Chiara Rossi
email: chiara.rossi@iisb.fraunhofer.de

Alexander Burenkov

Peter Pichler

Eberhard Bär

Jonas Müller

Guilhem Larrieu

Performance of vertical gate-all-around nanowire p-MOS transistors determined by boron depletion during oxidation

Keywords:

Vertical junctionless gate-all-around nanowire transistors show excellent electrical performance and can be fabricated using a top-down approach in conventional CMOS process technology. Thinning of the nanowires to the desired diameter is obtained by sacrificial wet oxidation. Then, the gate oxide is grown by dry oxidation. These oxidation steps deeply affect the doping distribution in the nanowire due to dopant segregation and selfinterstitial injection, especially for p-type dopants. This effect is more pronounced in 3D nanostructures with respect to bulk devices, due to geometry. Modeling the resulting doping distribution is a prerequisite for understanding the electrical properties of the devices and exploring their potential for optimization. In this work, 3D TCAD process and device simulations were performed using Synopsys Sentaurus and the results are compared with experimental data of devices fabricated at CNRS-LAAS. The impact of the implemented process and device models and their capability to predict nanowire properties and device behavior is assessed.

Introduction

The gate-all-around (GAA) nanowire (NW) field-effect transistor (FET) represents one of the most promising candidates for the Power-Performance-Area-and-Cost scaling that replaced Moore's law. GAA-NW-FETs show excellent performance, including a great immunity against short-channel effects and, thanks to their 3D layout configuration, can enhance integration and overcome some physical limitations resulting from 2D layout, such as contact placements and interconnect routing congestion [START_REF] Takato | High performance CMOS surrounding gate transistor (SGT) for ultra high density LSIs[END_REF][START_REF] Lu | Nanowire transistor performance limits and applications[END_REF][START_REF] Singh | High-performance fully depleted silicon nanowire (diameter ≤ 5 nm) gate-all-around CMOS devices[END_REF][START_REF] Yang | Vertical siliconnanowire formation and gate-all-around MOSFET[END_REF][START_REF] Ng | Vertically stacked silicon nanowire transistors fabricated by inductive plasma etching and stress-limited oxidation[END_REF][START_REF] Ritzenthaler | Vertically stacked gate-all-around Si nanowire CMOS transistors with reduced vertical nanowires separation, New Work function metal gate solutions, and DC/AC performance optimization[END_REF][START_REF] Veloso | Vertical nanowire FET integration and device aspects[END_REF]. Vertical GAA-NW-FETs can be fabricated using a top-down approach with conventional processes and very good control on dimensions and localization. Moreover, the gate length is given by the thickness of the deposited gate material, without the need of high-resolution lithography [START_REF] Moers | Turning the world vertical: MOSFETs with current flow perpendicular to the wafer surface[END_REF][START_REF] Guerfi | Vertical silicon nanowire field effect transistors with nanoscale gate-all-around[END_REF]. Thermal oxidation is a key process for the nanowire fabrication. For silicon nanowire, it can be used to shrink the diameter, remove residual damage from the etch process and to grow the gate oxide. Thermal oxidation is also used in SiGe nanowire fabrication to obtain core-shell structures, exploiting selective oxidation of the Si element of SiGe alloys and germanium condensation [START_REF] David | Tailoring strain and morphology of core-shell SiGe nanowires by low-temperature Ge condensation[END_REF][START_REF] Long | Nano-structuring in SiGe by oxidation induced anisotropic Ge selforganization[END_REF][START_REF] Fang | Vertically stacked SiGe nanowire array channel CMOS transistors[END_REF][START_REF] Müller | Engineering of Dense Arrays of Vertical Si1-xGex Nanostructures[END_REF].

During sacrificial oxidations of silicon nanowires for diameter thinning as well as during gate oxide growth, dopants segregate into the growing oxide and self-interstitials are injected into silicon [START_REF] Grove | Redistribution of acceptor and donor impurities during thermal oxidation of silicon[END_REF][START_REF] Fukata | Segregation behaviors and radial distribution of dopant atoms in silicon nanowires[END_REF]. Unlike for the case of oxidation of a bulk planar surface, in silicon nanowires, the self-interstitials cannot diffuse efficiently into the bulk. This increases dopant diffusion and segregation loss in the nanorod while the resupply of dopants from the bulk is less effective due to device geometry. With the inhomogeneous dopant distribution in the nanowire deeply affecting the electrical characteristics and performance of junctionless GAA-NW-FETs, dopant diffusion and segregation during oxidation needs to be correctly modeled. This work discusses 3D simulations of the full process flow for the fabrication of vertical nanowire p-MOS transistors, including the sacrificial oxidation step, with the Sentaurus Process program of Synopsys. Subsequent 3D device simulations, performed with Sentaurus Device, are compared with experimental data of devices fabricated at CNRS-LAAS, obtained by a large-scale CMOS compatible manufacturing process. Firstly, the vertical channel was fabricated by a top-down approach on 4-inch (1 0 0) wafer of boron-doped silicon. Dense circular nanopillars of negative resist were patterned by electron beam lithography [START_REF] Han | Realization of vertical silicon nanowire networks with an ultra high density using a top-down approach[END_REF][START_REF] Guerfi | High resolution HSQ nanopillar arrays with low energy electron beam lithography[END_REF] and transferred onto the substrate by anisotropic reactive ion etching (RIE) resulting in 210 nm high vertical silicon nanowires. Wet thermal oxidation at 850 • C grew 14 nm of sacrificial oxide that was selectively removed from the device in buffered oxide etchant. A second dry oxidation at 725 • C then created a thin gate oxide of nominally 5 nm on the NWs. The oxide on top of the substrate and NWs was removed by dry plasma etching. A 10 nm thick Pt layer was deposited and annealed by rapid thermal annealing (RTA) at 500 • C for 3 min forming Pt-silicide (PtSi). Unreacted Pt was selectively etched in aqua regia. The entire device was covered with a spin-on-glass (SOG) that was etched down to the middle of the channel height, creating the first dielectric spacer [START_REF] Guerfi | Thin-dielectric-layer engineering for 3D nanostructure integration using an innovative planarization approach[END_REF]. The metallic GAA layer was defined by depositing 18 nm of Cr [START_REF] Guerfi | Vertical silicon nanowire field effect transistors with nanoscale gate-all-around[END_REF][START_REF] Larrieu | Vertical nanowire array-based field effect transistors for ultimate scaling[END_REF][START_REF] Larrieu | Sub-15 nm gate-all-around field effect transistors on vertical silicon nanowires[END_REF]. A second SiO x spacer was defined up to the PtSi contacts. The bottom contacts and gate layer were connected to the surface with vias filled by a 400 nm thick aluminum deposition, also covering the PtSi top contacts. Selective wet chemical Al etch-back was used to define contact pads. Finally, an RTA process under forming gas atmosphere of N 2 /H 2 (4 %) at 200 • C for 4 min passivated interfacial defects.

