Arthur Lamouroux

Laure Sebaoun

Barbara Wicher

Brice Kauffmann

Yann Ferrand

Victor Maurizot

Ivan Huc

Controlling Dipole Orientation through Curvature: Aromatic Foldamer Bent β-Sheets and Helix-Sheet-Helix Architectures

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INTRODUCTION

The structures of biopolymers demonstrate the efficiency of folding as an approach to construct atomically precise arrays of chemical groups in space. Folding may counterbalance local repulsions and enforce the proximity of desired functionalities even in disfavored relative orientations. As exemplified by the catalytic properties of enzymes' active sites, unique functions may arise from folding-based conformational control. In the last two decades, this knowledge has been the source of inspiration for chemists to design a wide range of artificial folded architectures termed foldamers. 1 Chemically diverse foldamer backbones have been reported but, quite remarkably, their structures were found to be less diverse, with an overwhelming prevalence of the helix, turn and linear-strand secondary motifs common in biopolymers. In this context, foldamers that have aromatic groups in their main chain bear special interest in that they may give access to unusual folded patterns some of which are beyond the reach of biopolymers. Prominent examples include pillars, 2 knots and interlocked macrocycles, 3 spiral-like objects, 4 and herringbone helices. 5 A remarkable architecture is that of a helix whose diameter may increase and decrease along the sequence (Fig. 1E), thus creating a cavity within a secondary folded motif in which outstanding molecular recognition properties have been evidenced. 6 Just like peptidic and nucleotidic structures and functions differ because the backbones of peptides and nucleotides are chemically distinct, it may be envisaged that the chemically distinct backbones of aromatic foldamers give access to structures and functions different from those of peptides and nucleotides.

In the context of our investigations on aromatic oligoamide foldamers, we have characterized multi-stranded aromatic sheets. 7,8 Examples of artificial sheets in the foldamer literature are rare in comparison to helices. Unlike helices which fulfill their potential for non-covalent interactions intramolecularly, sheets may also engage in intermolecular interactions and form large aggregates that are prone to precipitation, e.g. amyloid fibrils, and thus complicate structural investigations. Hence, sheet models based on hydrogen bonds are often designed to be monomeric 9 or dimeric. 10 Our aromatic multiturn -sheets involved a balanced set of non-covalent interactions strong enough to operate intramolecularly to promote folding and yet weak enough to prevent aggregation. This first generation of aromatic -sheets was composed of stacks of linear aryl-amide segments connected by dinitro-phenyl turns (Figure 1A). The nitro groups hydrogen bond to neighbor aryl-amines while the methyl groups of two adjacent xylyl rings set their orientation parallel to each other in planes roughly perpendicular to the plane of the dinitrophenyl ring. Aryl moieties attached to the xylyl rings then stack in a face-to-face arrangement that, in principle, may be anti-parallel or parallel (Figure 1A). We found that anti-parallel organization was favored (Figure 1B) as, for example, in macrocycle 1 (Chart 1). This preference results from the effect of local dipoles, and is generally observed upon stacking polar aromatic rings in the solid state or in aggregates. 11 In the following, we present a study of the effect of bending on the conformation of these aromatic -sheets. We find that introducing curvature effectively enforces the formation of stacks of aryl rings in which local dipoles are parallel despite the repulsions associated with this arrangement (Figure 1C). In such bent structures an anti-parallel orientation would give rise to misfolding with reduced  overlap. Parallel orientations are favored as they maximize  overlap. This relationship between bending and polarity, a notion common in the field of liquid crystals, 12 is significant because parallel dipole arrangements are a much sought for property in materials, for example in the context of their non-linear optical responses. 13 Furthermore, we demonstrate that bent aromatic -sheets can be combined with aromatic helices. Aromatic helix-sheet-helix folded structures were generated that, to the best of our knowledge, constitute the first examples of aromatic foldamers composed of more than one kind of secondary motif and, significantly, possess an unprecedented architecture. Indeed, the introduction of aromatic -sheets within the wall of a helical capsule (Figure 1D vs. 1E) amounts to creating a basket-like object having a window of defined size and provides a means to fine tune guest binding and release pathways and kinetics to and from the basket cavity. Chart 1. Linear (1) or bent (2-4) macrocyclic and multi-turn aromatic oligoamide -sheets.

We bring proof of concept of this design feature by showing that an -hydroxy-acid guest exchanges rapidly on the NMR timescale from these open receptors whereas exchange was found to be slow in closed-shell helical receptors. Thus, the helix-turn-helix structures further demonstrate the ability of aromatic backbones to adopt shapes beyond the reach of peptides, nucleotides and related backbones.

