di alkylated 4,4'-di-tert-butyl-2,2'-bipyridines was obtained. The alkylation regioselectivity took place at C6 and C6'-positions. In 2020, Song, Xu, and co-workers reported one example of the oxidant-free, electrophotochemically driven C-H alkylation of 2,2'bipyridine with carboxylic acids through H2 evolution (Scheme 19, bottom). [41] The optimal reaction system [namely, 5 mol% of CeCl3•7H2O as the catalyst, 1 equivalent of HCl as an acid additive, 0.5 equivalent of nBu4NCl as the supporting electrolyte, and a mixed solvent of HFIP/TFE (3:2) in an undivided cell exposed to 392 nm LEDs] also gives a mixture of mono-and di-alkylated products, albeit the monoalkylated 2,2'bipyridne is obtained as the major product.

Scheme 19. Decarboxylative Alkylation of 2,2'-Bipyridines

In 2020, our group reported the synthesis of asymmetrically substituted 2,2'-bipyridines through the desymmetrization of 2,2'-bipyridines via two successive photoredox C-H bond functionalizations. The first step is the C-H bond arylation of 4,4'-disubstituted 2,2'-bipyridines using in-situ formed diazonium salts (from commercially available aromatic amines) in the presence of a photoredox catalyst under blue LEDs irradiation (Scheme 20). [42] Besides safety concerns avoiding the manipulation of explosive aryl diazonium salts, the other main advantage to employ in-situ diazotization of anilines is a slower generation of the aryl radical than reactions from aryl diazonium salts favorizing the monoarylation. The coupling reaction between 4,4'-di-tert-butyl-2,2'-bipyridine (4 equivalents) or 4,4'-dimethyl-2,2'bipyridine and diversely substituted aniline (1 equivalent) -in a mixture of CH3CN/TFA (8:1) solvent containing 1 mol% (Ir[dF(CF3)ppy]2(dtbpy))PF6and 1.2 equivalents of t-BuONO under blue LEDs irradiation-afforded the monoarylation of 2,2'-bipyridines in good yields.

Accepted Manuscript

After having succeeded the mono-arylation of 2-2'-bipyridines, our group also investigated the C-H bond alkylation of 6-aryl-2-2'-bipyridines through Minisci reaction using redox-active esters [N-(acyloxy)phthalimide] and phoptoredox catalysis (Scheme 21). [43] Under the optimized conditions [2.5 mol% fac-Ir(ppy)3, DMSO -TFA (4:1) as the solvent mixture and blue light irradiation over 72 h], 4,4'-di-tert-butyl-6-phenyl-2,2'-bipyridine was alkylated with 1º, 2º, and 3º alkyl group. The reaction also allows to introduce of some functional groups such as alkene, ester, and amine. Densely functionalized dehydrocholic acid and both enantiopure (-)-and (+)-menthyloxyacetic acid NHP esters derived from naturally occurring and bioactive chiral acid feedstocks were also usefully employed as alkyl radical precursors. The reaction also tolerates halogen atoms such as Cl or Br on the phenyl at the C6 positions. These functionalized 6'alkyl 4'-di-tert-butyl-6-phenyl-2,2'-bipyridines obtained from 4'-di-tert-butyl2,2'-bipyridines via two successive C-H bond functionalizations were later employed as cyclometalated ligand in the preparation of luminescent Pt(II) complexes. Very interestingly, we demonstrated that the incorporation of a chiral group onto a bipyridine scaffold enables the resulting Pt(II) complex to exhibit electronic circular dichroism (ECD) and circularly polarized luminescence (CPL) properties.

Scheme 21. Photoredox-Catalyzed Minisci Alkylation of 6-Aryl-2,2'-Bipyridines

C-H Bond Functionalization of Activated Bipyridines

As seen in section 3, the C-H bonds of pyridine N-oxides are more reactive than those of pyridines.