3D TCAD simulation of boron segregation during oxidation

TCAD simulation of the process flow used for device fabrication at CNRS-LAAS started with the geometrical definition of a silicon nanowire, on a silicon substrate, with a diameter as obtained by the initial etching (Fig. 1.a). The whole structure is initially homogeneously doped with boron with a concentration of 3.5•10 19 cm -3 . The first sacrificial wet oxidation (850 • C, 5 min) was simulated and the resulting structure is shown in Fig. 1.b. After the sacrificial oxide stripping (Fig. 1.c), dry oxidation for gate oxide formation (725 • C, 20 min) was performed and the result is reported in Fig. 1.d. Since the oxidation rate depends on the silicon orientation, a diamond-like shape in the nanowire cross-section is expected. This peculiar cross-section shape was initially observed in TCAD simulations also. However, SEM images of nanowires after oxidation like in Fig. 2 always show circular shapes. The circular crosssection was also reported in literature by previous works [START_REF] Ye | Precise fabrication of uniform sub-10-nm-diameter cylindrical silicon nanopillars via oxidation control[END_REF][START_REF] Fazzini | Modeling stress retarded self-limiting oxidation of suspended silicon nanowires for the development of silicon nanowire-based nanodevices[END_REF][START_REF] Uematsu | Two-dimensional simulation of pattern-dependent oxidation of silicon nanostructures on silicon-on-insulator substrates[END_REF][START_REF] Han | Modelling and engineering of stress based controlled oxidation effects for silicon nanostructure patterning[END_REF]. A possible explanation of this phenomenon has been reported by Ye et al. [START_REF] Ye | Precise fabrication of uniform sub-10-nm-diameter cylindrical silicon nanopillars via oxidation control[END_REF]: Due to the increased energy barrier against oxidation of the Si-Si bond at the edges, oxidation at edges is slower than that far from edges. As the oxidation proceeds and the regions far from edges are segmented into additional and finer adjacent surfaces, the energy at the center of the adjacent surfaces increases to the same high value as that at the edge. The same oxidation rate is obtained all around the outer surface of nanowire, leading to a circular shape. The reproduction of the circular shape could finally be obtained also in TCAD simulations, as shown in Fig. 1.e, but required a suitably fine mesh for the nanowire.

The simulated final doping distribution after the two oxidation steps is reported in Fig. 3 for two different nanowire geometries (final diameters of 37 and 28 nm). The doping concentration was computed using the ChargedPair diffusion model and the three-phase segregation model [START_REF] Lau | A model for phosphorus segregation at the silicon-silicon dioxide interface[END_REF] with the parameter set from Sentaurus TCAD AdvancedCalibration S-2021.06. Process simulations show that the doping distribution in the nanowire changes considerably during the oxidation steps. In particular, it is lowered by about one order of magnitude in comparison with the initial dopant concentration. Moreover, the decrease is more pronounced for thinner nanowires. In fact, for smaller nanowire diameters, the surface-to-volume ratio increases and so a higher fraction of the total number of dopants diffuses out from the silicon nanowire into the oxide. Plus, for thinner nanowires, the area through which the dopants can diffuse from the highly doped bulk region is smaller, so the resupply of dopants is increasingly suppressed. Thus, the effect of boron depletion will be even more prominent for future device generations.

Results of 3D device simulations and comparison with experimental data

Starting from the structure after dry oxidation, as depicted in Fig. 2. d and Fig. 3, the next steps of the fabrication of GAA-NW-FETs were subsequently simulated. The gate oxide at the bulk surface and nanowire top was anisotropically etched and nickel was anisotropically deposited there. Silicidation was then simulated. Models for platinum silicide formation are not available in Sentaurus Process, for this reason nickel silicide was selected. During device simulations, the correct work function for platinum silicide was imposed at source and drain contacts. Convergence of the silicidation simulation posed some challenges, and a mesh adjustment was needed. Moreover, silicidation simulation had to be stopped before the total consumption of the metal to avoid fatal errors with a stop of execution. The next process steps simulated were the deposition of spacers and the gate metal layer. The latter defines with its thickness of 18 nm the gate length. The final simulated structure, ready for 3D device simulations, is presented in Fig. 4. For comparison, a SEM image of a fabricated GAA-NW-FET is presented in Fig. 5.

Fig. 6 reports the transfer characteristic obtained by device simulation of the GAA-NW-FET with a diameter of 37 nm, compared with experimental data. For a better visualization, only each second data point of the measured characteristic is shown. The drain voltage is -0.6 V with the source at 0 V. As gate oxide thickness, 5 nm was taken as determined experimentally, and the gate metal work function was set to 3.5 eV. The experimental data exhibit a higher subthreshold swing (147 mV/dec) compared with simulation results (90 mV/dec) obtained with AdvancedCalibration S-2021.06 parameters. To investigate the reason behind this mismatch, different parameters were varied in TCAD simulations. The starting point was the analysis of the doping distribution in the nanowire. Fig. 6 shows, in addition to the I-V curve obtained for the doping distribution resulting from process simulation (Fig. 3.a), simulation results for nanowires with different doping concentrations. Namely, these doping distributions were obtained by multiplying or dividing the doping distribution obtained after process simulation with certain factors. The threshold voltage of the transistor changes considerably already for factors lower than 2. Moreover, the doping concentration also changes the subthreshold slope. The simulated threshold voltage and subthreshold slope as a function of doping multiplication factor are reported in Fig. 7. This confirms the importance of a precise modeling of boron segregation in nanowires. By increasing the doping concentration, a subthreshold slope as in the experiments can be obtained on the cost of a threshold voltage shift in direction of positive gate voltages. A second influence investigated comes from the models used for the source and drain contacts. Platinum silicide has a work function of about 4.9-5.2 eV [START_REF] Drummond | Work Functions of the transition Metals and Metal Silicides[END_REF]. Fig. 8 shows the effect of the silicide work function on the characteristics. It has virtually no influence on the threshold voltage or subthreshold slope but a big impact on the oncurrent. Another parameter that was studied is the concentration of charges at the silicon-oxide interface, which modifies the threshold voltage as it can be seen in Fig. 9. In summary, these results show that the main influence on the subthreshold slope comes from the doping of the nanowire, whereas the threshold voltage is affected by both the doping and interface charges. This means that boron depletion during oxidation plays an important role in the transistor behavior at and below Fig. 10 reports a comparison of the experimental data and TCAD simulations for a drain voltage of -0.6 V. The simulated I-V curves in Fig. 10 were obtained using a gate metal work function of 3.5 eV and interface charges with an area density of 3•10 12 cm -2 . The latter appears high considering the forming gas annealing performed and needs to be further investigated. To obtain the experimental subthreshold slope, the doping had to be multiplied by a factor of 1.5. The impact of the oxide thickness on the I-V characteristics is also shown. Considering that the increase of oxide thickness is 20 %, the effect on the transistor behavior is rather low. Experimental results do not show a flat current curve for negative gate voltages, so a silicide work function of 5.1 eV was used. Thanks to these results, we can conclude that TCAD model parameters for diffusion and segregation still need to be improved for a better agreement to the measured electrical data, as the doping distribution in the nanowire appears currently underestimated. In order to obtain a more precise estimation of doping concentration in 3D nanostructures, diffusion and segregation parameters need to be recalibrated.

Conclusion

We reported on results of full 3D TCAD process and device simulations of vertical junctionless GAA-NW-FETs, with particular emphasis on the effect of boron depletion after oxidation. Vertical nanowires were fabricated at CNRS-LAAS exploiting wet sacrificial oxidation for the thinning of the nanowires. Reproduction of the circular shape of the nanowire cross section after oxidation, as observed experimentally, was achieved in TCAD simulations, but it was only possible with a fine mesh. The doping distribution in the nanowire is deeply affected by thermal oxidation due to dopant segregation and self-interstitial injection. 3D simulation results show a decrease of boron doping in the nanowire of about one order of magnitude, with the decrease being more pronounced for thinner nanowires. 3D device simulations compared with experimental data showed that the fabricated GAA-NW-FETs have a higher subthreshold swing with respect to simulated devices using Advanced Calibration S-2021.06 Sentaurus TCAD parameters. The impact on the transfer characteristic of key technological parameters was analyzed, i.e. doping, silicon-oxide interface charges and work function of source and drain silicide. In particular, the lowering or increasing of the doping by a factor of 2 or less changes considerably the value of the threshold voltage and also affects the subthreshold slope. By choosing appropriate parameters, a fit to the experimental data was obtained. The doping distribution obtained during process simulation had to be modified to achieve the experimental subthreshold slope (multiplication by a factor of 1.5), meaning that the Sentaurus TCAD Advanced-Calibration S-2021.06 of boron segregation and diffusion during oxidation underestimate the doping distribution in the nanowire. In conclusion, these results show the importance of precise modeling of boron depletion during oxidation for a correct prediction of the electrical characteristic of junctionless GAA-NW-FET and call for further work to obtain a set of model parameters for a better estimation of doping in nanowires.