RESULTS AND DISCUSSION

Bent -sheet architectures. The induction of curvature within aromatic -sheets was made possible by connecting of dinitro-di(xylylamino)-phenyl turns using aromatics with substituents at an angle different from 180°. For this purpose, we used pyrido[3,2-g]quinolinedicarboxylic acids which we refer to as 1, 8-diaza-2,6-anthracenedicarboxylic acids. 11d,14 Prior to preparing multistranded turns, we tested the validity of our design in a one-step macrocyclization reaction. A diaza-anthracene diacid was coupled to a diamine-functionalized turn using PyBOP (benzotriazol-1-yl-oxytripyrrolidinophosphonium hexafluorophosphate) as an activating agent to produce macrocycle 2 in an unoptimized 24 % yield. A crystal structure of 2 demonstrated the expected parallel arrangement of the stacked diaza-anthracenes (Figure 2A,2B). The top view shows that aryl rings are slightly offset but that local dipoles are strictly parallel. An energy-minimized model of the antiparallel non-cyclic amino acid precursor of 2 illustrates the reduced aromatic  overlap and a large distance between amine and acid functions that presumably precludes cyclization in this conformation (Figure 2C). In solution, the sharp 1 H NMR spectrum of 2 (Figure 3A) was fully assigned using multi-dimensional experiments 15 (see Supporting Information). Hint protons as defined in Chart 1 are shielded down to 4.76 ppm, reflecting strong ring current effects by the xylyl rings above and below. CH2 protons of isobutoxy chains, H3 and H5 protons of xylyl groups (at 8.38 and 6.69 ppm) and even the protons of the two xylyl CH3 groups are all anisochronous, indicating slow dynamics of the structure. The anisochronicity of the xylyl CH3 protons, despite a small chemical shift difference ( = 0.02 ppm at 300 MHz) implies that exchange by rotation of the xylyl units is slower than 7.71 Hz. No exchange could be observed in ROESY spectra. Furthermore, no change of the spectrum was observed from -50 °C (in CDCl3) to + 80 °C (in C2D2Cl4, Figure S1). These results overall support the formation of a single conformer with remarkably slow dynamics. By comparison, slow exchange between the H3 and H5 xylyl protons of macrocycle 1 was only observed upon cooling. 7 We then proceeded with the synthesis of tri-stranded sheet 3 and tetra-stranded sheet 4 (Scheme S2). A key step was the desymmetrization of the diamine-functionalized turn using one equivalent of a diaza-anthracene mono acid mono ester. A crystal structure of 4 (Figure 2D-F) confirmed the features revealed by the bent-sheet structure of macrocycle 2. The non-cyclic nature of 4 offers additional conformational degrees of freedom and offsetting between aryl rings as well as tilting of turn units away from 90° are more pronounced than in the structure of 1. Nevertheless, the structure does consist of a stack of four diazaantracene rings with all their local dipoles parallel.

In solution the NMR spectra of 3 and 4 were both fully assigned using multidimensional experiments 15 (see Supporting information). The results corroborated the solid state structure of 4. Evidence for folding in multi-stranded bent sheet conformations came for NOE correlations between the different 9-methyl-antracene protons, both in 3 and 4 (Figure S4): proximity between these methyl groups is only allowed in the parallel arrangement of the diaza-anthracene units. NMR also indicates faster dynamics in the non-cyclic structures than in 2 and that sheet stability increases upon increasing strand length, suggesting a certain degree of cooperativity. Helix-sheet-helix architectures. The well-behaved folding of aromatic bent -sheets was further explored in combination with helically folded segments within longer sequences. We surmised that an aromatic helix connected at the end of a bent sheet would simply stack on top of it without any detrimental interference and decided to test this prediction using helical segments the diameter of which varies along the sequence. These variable-diameter segments are key components of aromatic foldamer-based helical capsules that recognize guest molecules in their cavity with high selectivity and affinity. 6b,c A capsule shape with its characteristic reduced diameter at both ends is schematized in Figure 1D. An interesting aspect of these molecular containers is that they completely isolate their guest from the solvent. For a guest to come in and out, the helix has to partially unfold, a given unit generally playing the role of a hinge to temporarily open a passage. 16 As a consequence, binding and release kinetics are all the more slow when the guest is large, e.g. slow exchange on the NMR time scale is reached below -55°C for a water molecule guest 17 and persists above 50°C for a monosaccharide. 6b The introduction of a bent sheet between two variable-diameter helix segments thus amounts to permanently opening a sort of window in a capsule wall. As shown below, the size of the window may be tuned according to the curvature imparted by the bent aromatic -sheet. This provides a means to enhance and fine tune guest and binding release kinetics.

Foldamers 5-8 (Chart 2) were prepared as described in the Supporting Information. They all contain one (8) or two (5-7) helical oligoamide segments composed of two 7-amino-1,8naphthyridine-2-carboxylic acid, one 2,6-diamino-pyridine and three 8-amino-2-quinolinecarboxylic acid at the more curved end, as well as one (8) or two (5-7) turn units. The connections between the turn units vary. In 5, the connecting element is itself a 1,8-diaza-2,6-antracenedicarboxylic acid. The central sheet of 5 is thus identical to compound 3. In 6, the central pyridine-pyridazine-pyridine motif is slightly longer than a diazaanthracene but it imparts a higher curvature because the angle between the two carboxylic acid functions is reduced. 16a,18 The urea function of 7 is a very short linkage imparting low curvaturetwo distinct conformers may be expected depending on whether the urea aligns itself one way or the other with respect to the anthracenes above and below.