During their investigation on photocatalyzed ortho-alkylation of pyridine N-oxides through alkene cleavage, Miura, Murakami, and co-workers reported one example of C-H bond alkylation of 2,2'bipyridines N-oxide (Scheme 22). [44] The reaction proceeded through a radical pathway using an acridinium-based photocatalyst. From 4,4'-dimethyl-[2,2'-bipyridine] N-oxide and cinnamyl propionate the mono-benzylated 4,4'-dimethyl-[2,2'-bipyridine] at C6-position was obtained in 58% yield. In 2020, Fu, Chen, and co-workers investigated selective C2-alkenylation of pyridines with internal alkynes via the pyridinium salt activation strategy using Cu/Pd as dual catalysis. [47] Besides the functionalization of pyridines, they also reported one example of the alkylation of 2,2'-bipyridine (Scheme 25). They employed a bimetallic system Pd(OAc)2 associated with CuBr to functionalized 1methyl-[2,2'-bipyridin]-1-ium salt. The reaction involves the C-H bond cleavage by Cu(I) to form 2pyridiniumyl-Cu(I) species, due to the increased acidity, followed a nucleophilic attack to the palladium coordinated alkynes. The N-alkyl group could be in situ removable via an N-dealkylation process, affording the functionalized 2,2'-bipyridine in one-pot.

Accepted Manuscript

Scheme 25. Pd/Cu Catalyzed C-H Alkenylation of 2,2'-Bipyridinium Salts with Alkynes

Conclusion

The growing interest in the field of material sciences as well as catalysis has led to the development of new eco-friendly protocols for the synthesis of various 2,2'-bipyridine derivatives. In this review, we account for the recent developments that have been achieved concerning the use of C-H bond functionalization for C-C bond formation that allow the selective formation of highly functionalized 2,2'-bipyridines. While the dimerization of pyridines is an established approach for the synthesis of 2,2'-bipyridine, only recently homogeneous catalysis has been employed to improve the functional group tolerance. However, this method remains limited to the synthesis of symmetrical 2,2'-bipyridines.

A catalytic system enabling the general cross-coupling of two different pyridines in high yield and selectivity has not been discovered yet. Asymmetrically 2,2'-bipyridine derivates can be prepared via C-H bond heteroarylation of pyridine derivatives with 2-halopyridines, but activated pyridines such as pyridines N-oxides or N-iminopyridinium ylides has to be employed to enhance the reactivity of the C2-H bond. The major drawback of this approach is the final deprotection to access 2,2'-bipyridines. The remarkable developments of the late-stage functionalization of bipyridines through regioselective C-H bond cleavage profoundly impact the synthesis of symmetrical and asymmetrical 2,2'-bipyridines holding versatile functional groups at different positions in an efficient and often high-yielding manner.

In the near future, the combination of all these strategies among with the development of new protocols for the C-H bond activation of pyridines and 2,2'-bipyridines shall open the way of on-demand synthesis of well-decorated ligands with the desired substitution patterns allowing chemists to design novel organometallics materials or catalysts displaying better properties.

Figure 1 .

 1 Figure 1. Representative 2,2'-Bipyridyl Ligands in Transition-Metal Catalysis and Material Sciences

Figure 2 .

 2 Figure 2. Synthesis of 2,2'-Bipyridines: Scope of this Review

Scheme 2 .

 2 Scheme 2. Polynuclear Organometallic Complexes-Catalyzed Dehydrogenative Coupling of 4-Substituted Pyridines

Scheme 3

 3 Scheme 3 Pd-Catalyzed Dehydrogenative Synthesis of 2,2'-Bipyridines via Regioselective Pyridine Dimerization The authors proposed that the reaction is initiated from a pyridine C2-H activation by palladium(II) acetate to give pyridyl palladium I. A second C-H bond palladation affords bis(pyridyl)palladium intermediate II, which undergoes reductive elimination to furnish the 2,2'-bipyridine and Pd(0). Finally, Ag salts play the role of oxidant to generate the Pd(II) catalytic active species (Figure 3).

Figure 3 .

 3 Figure 3. Proposed Mechanism for Pd-Catalyzed Pyridine Dimerization

Scheme 5 :

 5 Scheme 5: Pd-Catalyzed Homocoupling of Pyridine N-O oxide

  This step is followed by transmetallation with Cu(OAc)2 to give intermediate II, which undergoes oxidative coupling in the presence of air to yield the homocoupled 2,2'-bipyridine N,N'-dioxide. In the absence of copper acetate, the reaction follows an alternative nucleophilic aromatic substitution pathway (SNAr), where intermediate I attacks another molecule of pyridine N-oxide to yield intermediate III, which losses LiOH to produce the homocoupled 2,2'-bipyridine N-oxide.

Figure 4 .