Fig. 1 .Fig. 2 .

 12 Fig. 1. a) Initial structure after etching of the nanowire; b) nanowire after sacrificial wet oxidation (oxide in light blue); c) thinned nanowire after oxide stripping; d) nanowire after dry oxidation for gate oxide growth; e) cross section of the nanowire after gate oxidation. (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)

Fig. 3 .

 3 Fig. 3. Electrically active doping concentration in nanowires after sacrificial wet oxidation and gate dry oxidation for two different nanowire diameters: a) 37 nm, b) 28 nm. The graphs show a 2D cut-plane of the 3D simulated structures.

Fig. 4 .

 4 Fig. 4. Final 3D structure of GAA-NW-FET at the end of process simulation, on the right: cross section.

Fig. 5 .

 5 Fig. 5. TEM image of vertical GAA-NW-FET [15] fabricated at CNRS-LAAS.

Fig. 6 .

 6 Fig. 6. Simulated I-V characteristics (NW diameter of 37 nm) for different doping distribution and experimental I-V curve. "Doping conc." refers to the doping concentration as plotted in Fig. 3.a.

Fig. 7 .

 7 Fig. 7. Simulated subthreshold slope and threshold voltage as a function of the doping multiplication factor, i.e. the factor by which the doping distribution obtained by process simulation was multiplied before being used for device simulation.

Fig. 8 .

 8 Fig. 8. Simulated I-V characteristics (NW diameter of 37 nm) for different silicide work functions. The doping concentration from process simulation is considered.

Fig. 9 .

 9 Fig. 9. Simulated I-V characteristics (NW diameter of 37 nm) for different charges concentration at silicon-oxide interface. The doping concentration from process simulation is considered.

Fig. 10 .

 10 Fig. 10. Comparison of experimental and simulated I-V characteristics (NW diameter of 37 nm). Parameters for simulated curves: gate metal work function of 3.5 eV, interface charges of 3•10 12 cm -2 , doping multiplied by 1.5.

Acknowledgment

The research leading to these results has received funding from the European Union's Horizon 2020 research and innovation programme under grant agreement No. 871813 MUNDFAB. This work was partly supported by LAAS-CNRS micro and nanotechnologies platform, a member of the Renatech french national network.

Data availability

Data will be made available on request.

Declaration of Competing Interest

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.


Matthieu Mandard
email: matthieu.mandard@univ-rennes.fr

Anthropological Grounds of Climate Equity Principles

Keywords: Anthropology, climate equity principles, distributive justice, relational models theory

As repeatedly acknowledged since the ratification of the United Nations Framework Convention on Climate Change in 1992, the equitable allocation of climate burden among parties is at the heart of climate negotiations. But while equity principles have been extensively addressed in the literature on climate policy, their fundamental underlying values remain elusive. Drawing on anthropology and relational models theory, this Perspective paper aims to shed light on this topic by showing that any climate equity principle corresponds to one of the four relational models people in all cultures use to implement their exchanges -and thus corresponds to their associated principles of distributive justice. Such a clarification may facilitate academic research on this topic, as well as the definition, the acceptance and, finally, the effective implementation of climate policies.

Anthropological Grounds of Climate Equity Principles

Introduction

Since the ratification of the United Nations Framework Convention on Climate Change in 1992, the equitable allocation of climate burden among parties is a key issue in climate negotiations [START_REF] Dooley | Ethical choices behind quantifications of fair contributions under the Paris Agreement[END_REF][START_REF] Klinsky | Why equity is fundamental in climate change policy research[END_REF][START_REF] Pottier | A survey of global climate justice: from negotiation stances to moral stakes and back[END_REF]. The main reasons are twofold [START_REF] Frey | Why do climate change negotiations stall? Scientific evidence and solutions for some structural problems[END_REF][START_REF] Grecksch | Access and allocation in climate change adaptation[END_REF][START_REF] Höhne | Regional GHG reduction targets based on effort sharing: a comparison of studies[END_REF][START_REF] Powers | What's fair in international politics? Equity, equality, and foreign policy attitudes[END_REF]. On the one hand, perception of fairness positively affects the probability of reaching an agreement regarding how climate adaptation and mitigation measures should be distributed. On the other hand, agreements that conform to parties' interests are generally better accepted and thus implemented.

To accompany these political debates, the concept of equity has consequently been extensively addressed in the literature on climate policy and has been the subject of several attempts at classification, whether by grouping equity principles or by connecting them to theories of distributive justice (e.g., [START_REF] Davidson | How Fairness Principles in the Climate Debate Relate to Theories of Distributive Justice[END_REF][START_REF] Mattoo | Equity in climate change: an analytical review[END_REF][START_REF] Page | Distributing the burdens of climate change[END_REF][START_REF] Ringius | Burden sharing and fairness principles in international climate policy[END_REF][START_REF] Ryberg | Downscaling the planetary boundaries in absolute environmental sustainability assessments-A review[END_REF]. Despite these efforts, however, there is still no consensus regarding the key similarities and differences between these principles and, as pointed out by various scholars, their fundamental underlying values remain elusive [START_REF] Davidson | How Fairness Principles in the Climate Debate Relate to Theories of Distributive Justice[END_REF][START_REF] Dooley | Ethical choices behind quantifications of fair contributions under the Paris Agreement[END_REF][START_REF] Klinsky | Why equity is fundamental in climate change policy research[END_REF]. This is problematic, inasmuch fairness is a fundamental condition for institutions to envision climate actions [START_REF] Frey | Why do climate change negotiations stall? Scientific evidence and solutions for some structural problems[END_REF][START_REF] Grecksch | Access and allocation in climate change adaptation[END_REF][START_REF] Höhne | Regional GHG reduction targets based on effort sharing: a comparison of studies[END_REF][START_REF] Powers | What's fair in international politics? Equity, equality, and foreign policy attitudes[END_REF].

In line with a growing literature convoking anthropology to illuminate climate change (e.g., [START_REF] Baer | The Anthropology of Climate Change: An Integrated Critical Perspective[END_REF][START_REF] Crate | Climate and culture: anthropology in the era of contemporary climate change[END_REF]Crate and Nuttall, 2016;[START_REF] O'reilly | Climate change: Expanding anthropological possibilities[END_REF], the objective of this Perspective paper is to draw on relational models theory to show that any climate equity principle corresponds to one of the four relational models people in all cultures use to implement their exchanges -and therefore corresponds to their associated principles of distributive justice. Such a clarification may facilitate academic research on this topic, as well as the definition, the acceptance and, finally, the effective implementation of climate policies.