Crystal structures of 6a and 7 as well as an energy minimized molecular model of 5 confirmed the formation of the anticipated folding (Figure 4A-4O). In all structures, a central sheet segment is flanked by two helices, giving rise to basketlike objects the openings of which depends on the degree of curvature of the sheet. Thus, the structure of 6a has a deeper and narrower cavity, whilst the structure of 7 possesses the widest opening, the structure of 5 showing an intermediate situation.

In the structure of 7, the urea carbonyl oxygen points towards the cavity. It is worth noting that regardless of the sheet curvature, the overhanging helical segments do not collapse in the cavity and that all folded secondary motifs retain their integrity in these structures. A somewhat unexpected result came from the structure of 8 that revealed a dimerization process in which the peripheral strands of two sheets interdigitate (Figure 4P-4T). The outcome is a basket that much resembles the structures of 5-7 even though 8 itself is only about half the size of 5. Because they all comprise a two turn -sheet, all structures also possess an average C2 symmetry axis perpendicular to the axis of the helix segments. Consequently, the two helices of a given sequence possess the same handedness. However, one can predict that sheets with an odd number of turns would be flanked with helices having opposite handedness, the sheet triggering handedness inversion 19 and the overall structure being plane symmetrical.

In solution, sharp and well-resolved NMR spectra suggest that 5-8 adopt well-defined conformation (Figure 5). Amides and aromatic resonances are spread over a wide chemical shift range, reflecting the combined effects of hydrogen bonding involving amide protons which cause downfield shifts of their resonances, and ring current effects associated with aromatic stacking, which cause upfield shifts. The methyl groups of the xylyl units are under rapid exchange at room temperature, indicating fast conformational dynamics. Yet their resonances are found below 2 ppm, i.e. at lower chemical shift values than in macrocycle 2 (Figure 3A), as a consequence of the stacking of the helical segments. Similarly, Hint protons in 5-8 are even more shielded (signals below 4.5 ppm) than in -sheets 2-4 (Figure 3, Figure S5).

Chart 2. Formula of helix-sheet-helix oligomers. 8)2. In all representations except (P), the helical cones are shown in blue, the diaza-anthracene units are shown in red whereas the turn units are shown in white. In (K) the intercalated strands of the duplex (8)2 are highlighted in red and black. (E), (J), (O) and (T) show a top view of the helix-sheet-helix architectures in which both helical cones have been omitted. This view highlights the varying orientations of the turn units. In (A-T) isobutoxy side chains and included solvent molecules were omitted for clarity. Only the all P helical isomers are shown. The structures belong to centrosymmetrical space groups and thus also contain the all M isomers. Solution studies of 8 were carried out in CDCl3 and supported the existence in solution of the dimer observed in the solid state (Figure 4P-4T). Upon increasing concentration, a new set of resonance appeared assigned to the formation of duplex ( 8)2 (in red, Figure 6 A-E). DOSY experiments (Figure S7) allowed for the calculation of the diffusion coefficients of the monomer (4.84 -10 m²/s) and of the dimer (4.73 -10 m²/s). The monomeric and dimeric forms are in slow exchange on the NMR timescale (Figure 6) despite the limited overlap associated with the reciprocal intercalation observed in the solid state. Integration of the signals allows for the calculation of the di-merization constant Kdim = 47 L.mol -1 in CDCl3. The amide protons of the duplex are shielded compared to the monomeric form that prevails at lower concentration.

Another dynamic parameter concerns the orientation of the urea function of 7. In the solid state structure, the urea carbonyl group was found to point towards the cavity of the molecule, whereas the NH pointed towards the solvent. Modelling studies show that changes of this orientation may result in different curvatures of the molecule (Figures S15-S16). Although it was not possible to assign the NMR spectrum of 7 (Figure 5) to one or another conformer, its sharpness suggests that one conformation prevails or that different conformers exchange fast despite significant structural rearrangements. Sequences 5-8 were not designed to bind to any particular guest. Nevertheless, their quinoline-pyridine-naphthyridine helical segments have been shown before to hydrogen bond to -hydroxy-acids and capsules containing these segments may bind to tartaric or malic acid. 6c,16,18 Titrations were thus carried out using glycolic acid as a guest. However, no clear binding was detected in the case of 5, 6a and 7. This was attributed to the 9-methyl groups of the 1,8-diaza-anthracene units that protrude at the entrance of the site where hydroxy acids are expected to bind. Sequence 6b which lacks these methyl groups was thus synthesized (Chart 2). Titration with glycolic acid in CDCl3/D6-acetone (9:1 vol/vol) at 298K resulted in a chemical shift variations consistent with rapid binding and release on the NMR timescale (Figure 7). Using HypNmr (2008, Version 4.0.71) 20 chemical shift values could be fitted well to a 1:2 binding isotherm consistent with the two binding sites of 6b. The first and second association constants, K1 = 121 L.mol -1 and K2 = 162 L.mol -1 , albeit not very strong, revealed a certain degree of positive cooperative ( = 5). 21 3, 2.6, 3.9, 5.3, 7.9, 10.5, 13.1, 19.7, 26.3, 39.4, 59.2 and 78.9 equiv. of glycolic acid. (B) Experimental and calculated values using a 2:1 binding model for the NMR binding study of oligomer 6b vs. glycolic acid in CDCl3:D6-acetone (9:1, vol/vol). Resonance at 9.78 ppm was used for the analysis.