 4 Figure 4. Proposed Mechanisms for Homocoupling of 2-Arylpyridine N-Oxides.

  Scheme 8. t-BuOK-Mediated Deoxygenative Homocoupling of Pyridine N-OxidesThe authors suggested that the reaction is initiated by the deprotonation of DMF or the C2-H of pyridine N-oxide to give carbanions A and B (Figure5). Then, a single electron transfer (SET) reduction of pyridine N-oxide produces radical anion D, which adds to a second molecule of pyridine N-oxide to produce radical anion E that, after the recombination with radical C gives rise to intermediate F. Basemediated elimination of ROH and hydroxide can then produce 2,2'-bipyridine. Alternatively, a SET reduction of anion radical E can produce dianion G that is converted to 2,2'-bipyridine via a stepwise base-mediated elimination of hydroxide anion.

Figure 5 .

 5 Figure 5. Proposed Mechanisms for Homocoupling of 2-Arylpyridine N-Oxides.

Scheme 14 .

 14 Scheme 14. Rh-Catalyzed C-H Bond Bisalkylation of Bipyridines: Rollover Cyclometalation Pathway

Figure 7 .

 7 Figure 7. Proposed Mechanistic Pathway of the Rh(NHC)-Catalyzed Bishydroarylation Reaction.

Scheme 15 .

 15 Scheme 15. Rhodium-Catalyzed C-H Alkylation of NNN tridentate Chelating Compounds via a Rollover PathwayIn 2016, Cheng and co-workers employed this rollover cyclometalation pathway for the C-H bond arylation of 2,2-bipyridine-6-carboxamides with aryl iodides using palladium catalysis (Scheme 16).[36] The 2,2-bipyridine-6-carboxamides were regioselectively arylated at the C3'-position using 10 mol% Pd(OAc)2 associated with Cs2CO3 (4 equivalents) in mesitylene (2 mL) at 160 ºC. Notably, the primary amide group at the C6-position is required, as reaction from substrates bearing secondary amide carboxylic acid, carboxylic ester, or 2,2-bipyridine itself failed to afford the corresponding arylated bipyridine products. Both electron-rich and electron-poor of aryl iodides were efficiently coupled, and the reaction displayed a broad functional group tolerance such as halogen (Br, I), trifluoromethoxy, carboxylic acid ester, aldehyde, ketone, cyano, nitro, and trifluoromethyl groups.

Scheme 16 .

 16 Scheme 16. Palladium-Catalyzed Site-Selective C-H Arylation of 2,2′-Bipyridine-6-carboxamides via a Rollover Cyclometalation Pathway

  First, the coordination of 2,2′-bipyridine-6-carboxamides with Rh(III) forms Rh(III) complexes A, which undergo a reversible rollover cyclometalation to give complexes B. Then, the intermediates B are captured by the sulfoxonium ylides to generate Rh-carbene complexes C along with the extrusion of DMSO. The migratory insertion of complexes C leads to intermediates D. Finally, the protonolysis of D delivers the final product and closes the catalytic cycle.

Figure 8 .

 8 Figure 8. Plausible Catalytic Cycle for Rh-Catalyzed Regioselective C3-H Acylmethylation of [2,2′-Bipyridine]-6-carboxamides with Sulfoxonium Ylides

Scheme 18 .

 18 Scheme 18. Rh-Catalyzed C-H Bond Functionalization of 6-Substituted 2,2'-Bipyridines with Alkynes From experimental data and DFT calculation, the authors proposed a plausible reaction pathway (Figure 9). First, the coordination of 2,2'-bipyridine with Cp*Rh III generates Rh III complexes A, followed by rollover decomplexation (B) and cyclometalation to give complex C. Then, 1,2-insertion of alkyne into Rh-C bond forms intermediate D. Next, the N-Rh decomplexation with the help of the excess amount of Ag+ induces the C-H cleavage, leading to new five-membered rhodium complex E. This rhodacycle intermediate can insert the second alkyne to give seven-membered rhodacycle F or G. Finally, the reductive elimination of F or G affords the annulated pyridine, and Rh I is oxidized by AgOAc to regenerate the Rh III active species.

Figure 9 .