Relational models theory

Relational models theory is a framework for understanding social relationships proposed by the American anthropologist Alan Page Fiske in the mid-1980s and later published in a book entitled Structures of Social Life: The Four Elementary Forms Of Human Relations [START_REF] Fiske | Structures of social life: The four elementary forms of human relations: Communal sharing, authority ranking, equality matching, market pricing[END_REF] and in an article issued in the Psychological Review under the title "The Four Elementary Forms of Sociality:

Framework for a Unified Theory of Social Relations" [START_REF] Fiske | The four elementary forms of sociality: framework for a unified theory of social relations[END_REF]. The basic tenet of this theory is simple: social life is essentially based on four relational models upon which people organize their interactions [START_REF] Fiske | Relational models theory 2.0[END_REF][START_REF] Fiske | The four elementary forms of sociality: framework for a unified theory of social relations[END_REF][START_REF] Fiske | Structures of social life: The four elementary forms of human relations: Communal sharing, authority ranking, equality matching, market pricing[END_REF]. Labelled communal sharing, authority ranking, equality matching and market pricing, these models respectively posit that people may (1) treat each other as equivalent, (2) attend to their positions in ordered differences, (3) keep track of the imbalances among them, or (4) orient to ratio values [START_REF] Fiske | The four elementary forms of sociality: framework for a unified theory of social relations[END_REF][START_REF] Fiske | Structures of social life: The four elementary forms of human relations: Communal sharing, authority ranking, equality matching, market pricing[END_REF] -we further detail them in the next section. Of course, while analytically distincts, the theory acknowledges that these models are often employed in combination in everyday life.

Although originally developed in anthropology, this theoretical framework has garnered considerable attention and empirical support in various disciplines, such as psychology, sociology, neurology or management [START_REF] Haslam | Relational models theory: A contemporary overview[END_REF][START_REF] Vodosek | Relational Models Theory[END_REF], with a total of about 6,500 citations for its two seminal publications as of August 2023. However, surprisingly, to the best of our knowledge, it has not been transposed to the study of climate equity principles, though it turns out to be particularly useful on shedding light on their very nature. This is what we intend to do in the next section.

Relational models and climate equity principles

Several climate equity principles have been identified in the literature. We thus chose to focus on the most commonly used ones, which have been recently inventoried by Dooley et alii (2021). Here we show how each of them can be associated with one of the four relational models described by Fiske (Table 1).

- ---------------------------------------Please insert Table 1 about here ----------------------------------------

Communal sharing: to each according to his needs

According to the relational model of communal sharing, people are not differentiated [START_REF] Fiske | The four elementary forms of sociality: framework for a unified theory of social relations[END_REF].

Group members treat each other as equivalent, with no regards for their individual identities. As Fiske puts it, the functioning of such a collective is well illustrated by the Marxist maxim: "From each according to his ability to work, to each according to his needs" (Fiske, 1992, p. 693). This is the kind of behaviour one can observe in families, in hunting and gathering societies or, more generally, between people who share close connections.

Communal sharing appears to be the ethical principle behind need-based equity, which posits that all human beings should be allowed to emit minimal levels of pollution in order to satisfy their needs [START_REF] Dooley | Ethical choices behind quantifications of fair contributions under the Paris Agreement[END_REF][START_REF] Ringius | Burden sharing and fairness principles in international climate policy[END_REF]. These levels of pollution do not have to be equal: in line with the communal sharing model, the least capable actors (e.g., those who have less natural resource endowments) may have less ambitious climate burdens to secure a decent standard of living 1 . While criticized, as emitting greenhouse gas is seen by some authors as loosely correlated to the satisfaction of needs, such an approach is largely defended by less developed countries, mostly located in South Asia and Africa, which are particularly concerned with poverty issues [START_REF] Pottier | A survey of global climate justice: from negotiation stances to moral stakes and back[END_REF][START_REF] Tørstad | Fairness in the climate negotiations: what explains variation in parties' expressed conceptions?[END_REF].

Here, rights are not acquired because people belong to the same family or to the same tribe, but because they are part of the same humanity, which induces a sense of community. This is what is acknowledged in the preamble of the 2015 Paris agreement, which states that as "climate change is a common concern of humankind, Parties should, when taking action to address climate change, respect, promote and consider their respective obligations on human rights" (UNFCCC, 2015).

Authority ranking: to each according to his position

Authority ranking is characterized by the differentiation of people according to some social dimension [START_REF] Fiske | The four elementary forms of sociality: framework for a unified theory of social relations[END_REF][START_REF] Fiske | Structures of social life: The four elementary forms of human relations: Communal sharing, authority ranking, equality matching, market pricing[END_REF]. Individuals may be positioned above others because of privilege, status or traditions, and are thus entitled to preempt more resources accordingly. Such relationships can typically be observed in military organizations, in caste systems or in religious institutions [START_REF] Fiske | The four elementary forms of sociality: framework for a unified theory of social relations[END_REF].

Interestingly, this relational model is the ethical ground of the initial version of the controversial climate equity principle labelled grandfathering. Originating in 19 th century racist laws promulgated in the southern U.S. states, the historical purpose of grandfathering was to allow actors to continue an activity by exempting them from new regulatory requirements that were imposed on others -the objective was initially to reserve the right to vote for white people, while excluding former slaves and their descendants [START_REF] Damon | Grandfathering: environmental uses and impacts[END_REF][START_REF] Robertson | If Your Grandfather Could Pollute, So Can You: Environmental Grandfather Clauses and Their Role in Environmental Inequality[END_REF]. Transposed to environmental issues, the idea was to offer the possibility for existing polluters to pursue their emissions -at least, temporarily -while imposing more stringent standards on new actors [START_REF] Damon | Grandfathering: environmental uses and impacts[END_REF][START_REF] Robertson | If Your Grandfather Could Pollute, So Can You: Environmental Grandfather Clauses and Their Role in Environmental Inequality[END_REF]. For instance, the 1970 U.S. Clean Air Act allowed old facilities to operate according to less strict pollution standards than similar new facilities [START_REF] Nash | Grandfathering and environmental regulation: the law and economics of new source review[END_REF]. Similarly, the 1975 U.S. Corporate Average Fuel Economy imposed pollution control on new cars while exempting existing vehicles [START_REF] Levinson | Grandfather regulations, new source bias, and state air toxics regulations[END_REF].

In recent decades, however, another definition of grandfathering has emerged [START_REF] Damon | Grandfathering: environmental uses and impacts[END_REF]. More often, regarding climate policy, grandfathering is now no more considered as an exemption from regulatory requirements, but as an allocation of emissions proportional to historical pollution levels [START_REF] Damon | Grandfathering: environmental uses and impacts[END_REF][START_REF] Knight | What is grandfathering?[END_REF]. As a consequence, the associated ethical ground of this newly defined equity principle has shifted from authority ranking to market pricingsee section 3.4.

Nowadays, authority ranking is thus no longer a popular ethical ground of climate equity principles, with a notable exception: this is the rationale that can be found when some authors propose exempting poor countries from any contribution to climate change adaptation and mitigation -this is the so-called sufficientarianist perspective [START_REF] Davidson | How Fairness Principles in the Climate Debate Relate to Theories of Distributive Justice[END_REF]. Here, contrary to grandfathering, such a differentiation is not based on temporal grounds but on economic ones, as it considers whether actors have secured enough of some goods for a basic standard of living [START_REF] Shields | Sufficientarianism[END_REF].

Equality matching: to each the same

The model of equality matching posits that relationships should be evenly balanced [START_REF] Fiske | The four elementary forms of sociality: framework for a unified theory of social relations[END_REF][START_REF] Fiske | Structures of social life: The four elementary forms of human relations: Communal sharing, authority ranking, equality matching, market pricing[END_REF]. Following this line of reasoning, everyone should get the same amount of goods, compensations or punishments regardless of their status or properties. This is the kind of relationship that can be observed between people such as colleagues or acquaintances, who are not intimate and who want to establish trustful interactions [START_REF] Fiske | The four elementary forms of sociality: framework for a unified theory of social relations[END_REF].