Rapid guest exchange on the NMR timescale contrasts with the slow guest exchange observed in closed shell helical containers of the kind shown in Figure 1D. 6b,c,16 It suggests that the guest is bound through the open side of the basket-like host and therefore that both the binding pathway and the binding kinetics may be modulated through the design of the basket structural features.

CONCLUSION

Aromatic amide -sheets may adopt stable bent conformations upon introducing building blocks that impart curvature. In these structures, folding overcomes intramolecular electrostatic repulsions and forces local dipoles in each layer of the stacked strands to align in a parallel fashion. Furthermore, se-quences having helical segments flanking a central bent aromatic -sheet were shown to form well-defined helix-turn-helix architectures in which helical and sheet sub-components conserve their respective integrity. Our results thus provide new examples of how foldamers with aryl rings in their main chain may give access to original and engineerable folded structures that would be difficult to produce from peptides, nucleotides and related foldamer backbones. The possibility to tailor an opening of predictable size within the wall of a helically folded receptor represents a step towards controlling guest binding and release pathways and kinetics (Figure 1D,E), a prospect of interest considering recent achievements using aromatic foldamers in the field of molecular recognition. 6,16 Helix turn-helix motifs may also help access topologically complex objects through the winding of open-ended helices around dumbbell guests to form so-called foldaxanes. Foldaxanes have already been produced using rod-like guests and single helical or double helical hosts (Figure 8A). 22 When the host is a helix-turn-helix, foldaxane formation may proceed through original pathways and be compatible with T-shaped or X-shaped guests (Figure 8B,C). Finally, foldamer structures comprised of several and distinct secondary sub-components represent an important and challenging objective towards the implementation of sophisticated function. 10e,f,23 The helix-sheet-helix architectures described here constitute an original development in this respect and complements objects produced from helices only.

ASSOCIATED CONTENT

Experimental details for synthetic procedures, spectroscopic data, molecular modelling. Crystallographic information files for 2, , 6, 7, 8 (CIF). This material is available free of charge via the Internet at http://pubs.acs.org.

Figure 1 .

 1 Figure 1. (A) Structure of aromatic -sheet turn units and equilibrium between anti-parallel and parallel arrangements. (B) Anti-parallel arrangements of strands in linear aromatic sheets. (C) Parallel arrangements of strands in bent aromatic sheets. (D) Schematic representation of a C2 symmetrical helical capsule that must unfold to bind and release a guest. (E) Schematic representation of a helixsheet-helix architecture that can bind and realse a guest with minimal perturbation of its structure. Aromatic strands are represented as rectangular objects whereas turns and helix segments are shown as gray and blue tubes, respectively. White arrows in B and C schematize the orientation of local dipoles.

  For instance, H3 and H5 protons of each xylyl groups of 3 exchange rapidly on the NMR time scale at 50 °C and the signals are coalesced at 10 °C (Figure S2), whereas slow exchange is observed for 4 at 10 °C (Figure3C). The spectra of 4 (but not the spectra of 3) displays anisochronicity of the CH2 protons of isobutoxy chains and of the protons of the xylyl CH3 groups. Nevertheless, dynamics in 4 are faster than in macrocycle 2: exchange is fast on the NMR time scale at 80 °C (Figure3D) and correlations can be observed in ROESY maps in the slow exchange regime at lower temperatures.

Figure 2 .

 2 Figure 2. (A) Side view and (B) top view of the crystal structure of macrocycle 2; (C) Top view of the energy minimized model conformation (Maestro version 6.5 using the MMFFs force field) of macrocycle 2 precursor as an anti-parallel conformer. (D,E) Side views and (F) top view of the crystal structure of tridecamer 4. In (D) and (F) 4 is shown in tube representation whereas it appears in CPK representation in (E). In all the representations the turns are shown in light grey whereas the bent diaza-anthracenes appear in red. Methyl groups of diaza-anthracenes are shown as black spheres. In (C), acid and amine terminal groups are shown as red and blue spheres, respectively. Isobutoxy side chains have been omitted for clarity.

Figure 3 .

 3 Figure 3. Part of the 300 MHz 1 H NMR spectra in C2D2Cl4 at 283K of: (A) macrocycle 2; (B) nonamer 3; (C) tridecamer 4; and (D) tridecamer 4 at 353 K. Signals of Hext are marked with white circles. Signals of Hint are marked with black circles. H3 and H5 signals of xylene units are marked with white and black triangles, respectively. H3+H5 fast exchange average signal is marked by a grey triangle. Diastereotopic protons belonging to isobutoxy side chains are marked by solid arrows.