 9 Figure 9. Proposed Mechanism for Rh-Catalyzed C-H Bond Functionalization of 6-Substituted 2,2'-Bipyridines with Alkynes

  reaction begins with the generation of excited state catalyst *PC from ground state PC by visible-light, a single electron transfer (SET) occurs from *[PC] to the radical precursors leading to the formation of a free radical (R 1 ) among with the oxidized catalyst [PC + ]. Then, the radical adds to one of the protonated pyridine units at C2 or/and C4 position(s) -depending on its substitution patterns-yielding the 2,2'-dipyridine radical-cation intermediate (A). Then, A is subsequently transformed into a carbocation intermediate B via a SET from [PC + ] with the loss of a proton. Finally, the rearomatization gives the monofunctionalized 2,2'-bipyridine, which can undergo a second functionalization via a similar mechanism. As the two C6 and C6'-positions of the 2,2'-bipyridine derivatives display a similar reactivity, it is challenging to control the mono versus difunctionalizations.

Figure 10 .

 10 Figure 10. General Pathway for the C-H Bond Functionalization of 2,2'-Bipyridines using Photoredox catalysis

Scheme 20 .

 20 Scheme 20. Synthesis of 6-Aryl-2'2'-bipyridines via Photoredox-Induced Meerwein Arylation with in-situ Diazotization of Anilines

Scheme 24 .

 24 Scheme 24. Decarboxylative C-H Alkylation of 2,2'-Bipyridine-N-Oxide by Visible Light/Copper Catalysis

Acknowledgements

Financial supports from CNRS, Univ. Rennes 1, Rennes Métropole, and Région Bretagne (Boost'Europe 18003332 for WH) are acknowledged.

In 2019, Wu and co-workers described one example of alkylation of 2,2'-bipyridine N-oxide using ndec-5-yne as an alkylation reagent (Scheme 23). [45] Both starting materials were irradiated under blue LEDs in the presence of the photocatalyst 9-mesityl-10-methylacridinium perchlorate(Mes-Acr + ClO4 -).

The reaction proceeded with a formal N-O atom transfer according to the alkylated 2,2'-bipyridine in 60% yield.

Scheme 23. Photoinduced Alkylation of 2,2'-Bipyridine N-Oxide with Alkynes under Anaerobic and Aerobic Conditions

In 2020, Loh, Tian, and co-workers reported a highly site-selective alkylation of heteroarene N-oxides (one example of 2,2'-bipyridine N-oxide) using hypervalent iodine(III) carboxylates as an alkylating agent in the presence of a cheap copper catalyst under visible light conditions (Scheme 24). [46] In contrast with two previous protocols, the reaction afforded the C6-alkylated 2'-bipyridine N-oxide, not the reduced form. Cu


Islem Snoussi
email: islem.snoussi@etud.u-picardie.fr

Nadia Hamani

Nassim Mrabti
email: nassim.mrabti@u-picardie.frl.kermad@iut.univ-paris8.fr

Lyes Kermad

Robust Optimization for Collaborative Distribution Network Design Problem

Keywords: Hub Location Problem, Horizontal collaboration, Sustainability, Robust optimisation, Budget of uncertainty

Collaboration is an interesting solution adopted to improve the levels of sustainability by massifying flows. In this study, we present a new design of a distribution network under uncertainties at two levels in a collaborative context. The proposed network includes suppliers who collaborate to deliver their products to retailers through common platforms. On the other hand, uncertainty is treated in terms of two parameters, namely demands and unit transportation costs. The different used models are validated and analysed by a case study. Computational results provided by the robust model and those given by the deterministic one are compared to evaluate the importance of uncertainty. The effects of uncertainty level on the optimal network configuration are also highlighted.