As its name indicates, equality matching is the ethical foundation of the general equity principle called equality, which states that all parties should have the same obligations [START_REF] Ringius | Burden sharing and fairness principles in international climate policy[END_REF]. While its formulation is simple, the concrete implementation of this general principle is actually debated [START_REF] Dooley | Ethical choices behind quantifications of fair contributions under the Paris Agreement[END_REF].

The more common approach is equal-per-capita, which assumes that all humans should be seen as equal and that everyone should thus have the same rights to pollute [START_REF] Torpman | Isolationism and the equal per capita view[END_REF]. The Indian government was one of the first to defend this approach at the First Conference of the Parties that took place in Berlin in 1995, and it has since gained wide support among non-governing organizations [START_REF] Pottier | A survey of global climate justice: from negotiation stances to moral stakes and back[END_REF][START_REF] Rajamani | From Berlin to Bali and Beyond: Killing Kyoto Softly?[END_REF]. Although equal-per-capita may seem attractive, it has been criticized for several reasons, such as ignoring peoples' needs, disregarding historical responsibility or forgetting differences in benefits and costs related to climate change adaptation and mitigation [START_REF] Torpman | Isolationism and the equal per capita view[END_REF].

To respond to these criticisms, another equality-based approach, labelled cost-effectiveness, has been proposed. It aims to distribute commitments in such a way that all actors have the same marginal mitigation cost [START_REF] Davidson | How Fairness Principles in the Climate Debate Relate to Theories of Distributive Justice[END_REF]. Here, emissions in each country should be reduced up to a point where the monetary cost of an additional reduction becomes the same for all actors. In doing so, the objective is to achieve emission reductions at the global least cost [START_REF] Ringius | Burden sharing and fairness principles in international climate policy[END_REF].

Market pricing: to each in due proportion

Finally, market pricing is a model in which relationships are based on proportionality: exchanges are regulated through ratios and rates [START_REF] Fiske | The four elementary forms of sociality: framework for a unified theory of social relations[END_REF][START_REF] Fiske | Structures of social life: The four elementary forms of human relations: Communal sharing, authority ranking, equality matching, market pricing[END_REF]. To organize this kind of interaction, metrics such as prices, budgets, wages, interest rates or taxes have to be defined to serve as a reference [START_REF] Fiske | The four elementary forms of sociality: framework for a unified theory of social relations[END_REF]. Such relationships are particularly common in contemporary societies: prices are proportional to expected value, wages are proportional to working hours or to production, and punishments are proportional to crimes.

Market pricing is the ethical foundation of the various climate equity principles that are based on proportions. One of them is the capacity principle (also labelled ability to pay), which posits that climate burden should be distributed in proportion to economic resources -most of the time in proportion to GDP per capita [START_REF] Dooley | Ethical choices behind quantifications of fair contributions under the Paris Agreement[END_REF][START_REF] Ringius | Burden sharing and fairness principles in international climate policy[END_REF]. Wealthy actors should therefore contribute more to solve climatic issues than poor ones, irrespective of their role in these issues.

Market pricing is also the ethical perspective behind historical responsibility -also known as polluter pays. In contrast to the ability to pay approach, the general idea here is that climate burden should be assigned in proportion to the contribution to the situation [START_REF] Friman | Historical responsibility for climate change: science and the science-policy interface[END_REF][START_REF] Truccone-Borgogno | Climate Justice and the Duty of Restitution[END_REF] 2 . As developed countries have been responsible for about 75% of anthropogenic CO² emissions since the Industrial Revolution, this is a position that is massively defended by developing countries [START_REF] Page | Distributing the burdens of climate change[END_REF] 3 . It is interesting to note that both the capacity principle and historical responsibility are defended in the 1992 United Nations Framework Convention on Climate Change, which states in its third article that nations should "protect the climate system for the benefit of present and future generations of humankind, on the basis of equity and in accordance with their common but differentiated responsibilities and respective capabilities" (UNFCCC, 1992).

Likewise, market pricing is associated with the beneficiary pays principle, which holds that past pollution should be taken into account by considering the benefits derived from these activities [START_REF] Page | Distributing the burdens of climate change[END_REF][START_REF] Truccone-Borgogno | Climate Justice and the Duty of Restitution[END_REF]. Contrary to historical responsibility, the objective is not to compensate past pollution per se, but to contribute to the cost of combating climatic issues in proportion to the enrichment derived from these damages -beneficiary pays is thus concerned with the effects rather than the causes of climatic issues [START_REF] Page | Distributing the burdens of climate change[END_REF].

2 For an overview of the various operationalizations of historical responsibility, see [START_REF] Friman | Historical responsibility for climate change: science and the science-policy interface[END_REF]. 3 It should be noted that historical responsibility is sometimes distinguished from polluter pays in that historical responsibility is specifically related to past emissions while polluter pays is associated with current emissions [START_REF] Tørstad | Fairness in the climate negotiations: what explains variation in parties' expressed conceptions?[END_REF].

Last, but not least, market pricing is also the ethical foundation for the current meaning of grandfathering. While grandfathering was originally defined as an exemption from a given regulation, the most common approach now holds that past emissions should determine the share of emissions actors are subsequently allowed to retain [START_REF] Damon | Grandfathering: environmental uses and impacts[END_REF][START_REF] Knight | What is grandfathering?[END_REF]. Here, allocations are supposed to be proportional to historical emissions. Such an approach of grandfathering is a key equity principle behind emissions reductions of the 1997 Kyoto protocol, which posits that countries should reduce their greenhouse gas emissions considering 1990 emission levels as an historic baseline [START_REF] Dooley | Ethical choices behind quantifications of fair contributions under the Paris Agreement[END_REF][START_REF] Knight | What is grandfathering?[END_REF]. As it is particularly favourable to industrialized countries, this principle is highly criticized -and often considered as unfair4 .

Conclusion: scientific and practical implications

The objective of this paper was to shed new light on climate equity principles by showing that they correspond to the four relational models previously identified by Fiske, and to their associated principles of distributive justice [START_REF] Fiske | The four elementary forms of sociality: framework for a unified theory of social relations[END_REF][START_REF] Fiske | Structures of social life: The four elementary forms of human relations: Communal sharing, authority ranking, equality matching, market pricing[END_REF]. In doing so, we do not claim to offer a radical departure from existing understanding of equity principles. Rather, we hope to refine our comprehension of this topic by pointing out that, beyond the apparent diversity and complexity of these principles, a quite simple conceptual apparatus may suffice to describe their very nature.

Naturally, as the four relational models are often employed in combination to structure social life, so climate equity principles are also often regularly employed jointly to organise burden sharing: the idea is to overcome the shortcomings associated with the use of isolated principles [START_REF] Dooley | Ethical choices behind quantifications of fair contributions under the Paris Agreement[END_REF]. For instance, the approach labelled equal cumulative per capita emissions is a combination of proportionality and equality, as climate burden will be first distributed in proportion to past emissions for a given period of time and then assigned on an equal-per-capita basis [START_REF] Höhne | Regional GHG reduction targets based on effort sharing: a comparison of studies[END_REF]. Similarly, various staged approaches have been proposed by which developing countries start with few restrictions following a need-based approach, and then progressively get burdens in proportion to their increasing economic resources and emissions [START_REF] Baer | The greenhouse development rights framework for global burden sharing: reflection on principles and prospects[END_REF].

We conclude here by highlighting both the scientific and the practical implications of such a clarification of climate equity principles.