Figure 4 .

 4 Figure 4. (A-E) Energy minimized molecular model using the Merck Molecular Force Field static (MMFFs) of oligomer 5. Structures in the solid state analyzed by single crystal x-ray crystallography of: (F-J) oligomer 6a; (K-O) oligomer 7; (P-T) duplex (8)2. In all representations except (P), the helical cones are shown in blue, the diaza-anthracene units are shown in red whereas the turn units are shown in white. In (K) the intercalated strands of the duplex (8)2 are highlighted in red and black. (E), (J), (O) and (T) show a top view of the helix-sheet-helix architectures in which both helical cones have been omitted. This view highlights the varying orientations of the turn units. In (A-T) isobutoxy side chains and included solvent molecules were omitted for clarity. Only the all P helical isomers are shown. The structures belong to centrosymmetrical space groups and thus also contain the all M isomers.

Figure 5 .

 5 Figure 5. Excerpts of 1 H NMR spectra (400 MHz, 298 K) showing the the amide resonances in CDCl3 of: 5 (1 mM), 6a (1 mM), 6b (1 mM), 7 (1 mM) and the monomeric form of oligomer 8 (0.5 mM). Stars denote aromatic resonances.

Figure 6 .

 6 Figure 6. Excerpts of the 400 MHz 1 H NMR spectra in CDCl3 at 298 K of 8 at: (A) 10 mM; (B) 5 mM; (C) 2 mM; (D) 1 mM; and (E) 0.5 mM. Signals of the amide protons belonging to single stranded 8 are marked with full black circles those belonging to the duplex (8)2 are marked with empty red circles.

Figure 7 .

 7 Figure 7. (A) 1 H NMR titration experiment (400 MHz, 298 K) in CDCl3:D6-acetone (9:1, vol/vol) of 6b (1 mM) in the presence of: 0, 1.3, 2.6, 3.9, 5.3, 7.9, 10.5, 13.1, 19.7, 26.3, 39.4, 59.2 and 78.9 equiv. of glycolic acid. (B) Experimental and calculated values using a 2:1 binding model for the NMR binding study of oligomer 6b vs. glycolic acid in CDCl3:D6-acetone (9:1, vol/vol). Resonance at 9.78 ppm was used for the analysis.

  23

Figure 8 .

 8 Figure 8. Host-guest topologies that can be expected from the association of dumbbell guests and an open-ended helix 22 (A), a helix-turn-helix (B), or a duplex formed from two helix-turnhelix structures (C).

ACKNOWLEDGMENT

This work was supported by the European Research Council under the European Union's Seventh Framework Programme (grant agreement no. ERC-2012-AdG-320892). We thank Dr. G. Lautrette for providing advanced synthetic precursor 19. We gratefully acknowledge the ESRF for beam time on ID29 and Christoph Muller-Dieckmann for technical assistance. This work has benefited from the facilities and expertise of the Biophysical and Structural Chemistry platform at IECB, CNRS UMS3033, INSERM US001, Bordeaux University, France.

AUTHOR INFORMATION

Corresponding Author * ivan.huc@cup.lmu.de Author Contributions ‡ These authors contributed equally to this work.

Present Address † Department of Chemical Technology of Drugs, Poznan University of Medical Sciences, Grunwaldzka 6, 60-780 Poznan, Poland. ‖ Department of Pharmacy. Ludwig-Maximilians-Universität, Butenandtstr. 5, 81377 München, Germany.

Notes

The authors declare no competing financial interest.


Soumen De

Bo Chi

Thierry Granier

Ting Qi

Victor Maurizot

Ivan Huc
email: i.huc@iecb.u-bordeaux.fr.

Designing cooperatively folded abiotic uni-and multimolecular helix bundles

Abiotic foldamers, i.e. foldamers that have backbones chemically remote from peptidic and nucleotidic skeletons, may give access to shapes and functions different from those of peptides and nucleotides, just like peptides and nucleotides differ from each. However, design methodologies towards abiotic tertiary and quaternary structures are yet to be developed. Here we report rationally designed interactional patterns to guide the folding and assembly of abiotic helix-bundles. Computations allowed to introduce hydrogen bonding functionalities at defined locations on aromatic amide backbones that eventually promote cooperative folding into helix-turn-helix motifs in organic solvents. The hydrogen bond-directed aggregation of helices not linked by a turn unit produced several thermodynamically and kinetically stable homochiral dimeric and trimeric bundles having structures distinct from the designed helix-turn-helix. Relative helix orientation within the bundles may be changed from parallel to tilted upon subtle solvent variations. Altogether, these results prefigure the richness and uniqueness of abiotic tertiary structure behavior.