Introduction

Nowadays, faced with high customer's demands, increasing transport costs and the drawbacks of Covid-19 health crisis on the market, companies find themselves in complex situations. Therefore, they must optimize their logistics operations by opting for a strategy that creates an efficient logistics system. According to [START_REF] Habibi | Collaborative Hub Location Problem under Cost Uncertainty[END_REF], [START_REF] Ferrell | Horizontal Collaboration: Opportunities for Improved Logistics Planning[END_REF] and [START_REF] Aloui | Systematic Literature Review on Collaborative Sustainable Transportation: Overview, Analysis and Perspectives[END_REF], the best solution that should be adopted to solve these problems is the integration of collaboration. In the literature, the latter can be Vertical (VC) or Horizontal (HC) [START_REF] Ouhader | Assessing the Economic and Environmental Benefits of Horizontal Cooperation in Delivery: Performance and Scenario Analysis[END_REF]. Besides, [START_REF] Moutaoukil | Pooling Supply Chain: Literature Review of Collaborative Strategies[END_REF] asserted that vertical collaboration takes place between partners who belong to the same logistic chain and are not at the same level. This type of collaboration is mainly limited to the sharing of information between partners, while in horizontal collaboration, also named ''pooling'', means and resources are shared between partners who are at the same level and do not belong to the same logistic chain [START_REF] Mrabti | The Pooling of Sustainable Freight Transport[END_REF]. In fact, most of the existing studies focused on VC. However, its performances can be improved only by pooling. HC involves massification of flows through concentrating certain flows on the same site to optimize the supply and distribution circuits. It allows increasing the frequency of delivery, augmenting service rate, Snoussi I., Hamani N., Mrabti improving the vehicle fill rate and, thus, reducing the logistics costs and the greenhouse gas (GHG) emissions. An important issue that arises while treating a Distribution Network Design Problem is how to cope with data uncertainty because the parameters of the logistic system are variable. For this reason, the deterministic approach is unfavourable to achieve the mission, such as for long-term strategic decisions like the location of hub facilities related to some parameters (e.g. demands, transportation costs, etc.). In cases where these parameters can be estimated, their probability distributions are determined by using stochastic programming techniques. However, in cases where the only available information is the specification of intervals containing the uncertain values of these parameters, the solution is the robust optimization techniques that can perform well even in the worst-case scenarios [START_REF] Zetina | Robust Uncapacitated Hub Location[END_REF]. In this paper, we focus on the collaborative distribution network design problem incorporating demand and unit transportation costs uncertainties when the only available information is an interval of uncertainty.

The remainder of the paper is structured as follows. Section 2 is a literature review. Section 3 represents the problem description. Section 4 shows the results produced by each model. Finally, Section 5 is dedicated to the conclusion and our perspectives.

Literature Review

The distribution network design problem is known as the "Hub Location Problem" (HLP) or "Hub Median Problem" (HMP) [START_REF] Hu | Joint Design of Fleet Size, Hub Locations, and Hub Capacities for Third-Party Logistics Networks with Road Congestion Constraints[END_REF]. The combination between this problem and the horizontal collaboration is still underdeveloped in the literature [START_REF] Mrabti | The Pooling of Sustainable Freight Transport[END_REF].