Scientific implications

As recently highlighted, many studies examining climate efforts pretend to be value-neutral and fail to clarify the ethical choices behind their indicators [START_REF] Dooley | Ethical choices behind quantifications of fair contributions under the Paris Agreement[END_REF][START_REF] Rivadeneira | In) justice in modelled climate futures: A review of integrated assessment modelling critiques through a justice lens[END_REF]. Such a practice induces a sense of objectivity and effort-sharing practices are thus presented as inherently fair. Yet, climate equity principles are always based on normative assumptions that need to be clarified.

By unveiling the ethical foundations of climate equity principles and by classifying these principles, this paper may enable the development of more transparent studies of climate efforts. In particular, integrated assessment models that are currently employed to quantitatively evaluate the impact of policies on decarbonization trajectories could be refined with a more explicit consideration of climate equity principles [START_REF] Rivadeneira | In) justice in modelled climate futures: A review of integrated assessment modelling critiques through a justice lens[END_REF]. The framework we propose may also favour more critical scholarly engagements with ethical choices that are made by the various actors concerned with climatic issues, as recently requested by some authors [START_REF] Dooley | Ethical choices behind quantifications of fair contributions under the Paris Agreement[END_REF][START_REF] Tilsted | Corporate climate futures in the making: Why we need research on the politics of Science-Based Targets[END_REF]. For instance, this framework may be employed to challenge the apolitical depiction of climate efforts that is employed by the Science-based targets initiative, which helps companies set climate targets in line with the Paris agreement and which has recently been criticized for obscuring the politics embedded within these targets [START_REF] Tilsted | Corporate climate futures in the making: Why we need research on the politics of Science-Based Targets[END_REF].

Practical implications

From a practical standpoint, inequity in burden sharing is known to be a key factor in explaining the failure to bend the global emissions curve, as it undermines trust required for successful climate negotiations and for the execution of subsequent actions [START_REF] Dooley | Ethical choices behind quantifications of fair contributions under the Paris Agreement[END_REF][START_REF] Stoddard | Three decades of climate mitigation: why haven't we bent the global emissions curve?[END_REF].

As a consequence, the concrete implications of our analysis are twofold.

On the one hand, by organizing information on equity, the framework we propose may facilitate the definition of climate efforts [START_REF] Markkanen | Social impacts of climate change mitigation policies and their implications for inequality[END_REF][START_REF] Ringius | Burden sharing and fairness principles in international climate policy[END_REF][START_REF] Tørstad | Fairness in the climate negotiations: what explains variation in parties' expressed conceptions?[END_REF]. Since the 2015 Paris agreement, countries have to propose voluntary intended nationally determined contributions every five years and to explain how fair these contributions are. However, to date, information on equity provided by each country has not been precisely structured, rendering their efforts difficult to assess and to compare [START_REF] Pauw | Beyond ambition: increasing the transparency, coherence and implementability of Nationally Determined Contributions[END_REF][START_REF] Saelen | Fairness conceptions and self-determined mitigation ambition under the Paris Agreement: Is there a relationship?[END_REF][START_REF] Winkler | Countries start to explain how their climate contributions are fair: more rigour needed[END_REF]. Similarly, companies are increasingly asked to set climate targets in order to reduce the environmental impact of their activities. Yet, the methodological choices they make are not always publicly available [START_REF] Giesekam | Science-based targets: on target?[END_REF][START_REF] Tilsted | Corporate climate futures in the making: Why we need research on the politics of Science-Based Targets[END_REF]. We think that the conceptual framework we propose to adopt in this paper -and, in particular, the distinction between need, position, equality and proportion -may help parties clarify their position, identify compromises, and communicate very simply about the nature of their climate efforts which, by generating perception of fairness and mutual trust, may consequently encourage more ambitious actions.

On the other hand, the clarification of the ethical foundations of climate equity principles may ease the acceptance and, finally, the effective implementation of climate policies [START_REF] Drews | What explains public support for climate policies? A review of empirical and experimental studies[END_REF][START_REF] Tørstad | Fairness in the climate negotiations: what explains variation in parties' expressed conceptions?[END_REF]. Even if the transition towards low-emissions economies is technically possible, the lack of public support is a major barrier to climate change adaptation and mitigation [START_REF] Bernauer | Simple reframing unlikely to boost public support for climate policy[END_REF]. To gain such a support, perceived fairness is considered to be key: policies that are in line with citizens' values and beliefs are more easily accepted by populations [START_REF] Drews | What explains public support for climate policies? A review of empirical and experimental studies[END_REF]. By simplifying the diffusion of climate

Table 1 . Relational models and climate equity principles

 1

	Relational model	Nature of relationships within groups	Principle of distributive justice	Corresponding equity principle general climate	Key examples
	Communal sharing	Members of a group are not differentiated from each other	To each according to his needs	Climate burden according to needs should be shared	-Need
	Authority ranking	Members of a group are differentiated according to some social dimensions	To each according to his position	Climate burden should be shared according to positions	-Grandfathering (original meaning) -Sufficientarianism
	Equality matching	Members of a group are treated as equals	To each the same	Climate burden should be shared equally	-Equal per capita -Cost-effectiveness
	Market pricing	Members of a group are treated according to ratios	To each in due proportion	Climate burden should be shared proportionally	-Capacity (or ability to pay) -Historical responsibility (or polluter pays) -Beneficiary pays -Grandfathering (current common meaning)

Subsistence emissions that are necessary to satisfy basic needs are thus often distinguished from luxury emissions, which are those generated to satisfy non-basic needs(Ringius et al.,

[START_REF] Ringius | Burden sharing and fairness principles in international climate policy[END_REF][START_REF] Shue | Subsistence protection and mitigation ambition: Necessities, economic and climatic[END_REF].

For a nuanced presentation of the current meaning of grandfathering, see[START_REF] Damon | Grandfathering: environmental uses and impacts[END_REF] and[START_REF] Knight | What is grandfathering?[END_REF].

policies to this audience, we think that the framework we propose may also help securing their approval.


Dr Victor Koehler

Gabrielle Bruschera

Dr Eric Merlet

Dr Pradeep K Mandal

Dr Estelle Morvan

Frédéric Rosu

Dr Céline Douat

Dr Lucile Fischer

Ivan Huc
email: ivan.huc@cup.lmu.de

Yann Ferrand
email: yann.ferrand@u-bordeaux.fr

Y Ferrand Cbmn

Prof. I Huc

High-Affinity Hybridization of Complementary Aromatic Oligoamide Strands in Water

Keywords: aromatic oligoamide, artificial hybridization, double helix, foldamer, self-assembly

We prepared a series of water-soluble aromatic oligoamide sequences all composed of a segment prone to form a single helix and a segment prone to dimerize into a double helix. These sequences exclusively assemble as antiparallel duplexes. The modification of the duplex inner rim by varying the nature of the substituents borne by the aromatic monomers allowed us to identify sequences that can hybridize by combining two chemically different strands, with high affinity and complete selectivity in water. X-ray crystallography confirmed the expected antiparallel configuration of the duplexes whereas NMR spectroscopy and mass spectrometry allowed us to assess precisely the extent of the hybridization. The hybridization kinetics of the aromatic strands was shown to depend on both the nature of the substituents responsible for strand complementarity and the length of the aromatic strand. These results highlight the great potential of aromatic hetero-duplexes as a tool to construct non-symmetrical dynamic supramolecular assemblies.