INTRODUCTION

In the last twenty years, chemists have shown that numerous non-natural foldamer backbones adopt well-defined conformations analogous to the secondary motifs of biopolymers, such as helices, sheets, and turns [START_REF] Guichard | Synthetic foldamers[END_REF][START_REF] Zhang | Aromatic amide foldamers: Structures, properties, and functions[END_REF][START_REF] Gellman | A Manifesto[END_REF][START_REF] Goodman | Foldamers as versatile frameworks for the design and evolution of function[END_REF][START_REF] Nair | Heterogeneous foldamers from aliphatic-aromatic amino acid building blocks: current trends and future prospects[END_REF] This new paradigm holds great promise for molecular design. Indeed, biopolymers' efficient and sophisticated functions in terms of recognition, catalysis, information storage and transfer, or energy conversion, rest on their specific shapes obtained through folding of their primary sequence. Folding emerges as the most efficient strategy to organize large and complex arrays of chemical groups in space with atomic precision. In addition, molecules consisting of a linear sequence of building blocks possess the advantage of being inherently modular and amenable to optimization through iterative modifications, e.g. mutations, and deletions or additions of monomers or longer segments. In this context, abiotic foldamers that have backbones chemically remote from the natural peptidic and nucleotidic skeletons are particularly interesting because they may give access to shapes, and thus to functions, different from those of peptides and nucleotides, just like the shapes and functions of peptides and nucleotides differ from each other because of their distinct chemical compositions. For example, foldamers having aromatic moieties in their main chain constitute a distinct grouping that has been shown to form unusual structures such as pilars [START_REF] Scott Lokey | Synthetic molecules that fold into a pleated secondary structure in solution[END_REF] , knots [START_REF] Brüggemann | Spontaneous knotting -From oligoamide threads to trefoil knots[END_REF][START_REF] Ponnuswamy | Discovery of an organic trefoil knot[END_REF] , or helices having a variable diameter [START_REF] Hua | Hydrophobic collapse of foldamer capsules drives picomolar-level chloride binding in aqueous acetonitrile solutions[END_REF][START_REF] Chandramouli | Iterative design of a helically folded aromatic oligoamide sequence for the selective encapsulation of fructose[END_REF] . However, most of the functions of biopolymers, especially of proteins, emerge at the level of their tertiary structures and would not be achieved by an isolated -helix or -sheet. A major challenge thus resides in establishing design principles of artificial tertiary folds comprised of several secondary structural elements. Because of the highly cooperative nature of tertiary folds, protein design itself remains a very challenging endeavour despite great advances [START_REF] Huang | De novo design of a four-fold symmetric TIM-barrel protein with atomiclevel accuracy[END_REF][START_REF] Boyken | De novo design of protein homo-oligomers with modular hydrogenbond network-mediated specificity[END_REF][START_REF] Hilvert | Design of protein catalysts[END_REF][START_REF] Khoury | Protein folding and de novo protein design for biotechnological applications[END_REF] including the development of dedicated computational tools [START_REF] Parmeggiani | A general computational approach for repeat protein design[END_REF] . Changing an -amino acid sequence to optimize interactions between secondary motifs may also alter interactions within them and thus decrease overall conformational stability. Nevertheless, helix bundling is a specific pattern that can largely be predicted, allowing the design of selfassembled quaternary bundles 12 Erreur ! Signet non défini. ,16,17,18,19 as well as novel tertiary structures [START_REF] Gradišar | Design of a single-chain polypeptide tetrahedron assembled from coiledcoil segments[END_REF] . Yet, the exploitation of non-natural monomers in artificial tertiary folds is limited.

Several -amino acids may be tolerated into known peptidic bundles or tertiary structures [START_REF] Reinert | Protein-like tertiary folding behavior from heterogeneous backbones[END_REF][START_REF] Mayer | Building proficient enzymes with foldamer prostheses[END_REF][START_REF] Tavenor | Comparison of design strategies for α-helix backbone modification in a protein tertiary fold[END_REF][START_REF] Horne | Helix-Bundle Quaternary Structure from α/β-Peptide Foldamers[END_REF] . Sheets containing non-natural units have been reported as well [START_REF] Cheng | The supramolecular chemistry of -sheets[END_REF][START_REF] Kreutzer | X-ray Crystallographic Structures of a Trimer, Dodecamer, and Annular Pore Formed by an A17-36 -Hairpin[END_REF] . As key milestones, self-assembled helix bundles of -peptides [START_REF] Daniels | High-resolution structure of a peptide bundle[END_REF][START_REF] Petersson | Biophysical characterization of a -peptide bundle: Comparison to natural proteins[END_REF] and of -ureas [START_REF] Collie | Shaping quaternary assemblies of water-soluble non-peptide helical foldamers by sequence manipulation[END_REF] have been reported.