To our knowledge, only Habibi et al. dealt, in [START_REF] Habibi | Collaborative Hub Location Problem under Cost Uncertainty[END_REF], with this combination under uncertainties. They studied a robust capacitated hub location problem under the uncertainty of installation costs for two distribution networks. Their objective was to reduce the costs generated by transportation and hub installations treating three cases of collaboration and four cost-sharing strategies. Moreover, Habibzadeh Boukani et al. [START_REF] Habibzadeh Boukani | Robust Optimization Approach to Capacitated Single and Multiple Allocation Hub Location Problems[END_REF] examined the robust capacitated single allocation and multiple allocation hub location problems under the uncertainty of fixed setup cost and the capacity of each hub using a minimax regret model. The latter is used to minimise the setup and transportation costs. The authors showed that neglecting uncertainty can cause large losses and expenses. Furthermore, Zetina et al. [START_REF] Zetina | Robust Uncapacitated Hub Location[END_REF] introduced robust counterparts for uncapacitated multiple allocation HLP employing a budget uncertainty model. They presented three different cases of uncertain parameters, namely demands, and transportation costs. The third case was solved using a branch-and-cut algorithm implemented on a commercial solver. Talbi and Todosijević [START_REF] Talbi | The Robust Uncapacitated Multiple Allocation p -Hub Median Problem[END_REF] investigated the same problem and proposed a new approach to quantify the robustness of a solution in the presence of uncertainties as it is the case of uncertain demands. So, the approach developed by Sim [START_REF] Bertsimas | Robust Discrete Optimization and Network Flows[END_REF] is, now, considered as a special case. The research work aimed at minimising the transportation costs in the presence of uncertain flows and the model was solved using a heuristic approach which is the VNS (Variable Neighbourhood Search). Correia et al. [START_REF] Correia | A Stochastic Multi-Period Capacitated Multiple Allocation Hub Location Problem: Formulation and Inequalities[END_REF] studied a modelling framework for stochastic Capacitated Multiple Allocation Hub Location Problem with multi-period, dealing with uncertain demands to minimise logistics costs. Martins de Sá et al. [START_REF] Martins De Sá | Benders Decomposition Applied to a Robust Multiple Allocation Incomplete Hub Location Problem[END_REF] proposed a robust optimization for multiple allocation hub location problem with uncertain demand flows and fixed setup costs. In their study, the level of conservatism was adjusted by an uncertainty budget. Authors also evaluated the transportation and setup costs and used benders decomposition and a hybrid heuristic approach to solve large scale problems. Rahmati and Bashiri [START_REF] Rahmati | Robust Hub Location Problem with Uncertain Inter Hub Flow Discount Factor[END_REF] introduced a robust uncapacitated multiple allocation hub location problem with uncertainty of demands, hub establishment fixed cost and inter hub flow discount factor by applying an uncertainty budget model employed to assess the costs generated by transportation and establishing hubs. The same problem was treated by Lozkins et al. [START_REF] Lozkins | Robust Uncapacitated Multiple Allocation Hub Location Problem under Demand Uncertainty: Minimization of Cost Deviations[END_REF], but with only uncertain demands by using a set of scenarios and assigning for each one a probability of occurrence. The problem was formulated as a nonlinear stochastic optimisation problem to reduce the hub installation costs, expected transportation costs, and estimated absolute deviation of transportation costs. Besides, two Benders Decomposition strategies were suggested to solve the obtained mathematical model and then, results were compared to each other. Peiro et al. [START_REF] Peiró | Heuristic Solutions for a Class of Stochastic Uncapacitated P-Hub Median Problems[END_REF] introduced a heuristic procedure for stochastic Uncapacitated r-Allocation p-Hub Location Problem under demand and cost uncertainties. They also developed a heuristic approach for the deterministic part. The objective of the study was to minimise the total cost by reducing the allocation and the transportation costs. Ben Mohamed et al. [START_REF] Ben Mohamed | Designing a Two-Echelon Distribution Network under Demand Uncertainty[END_REF] suggested a methodology for the stochastic design problem of two-stage distribution networks by integrating demand uncertainty.

In summary, most studies dealing with the uncertain distribution network design problem focused on the economic objective, by reducing logistics costs, and evaluated uncertainties related to the latter and the flows. In this paper, we examine the distribution network design problem with robust optimization to evaluate not only the economic dimension, but also the environmental one under demand and unit transportation costs uncertainties.

Problem Description

We cite, in the following sub-section, the objectives of the deterministic mathematical model and explain the uncertainty parameters.

Objectives of the Deterministic Model

The mathematical model proposed by Mrabti et al. [START_REF] Mrabti | The Pooling of Sustainable Freight Transport[END_REF] uses a three-echelon pooled distribution network represented in Figure 1. This distribution network consists of suppliers who collaborate to deliver their products to retailers through shared warehouses and distribution centres. Two objective functions are investigated: the economic function, which minimises logistics costs, and the environmental one that reduces CO2 emissions from vehicles and hubs. The economic objective function, represented by (1), aims to lowering the various logistical costs related to transportation CT, storage CS, late delivery penalties CD, opening hubs CO, and handling CH. The transportation cost CT v ijt, represented by (2), depends on the quantity of product p transported between the origin i and the destination j (qijt pv ), the type of vehicle (capacity Q v , unit costs of empty vehicles C0 v and full vehicles Cq v ), the travelled distance dij and the number of the required vehicles or trips Nijt v . It is important to consider the delay in delivery when designing a pooled distribution network. Indeed, this delay directly affects the service rate and the retailer loyalty. In some above-mentioned studies, the opening cost was predefined since there is a list of capacity choices. However, in this research work, the cost of installing a hub m (CWm) depends on its area Am and the unit opening cost. It is given by equation ( 3). The area of the hub m, depending on the capacity of the hub Cm, the unit area of a pallet AP and a coefficient α, is obtained by (4).

F1 = CT + CS + CD + CO + CH .