Molecular self-sorting is a process through which molecules self-organize into distinct entities. [1] This molecular programming achieves high-fidelity recognition and potentially function within complex mixtures. Over the years, chemists have established molecular codes that allow synthetic molecules to preferentially engage in interactions with specific partners. In nature, selfsorting of oligomeric peptidic and nucleotidic strands into duplexes is both common and essential. [2] In particular, social selfsorting, that is, the selective recognition of different yet complementary entities is critical to information storage, replication and transfer in living systems. This has inspired chemists to design artificial oligomers that form sequenceselective duplexes mimicking their natural counterparts. For example, linear oligomers may assemble when they possess sequence-complementary arrangements of hydrogen bond donors and acceptors. [3] However, combining such sequence complementarity with shape control proved to be more challenging. Some self-assembled homomeric double helices have been reported eventually nucleated by anions or cations, [4] yet heteromeric double helices (i.e. folded duplexes composed of two different strands) are far less common and often not produced quantitatively. [5] In other words, establishing a reliable pairing mode between two different folded strands distinct from nucleotidic duplexes or peptidic assemblies has been a rarely met challenge, even more so in water. In this context, the heteroduplexes based on aminidium-carboxylate [6] (or phosphate [7] ) pairing in organic solvents developed by Yashima et al constitute a unique and significant example. We describe herein the selective and quantitative hybridization of two chemically inequivalent artificial molecular strands in water to form heteroduplexes. Our design is based on aromatic oligoamide double helices that rely on two sorts of intermolecular interactions, one perpendicular to the helix axis and dominated by steric effects, and the other parallel to the helix axis and associated with aromatic staking.

The association of aromatic helices into helical dimers in organic solvents is well documented. [5,8] We have also shown that water may enhance double helix formation. [9] This was assigned to the hydrophobic effect via a lesser exposure of the aromatic surfaces to water in the duplex than in two single helices. The present study focuses on oligoamide sequences combining 8amino-2-quinolinecarboxylic acid (Q), which codes for a narrow helix diameter and 7-amino-2-quinolinecarboxylic acid bearing a substituent X in position 8 (Q X ), which codes for a wider helix diameter (Figure 1b-e). Qn segments fold into single helices whereas Q X m segments have a strong propensity to form helical dimers in which all X groups converge towards the helix cavity. Q X mQn may thus self-associate as an anti-parallel double helix of two Q X m segments with a Qn single helix at each end (Figure 1a). [9] We devised that modifications of the double helix inner cavity, i.e. varying X on the edge of the aromatic core, may allow for the hybridization of two distinct strands with different X groups able to engage in attractive contacts or able to minimize repulsions.

Synthetic methods and protocols for new monomers and oligomers are described in detail in the supporting information (SI). We first designed a series of Q x 4Q3 sequences equipped with cationic water-solubilizing side chains in position 4 of the monomers. Thus, 1-4 differ only by the nature of the X group of Q X monomers. X can be more or less bulky (F < Cl < OMe < OEt) and can present an electron donating (OMe, OEt) or withdrawing (F, Cl) character. Typically, the sequences were synthesized either manually, or with an automated synthesizer, on low-loading ProTide® PEG-PS resin using conventional Fmoc-based chemistry under microwave irradiation or by controlled induction heating. [10] The acids were activated using either Ghosez reagent before the coupling or in situ using Appel coupling conditions. Fmoc deprotection with DBU was preferred over piperidine in DMF as some Q X monomers were poorly reactive to the latter. All sequences were obtained as their respective trifluoroacetate ammonium salts with excellent purity and very good amounts (82-94% with respect to theoretical loading, see SI) without HPLC purification. The propensity of sequences 1-4 to form double helices was assessed by 1 H NMR. In DMSO-d6, a solvent known to disfavor double helix formation, sequences 1, 3, and 4 revealed a unique set of resonances that were assigned to a single helix (Figures 2a,2c, S1-S4). In contrast, still in DMSO-d6, 2 weakly aggregates with a Kdim = 1.6x10 3 L.mol -1 . Upon adding water to the DMSO-d6 solutions of 1-4, a new set of well-resolved high-field-shifted resonances, characteristic of double helix formation, emerged (Figures 2b,2d,). The number of resonances indicated that the double helices have two equivalent strands. At 1 mM, the amount of water required for the onset of aggregation varied with X, from 0 %, less than 15 %, less than 35 %, and less than 55 % for 2, 1, 3 and 4, respectively, indicating different duplex stabilities. However, in pure water, the duplex was the only detectable species for all compounds. Upon diluting solutions of (1)2 or (3)2 in water down to 20 µM, no monomer could be observed providing a minimal estimate of Kdim > 10 8 L mol -1 (Figures S6-S9). This behavior was unchanged in the presence of a 50 mM ammonium acetate buffer. Native electrospray mass spectrometry (ESI-MS) using soft instrumental conditions confirmed the formation of (1)2 and (3)2 albeit mixed with monomers 1 and 3. We inferred that some duplex dissociation occurred during ionization. Additionally, ion mobility MS analysis confirmed collision cross-section profiles that matched well with the double and single helices (Figures S10-S11).

Structural information at the atomic scale on the homomeric double helix was obtained from the solid state structures [11] of sequences 5 and 6, two analogues of 3 bearing short neutral and anionic side chains more prone to crystallization than the flexible ornithine-like side chains of 3 (Figure 1e). The structures confirmed the anticipated anti-parallel arrangement of the strands as well as the single helical and double helical nature of the Q3 and Q OMe 4 segments, respectively (Figure 2e,2f). In both structures, the helix cavity is filled with a crowded arrangement of eight methoxy groups, which pack more tightly in (5)2 than in (6)2 (Figure S12). Next, we assessed the propensity of homodimers 1-4 to form heteromeric double helices. Hybridization was followed using 1 H NMR and also 19 F NMR for 1. Upon incubating (1)2 with (2)2 in water (H2O/D2O -95:5 vol/vol) for 12 hours at 298 K no change could be observed (Figure S13). Heating to 323 K led to the appearance of a new set of sharp signals with twice the number of resonances (12 amide NHs) as (1)2 or (2)2 (6 amide NHs each) suggesting the formation of heteroduplex 1•2 (Figures S13d,S14b). Thermodynamic equilibrium was reached after 12 hours and revealed that the proportion of 1•2 reached 78 %. Similarly, mixing (2)2 with (3)2 yielded 2•3 up to a proportion of 87 % (Figure S15). In both cases, the proportion of heteroduplex exceeded that of a statistical distribution (50 %), hinting at a better steric or electronic complementarity than in homomeric double helices. Remarkably, upon mixing (1)2 with (3)2, the 1 H and 19 F resonances of the homomeric double helices completely vanished within 5 minutes at 323 K, and were fully replaced by a unique set of signals corresponding to 1•3 (Figures 3a,b,d and S16, S17). Consistent with the high stability of (1)2 and (3)2, no dissociation of 1•3 was detected upon dilution down to 20 µM in water or buffered water (Figures S18,S19). The quantitative formation of 1•3 indicates that the equilibrium constant K' between (1)2 + (3)2 and 2 × 1•3 is larger than 2500 (see the SI). Accordingly, a minimum estimate for the stabilization of the heteroduplex 1•3 was determined as ΔG < -19.4 kJ/mol. Finally the self-sorting ability of the aromatic sequences bearing fluorine, methoxy or chlorine substituents was evaluated by 1 H NMR. Mixing (1)2, (2)2 and (3)2 in a 1:1:1 ratio in CDCl3 showed the progressive disappearance of (1)2 and (3)2 -the concentration of (2)2 remained unchangedtogether with the quantitative formation of the heteroduplex 1•3, without detection of 1.2 or 2.3 (Figure S31).

The hybridization of 1•3 was also confirmed by native electrospray mass spectrometry. The heteromeric double helix was unambiguously detected at m/z = 1858.6, 1239.4, 929.8 (2+, 3+ and 4+ species) (Figures 3f, 3g and S22). Single helices 1 and 3 were also detected as a consequence of dissociation during ionization and despite the smooth conditions.