Stitching helices together within these bundles may in principle yield an artificial tertiary fold but this remains to be implemented. To the best of our knowledge, no genuine tertiary structures are known that do not contain -amino acids. Approaching examples simply consist in connecting several abiotic helices through short or rigid linkers [START_REF] Sharma | Design and synthesis of peptides with hybrid helix-turn-helix (HTH) motif and their conformational study[END_REF][START_REF] Delsuc | Relative helix-helix conformations in branched aromatic oligoamide foldamers[END_REF][START_REF] Ichinose | Tetrameric  aggregate formation by stereoisomeric bidomain helicene oligomers[END_REF] , or non-covalently threading them onto rod-like templates [START_REF] Gan | Translation of rod-like template sequences into homochiral assemblies of stacked helical oligomers[END_REF] . Going beyond the world of peptides, tertiary folds based on abiotic backbones remain an unexplored area. Given current standards in the fabrication of long sequences through solid phase synthesis and ligation methods [START_REF] Kent | Total synthesis of proteins[END_REF][START_REF] Pusterla | An oxazetidine amino acid for chemical protein synthesis by rapid, serine-forming ligations[END_REF][START_REF] Haj-Yahya | Synthetic polyubiquitinated α-Synuclein reveals important insights into the roles of the ubiquitin chain in regulating its pathophysiology[END_REF] , synthesis itself may not constitute a major stumble block towards artificial tertiary structures.

It then appears that, even though interactions within abiotic secondary structural motifs are well-understood and often predictable, interactions between secondary elements have escaped our prediction capabilities. Furthermore, tertiary folding and helix bundling have until now been limited to water.

In the following, we report our success at rationally designing helix-helix interactional patterns to form tertiary folds within abiotic aromatic amide sequences and the subsequent characterization of different quaternary folds based on the same patterns. The approach rested on rather simple structure-based considerations and computational tools. We exploited the high stability of aromatic amide helices [START_REF] Jiang | Aromatic -peptides[END_REF][START_REF] Delsuc | Kinetics of helix-handedness inversion: Folding and unfolding in aromatic amide oligomers[END_REF] that made a hierarchical approach to tertiary structure design possible: helices may not unfold upon interacting with each other and may thus be used as stable construction modules. Nevertheless, we observed the typical cooperative behavior associated to the simultaneous occurrence of a large number of interactions. We also demonstrate that tertiary folding may occur in apolar solvents and be driven exclusively by hydrogen bonds. These results demonstrate that rational design is a viable approach to artificial tertiary structures and pave the way to engineering more complex and larger molecular shapes with atomic precision. Although function was not considered in the context of this study, abiotic tertiary folded objects prefigure the implementation of sophisticated molecular machineries including controlled motion, recognition, and catalysis in other media than water.

RESULTS AND DISCUSSION

Design of tertiary structures

Oligoamides of 8-amino-2-quinoline-carboxylic acid Q (Figure 1a) have been shown to adopt helical conformations stabilized by hydrogen bonds, and electrostatic repulsions, between quinoline nitrogen atoms and adjacent amide NH and CO groups, respectively, as well as by solvophobic forces associated with aromatic stacking. 2 Erreur ! Signet non défini. , Erreur ! Signet non défini. ,39 Non covalent interactions within these helices thus generate well understood motifs in which, unlike in peptides, amides NH point towards, and amide CO point away from, the helix axis. However, at the start of this work, there was no clue on how to orchestrate interactions between two helices and generate more complex architectures. We set to design a tertiary helix-turn-helix structure ab initio by simply examining computer models.