(1)

CT ijt v = d ij ( Q v . C q v -C o v Q v . ∑ q ijt pv + 2 . C o v . N ijt v ) p ; ∀ t ∈ T 2 , v ∈ V, (i, j) ∈ A . ( 2
)
CW m = A m . Cw m ; ∀ m ∈ H . (3)
A m = α . AP. C m ; ∀ m ∈ H . (4)
The environmental objective function, represented in [START_REF] Moutaoukil | Pooling Supply Chain: Literature Review of Collaborative Strategies[END_REF], is applied to minimise the CO2 emissions due to vehicles EV (caused by their manufacturing, use/depreciation and freight transportation) and to hubs operation (resulting from their constructions EC, operations EO, heating, cooling, ventilation and auxiliaries, production of domestic hot water, lighting of premises, as well as upkeep and maintenance).

F 2 = EV + EO + EC . (5)
In this study, we consider the collaborative scenario shown in Figure 1. We assume that each supplier is assigned to a single warehouse and that this warehouse can serve multiple distribution centres, while each retailer can only be served by a single distribution centre.

Robust Proposed Uncertainties

Giving that the weaknesses of the stochastic programming make it inadequately used to deal with problems where there is no information about the parameter's distributions, we opt for the robust approach already used in several works [START_REF] Habibzadeh Boukani | Robust Optimization Approach to Capacitated Single and Multiple Allocation Hub Location Problems[END_REF], [START_REF] Talbi | The Robust Uncapacitated Multiple Allocation p -Hub Median Problem[END_REF] and [START_REF] Rahmati | Robust Hub Location Problem with Uncertain Inter Hub Flow Discount Factor[END_REF] to design a collaborative distribution network that can operate in worst-case scenarios. Our model incorporates several uncertainty levels to face the different variations caused by hazards, especially in this period of health crisis caused by COVID-19.

Nowadays, customer's demands cannot be estimated since, during this epidemic, the demands of some companies changed considerably (decrease or increase in sales). Indeed, unit transportation costs vary according to the price of fuel which is very sensitive to several external factors such as geopolitical tensions, epidemics and global growth. Thus, we study the two cases for which the parameters are subject to interval uncertainty. We choose the budget model already applied by [START_REF] Zetina | Robust Uncapacitated Hub Location[END_REF], [START_REF] Martins De Sá | Benders Decomposition Applied to a Robust Multiple Allocation Incomplete Hub Location Problem[END_REF] and [19] as it offers a certain flexibility to decision-makers choose more or less efficient solutions through the uncertainty budget that determines the maximum used number of demand (or unit transportation costs). This budget is defined as uncertain parameter. As a result, each uncertain parameter is assumed to have an interval of uncertainty. Demands are defined as 𝑊 𝑗𝑡 𝑝 ∈ [𝑊 𝑗𝑡 𝑝𝐿 , 𝑊 𝑗𝑡 𝑝𝐿 + 𝑊 𝑗𝑡 𝑝∆ ] where 𝑊 𝑗𝑡 𝑝𝐿 and 𝑊 𝑗𝑡 𝑝∆ ≥ 0 are their nominal and deviation values, respectively. In addition, unit transportation costs are defined as 𝐶 𝑞 𝑣 ∈ [𝐶 𝑞 𝑣𝐿 , 𝐶 𝑞 𝑣𝐿 + 𝐶 𝑞 𝑣∆ ] for the full-loaded vehicles and 𝐶 0 𝑣 ∈ [𝐶 0 𝑣𝐿 , 𝐶 0 𝑣𝐿 + 𝐶 0 𝑣∆ ] for empty vehicles where 𝐶 𝑞 𝑣𝐿 , 𝐶 𝑞 𝑣∆ ≥ 0, 𝐶 0 𝑣𝐿 , and 𝐶 0 𝑣∆ ≥ 0 are their nominal and deviation values for each case, respectively. Designing a collaborative distribution network under the mentioned uncertainties causes the nonlinearity of the mathematical models. As a solution, the dual procedure should be used to linearise them and provide a mixed integer linear programming (MILP) optimization problem for each robust counterpart.

Computational Experiments

In this section, we examine a case study of a French distribution network represented in Figure 2. This network contains 34 nodes that consist of seven suppliers delivering seven products to thirteen retailers via shared warehouses and distribution centres for six weeks. The maximum number of warehouses and distribution centres to be opened is seven for each set. The number of hubs and their storage capacities are determined by the employed model. The resulting MILP formulations are solved exactly using CPLEX and we use 5% of the uncertain parameters as the initial value of the uncertainty budget. Besides, the latter limits the number of uncertain parameters allowed to deviate from their nominal values. This behaviour is justified by the fact that the case where all parameters deviate from their nominal values are rare.