Detailed structural information could not be gathered using crystallography. Nevertheless, hybridization was confirmed using 1 H-19 F Heteronuclear Overhauser Effect SpectroscopY (HOESY) which can assess the spatial proximity between 1 H and 19 F nuclei. For these experiments, we used dimer 1•7, an analogue of 1•3 bearing seventeen hydro-solubilizing ammonium side chains vs fourteen for 1•3. The better solubility of 1•7 allowed for a higher concentration in water and increase signal-to-noise ratio for HOESY experiments. The 1 H-19 F HOESY spectrum demonstrated dipolar couplings between fluorine atoms of the Q F units and the methyl groups of the Q OMe units, thus confirming the proximity of these functional groups in the cavity of the heteromeric double helix (Figures 3h,S20). A molecular model of 1•3 was built, revealing shape complementarity between the small fluorine atoms and the methoxy groups whose position may vary through rotation about Car-O bonds (Figure 3i). The crowding of methoxy groups in (3)2, (5)2, and (6)2 (Figures 2, S21, S30 and S43) is released in 1•3, suggesting that sterics contribute to hybridization (Figure S12). The absence of steric bulk within (2)2 would explain why it hybridizes less efficiently with (1)2. The weak hybridization between (2)2 and (3)2 could also reflect a weaker steric drive (crowding within (3)2 is less efficiently released with chlorine than fluorine), or that other effects than sterics favor for the formation of 1.3, for example electronic effects specific to fluorine.

The hybridization kinetics were examined further by increasing the length of the C-terminal single helical Qn segments, or by increasing the substituent size in position 8 of the Q X units. Sequences 4, 7, and 8 served that purpose (Figure 1e). Upon mixing (1)2 and ( 7)2, the elongated version of (3)2, the formation of 1•7 was followed by 1 H and 19 F NMR (Figures 4b,c and S23-25). We found that the heteroduplex formed quantitatively over 90 minutes at 323 K in H2O / D2O (95:5 vol/vol). By comparison, the formation of 1•3 took less than five minutes (Table S1). The slower kinetics with 7 suggest a remote effect of the single helical segments which may stabilize the central duplex. Such remote effects have been observed before. [12] Effects were also strong with Q OEt -containing sequences 4 and 8. Mixing (1)2 with (4)2 produced 1•4 at a rate too fast to be monitored, as observed with 1•3. When using longer oligomer (8)2, the formation of 1•8 was slower, yet it was complete after only 15 minutes, that is, six times quicker than for 1•7 (Figures 4, S26-S29). Molecular models of heteromeric 1•7 and 1•8 (Figure S30) showed a good shape-complementarity, and compact well-folded helices for both hetroduplexes. In contrast, steric crowding of the ethoxy groups within (8)2 was considerable, leading to distortions of the double helix and ethoxy groups protruding from the cavity (Figure S30). The faster kinetics with (8)2 may thus result from its lower stability with respect to (7)2, facilitating the exchange with complementary strands. Nevertheless, the prevalent exchange mechanism is yet to be established, namely, completely dissociative (between two single helices), through direct exchange (between two double helices), or hybrid (between a dissociated single and a double helix). Double helix formation is presumed to proceed with limited perturbation of the helical structure via the slipping of the two single helices into one another in an eddy-like process. [13] Whether this allows for a direct exchange of strands, as seen in other systems, [6c] is not known. In any case, helix length and the bulk in the cavity emerge as convenient parameters to fine-tune the kinetics of exchange. In summary, we established quantitative hybridization of aromatic helices in water using interacting groups within the duplex cavity. The heterodimers show high stability even for the short sequences studied and their kinetics of formation may be adjusted. We believe that this high level of self-sorting in water will be useful when combined with biomolecules for biomaterial engineering. The hybridization may give access to advanced biomaterials with low symmetry.

Figure 1 .

 1 Figure 1. a) Schematic representation of: (left) the self-assembly of a single helix into an antiparallel homomeric double helix followed by (right) its hybridization with a chemically different strand forming a heteromeric double helix. b) Color coded formulae and associated abbreviations of aromatic amino acid monomers. Representation of the helical curvature of: c) a Q2 dimer (2.5 units/turn); and d) a Q X 2 dimer (X = F, Cl, OMe or OEt, 4 units/turns). e) Oligoamide sequences 1-8.

Figure 2 .

 2 Figure 2. Excerpts of 1 H NMR spectra (700 MHz, 323 K) of a) 1 (1 mM, DMSO-d6); b) (1)2 (0.5 mM, H2O/D2O -95:5 vol/vol); c) 3 (1 mM, DMSO-d6); d) (3)2 (0.5 mM, H2O/D2O -95:5 vol/vol). X-ray structures of the antiparallel homomeric double helices: e) (5)2 and f) (6)2. Methoxy groups are shown in space filling representation. In the side views, Q and Q OMe are colored in grey and orange, respectively. In the top view (slice), each strand is shown with a distinct color

Figure 3 .

 3 Figure 3. Excerpts of 1 H NMR spectra (H2O/D2O -95:5 vol/vol, 700 MHz, 323 K) of: a) (1)2 (0.5 mM); b) (3)2 (0.5 mM); c) (4)2 (0.5 mM); d) 1•3 (1 mM); e) 1•4 (1 mM). Amide resonances of (1)2, (3)2, (4)2, 1•3 and 1•4 are marked with blue, orange, green, blue/orange and blue/green circles, respectively. f,g) High resolution ElectroSpray Ionization Mass Spectrum (ESI-MS) of 1•3. h) Zoom of 1 H-19 F HOESY NMR spectra (H2O/D2O -95:5 vol/vol, 400 MHz, 323 K) of 1•7. Resonances corresponding to fluorine atoms are denoted in blue whereas those of methoxy groups (OCH3) are shown in orange, respectively. i) Energy minimized molecular models (MMFFs) of 1•3. Fluorine atoms and methoxy groups are shown in representation. Q, Q F and Q OMe are colored in grey, blue, and orange, respectively. Side chains have been removed for clarity.

Figure 4 .

 4 Figure 4. a) Schematic representation of the hybridization between two strands of different lengths. Excerpts of 1 H NMR spectra (H2O/D2O -95:5 vol/vol, 700 MHz, 323 K) of an equimolar mixture of (1)2 (0.5 mM) and (7)2 (0.5 mM) after: b) 3 min;(c) 90 min (thermodynamic equilibrium). Same experiment with (1)2 (0.5 mM) and (8)2 (0.5 mM) after: d) 3 min; e) 15 min (thermodynamic equilibrium). Amide signals of (1)2, (8)2, 1•7 and 1•8 are marked with blue, green, blue/orange and blue/green circles, respectively. f) Time traces of the hybridization process in water of 1•7 (orange circles) and 1•8 (green circles).

Acknowledgements

We thank the "Agence Nationale de la Recherche" for funding the project ANR-19-CE07-0034 and the France-Germany International Research Project "Foldamers Structures and Functions" (IRP FoldSFun). This work benefited from the facilities of the Biophysical and Structural Chemistry platform at IECB, CNRS UAR3033, and Bordeaux University, France. We thank G. Leonard (ID30b, ESRF, Grenoble) and S. Panneerselvam (P14 operated by EMBL at PETRA III, DESY, Hamburg) for assistance during data collection at synchrotron beamlines.

Conflict of interest

The authors declare no conflict of interest.


Marc-Antoine Turcotte

François Bolduc

Anaïs Vannutelli

Jérémie Mitteaux

David Monchaud

Jean-Pierre Perreault

Development of a highly optimized procedure for the discovery of RNA G-quadruplexes by combining several strategies

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