Table 1 presents a comparative study of the robust optimisation results and the fixed deterministic findings. The fixed deterministic case means that the used parameter values are those found in the worst case [START_REF] Martins De Sá | Benders Decomposition Applied to a Robust Multiple Allocation Incomplete Hub Location Problem[END_REF]. Therefore, the values of these parameters are equal to the nominal ones plus the deviations obtained by solving the robust model. The S.eco and S.env are the economic and environmental objective scenarios, respectively. We note that when 5% of the uncertain parameters can take their worst-case values, the total logistics cost of the robust approach is lower than that of the deterministic one. Table 2 shows that the total capacities of the hubs in the robust case are lower than those in the fixed deterministic case. We can also conclude that the integration of uncertainty allows saving the space dedicated to storage, which reduces the storage costs and, therefore, minimises the logistic costs, compared to the fixed deterministic case. One of the parameters that highlight the impacts of uncertainty is the number of vehicles in use (See Table 3). Obviously, the robust approach gives better result than the obtained one in the deterministic case, especially for the third type of vehicles that has the highest logistics costs and CO2 emissions. As a result, logistics costs and CO2 emissions are minimised.

Figure 3 demonstrates that, by presenting more opened distribution centres, the optimal configuration obtained in the S.eco scenario for the uncertain demands case is different from the costs case. This behaviour is due to the use of more resources when demands increase.

Conclusion and Perspectives

This study introduced a collaborative distribution network design problem under uncertainties of demands and unit transportation costs. It minimises logistics costs and CO2 emissions. A case study was examined via a distribution network in France. The obtained results show that the robust optimisation case is better than the fixed deterministic one in terms of costs and CO2 emissions by saving storage space and using a smaller number of vehicles. However, solving the problem exactly with CPLEX limits its size. Thus, it is preferable, in this case, to provide a heuristic method for the resolution of larger instances. We also intend to propose, in our future works, a robust counterpart that considers the uncertainty of the maximum number of vehicles in use and another case that combines demand, unit transportation costs and maximum number of vehicles in use uncertainties.

Fig. 1 .

 1 Fig. 1. Example of a collaborative distribution network.

Fig. 1 .

 1 Fig. 1. Set of selected nodes on the map of France.

Fig. 3 .

 3 Fig. 3. Optimal network configurations of the S.eco scenario.

  N., Kermad L. (2021) Robust Optimization for Collaborative Distribution Network Design Problem. In: Camarinha-Matos L.M., Boucher X., Afsarmanesh H. (eds) Smart and Sustainable Collaborative Networks 4.0. PRO-VE 2021. IFIP Advances in Information and Communication Technology, vol 629. Springer, Cham. https://doi.org/10.1007/978-3-030-85969-5_25

Table 1 .

 1 Summary of the results of uncertain demands and unit transportation costs cases.

	Sustainability indicators	Demands case		Unit transportation costs case
		Fixed	Robust	Fixed	Robust
		deterministic		deterministic
		S.eco S.env S.eco S.env S.eco S.env S.eco S.env
	Costs (10 7 €)	2.14 2.48 1.46 1.66 1.67 1.95 1.48 1.62
	CO2 emissions (10 9 g CO2) 4.33 4.37 3.22 3.20 3.28 3.21 3.18 3.14

Table 2 .

 2 Summary Total capacities of hubs.

	Parameter	Approach	Scenario Total capacity of hubs
				Upstream hubs Downstream hubs
	Demands	Robust	S.eco	3040	3225
			S.env	3029	3219
		Fixed	S.eco	3921	4447
		deterministic	S.env	4261	4185
	Unit	Robust	S.eco	3040	3219
	transportation		S.env	3028	3219
	costs	Fixed	S.eco	2741	2806
		deterministic	S.env	2705	2849

a: Distribution network of demands uncertainty case. b: Distribution network of unit transportation costs uncertainty case.


Olivia Zurita Miranda

Coralie Fourcade Dutin

Pierre Béjot

Frédéric Fauquet

Hervé Maillotte

Patrick Mounaix

Damien Bigourd

Four-wave mixing process induced by a self-phase modulated pulse in a hollow core capillary

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.