We eventually found that a terephthalic acid bis-hydrazide T had an appropriate length to constitute a turn unit linking two helical Qn segments. T inserts itself in the network of hydrogen bonds and prevents steric clashes between short helices while preserving enough proximity to be compatible with helix-helix interactions. Sequences such as Q4-T-Q4 in which quinoline rings are deprived of side chains may thus adopt two conformations: a C2 symmetrical conformer where both Q4 helices have right-handed (P) helicity (or left-handed (M) helicity) and are on the same side of the turn; and a centro-symmetrical PM conformation in which helices are on opposite sides of the turn (Figure 1e, 1f) [START_REF] Maurizot | Design of an inversion center between two helical segments[END_REF] . In the absence of helix-helix interactions, the PP/MM and PM conformers are expected to be equiprobable. A notable feature of the PP/MM conformer of Q4-T-Q4 is the relative proximity (3.5 to 4 Å) between the carbonyl amide of the second quinoline of each Q4 segment (counting starts from N-terminus), and positions 3 and 4 of the second quinoline ring of the other Q4 segment (Figure 1f). Based on this proximity we inferred that a hydrogen bond donor at these positions might reach the carbonyl group and generate attractive interactions between the two helices. The 4-hydroxygroup of unit X was thus added and energy minimized models predicted the formation of interhelix hydrogen bonds stabilizing the start of a helix-turn-helix motif in the PP/MM conformer (Figure 1b,1g). N-terminus of a Qn helix. The actual helix handedness depends on whether the helix is above or below the plane. For sequences terminated with 8-nitro groups, this group is noted in replacement of the NH group of Q or X units. b-d) hydrogen bonds or steric hindrance critical in the models shown in g-i), respectively. e-i) Energy minimised models. In e) and f), P and M helices are shown in blue and red, respectively. In g), h) and i), the units of interest at the top of the structures are shown as thick blue tubes. In the models, the isobutoxy groups of T were replaced by methoxy groups. Protons and oxygen atoms involved in hydrogen bonds are shown as white and red balls, respectively. A pink surface indicate a steric clash in c) and h). This initial prediction was verified by experiments. Sequence 1b was prepared (for synthetic Schemes see Supplementary Figures 1234). Its 1 H NMR spectrum in CDCl3 at 25 °C shows slightly broadened resonances that split upon cooling, consistent with the coexistence of PP/MM and PM conformers (Figure 2a, 2b) [START_REF] Delsuc | Relative helix-helix conformations in branched aromatic oligoamide foldamers[END_REF] . One species is favored which we assume to be PM (see below). On the contrary, the deprotected sequence 1a shows one set of sharp NMR signals. Diastereotopic motifs of the signals of iBuO side chains indicate slow helix handedness inversion on the NMR time scale even at 25 °C (Supplementary Figure 5), implying that only one conformer could be detected. A crystal structure of 1a was obtained (Figure 3a) which shows a PP/MM helix-turn-helix essentially superimposable to the computer prediction. Models of elongated sequences showed that an additional Q monomer leads to steric clashes between the helices: CH groups in position 6 and 7 of the most peripheral quinoline rings hinder each other (Figure 1c, 1h). Instead, monomer Y (Figure 1a) brings the same contribution to helix curvature as X yet lacks the aryl CH groups responsible for steric hindrance [START_REF] Sánchez-García | Nanosized hybrid oligoamide foldamers: Aromatic templates for the folding of multiple aliphatic units[END_REF] . Modelling predicts that this monomer may allow the formation of two additional inter-helix hydrogen bonds involving the amide carbonyl of the next unit (Figure 1d,1i). Sequence 2a was then prepared whose two helices each consists of two contiguous YQXQQ segments. Qn sequences have been shown to span about five units per two turns Erreur ! Signet non défini. , suggesting that inter-helix hydrogen-bonds of consecutive pentads may be in register. This assumption is to be related to the behavior of heptad repeat motifs of -peptidic zippers [START_REF] Woolfson | The Design of Coiled-Coil Structures and Assemblies[END_REF] . Heptads would be in perfect register if the -helix spanned 3.5 units per turn. The fact that curvature is closer to 3.6 units per turn forces some adjustments, i.e. coil-coiling, to maintain helix-helix interactions over long distances [START_REF] Woolfson | The Design of Coiled-Coil Structures and Assemblies[END_REF] . Sequence 2a contains one turn unit and twenty -amino-acids each of which has the same backbone length as a dipeptide. In terms of molecular weight, sequence 2a is to be compared to a 42mer peptide which is larger than required for small tertiary folds to form [START_REF] Neidigh | Designing a 20-residue protein[END_REF][START_REF] Hodges | Engineering a Monomeric Miniature Protein[END_REF][START_REF] Craven | A Miniature Protein Stabilized by a Cation-π Interaction Network[END_REF] . The 1 H NMR of spectrum of protected sequence 2b again shows two sets of signals reflecting the coexistence of PP/MM and PM species (Figure 2d). In this case, one species is strongly favored that can be assigned to the PM conformer by a crystal structure and the NMR spectrum of freshly dissolved crystals (Figure 3b, Supplementary Figure 6). The crystal structure shows that four tBu and four PMB protecting groups would be in contact and generate hindrance if the helices were on the same side of the turn, i.e. in the PP/MM conformer. Upon deprotection into 2a, only one species is observed (Figure 2e) showing that the two helices interact so as to completely bias the equilibrium. DOSY NMR testified that 2a is monomeric in solution: 2a and 2b have similar diffusion coefficients (Supplementary Figures 7,8). Assignment of OH protons to downfield shifted resonances between 9 and 10 ppm indicate their involvement in hydrogen bonds (Supplementary Figure 9). As our initial attempts to crystallize 2a failed, sequence 3a was prepared. Its helical segments are both one unit shorter than in 2a and are terminated by nitroquinoline rings instead of acetyl-aminomethyl groups, a feature that we have observed to favour crystallization. The crystal structure of 3a (Figure 3c,d, Supplementary Video 1) validated the predicted helix-turn-helix motif: no steric clash occurred and all six expected inter-helix hydrogen bonds were established (Figure 1b,1d), maintaining the two helices in a parallel orientation with the same handedness. In CDCl3, 3a shows spectral features similar to those of 2a: one set of sharp signals whereas its protected precursor 3b shows two sets (Supplementary Figure 10).

Eventually, diffracting crystals of 2a were obtained from DMF. From this polar solvent, an open, "unfolded", PP/MM form crystallized (Figure 3e). This observation triggered the investigation of the solvent dependence of the helix-turn-helix motif. No denaturation of 2 was observed by NMR upon heating in C2D2Cl4 up to 70 °C, reflecting high stability and hinting at a small entropy of folding. However, the addition of hydrogen bonding solvents disrupted the tertiary fold. No folding occurs in DMSO-d6 or pyridine-d5 as judged by the presence of both PP/MM and PM species (Supplementary Figure 11). Remarkably, DMSO-induced unfolding of 2a occurs through a sharp transition between 18 and 23% (vol/vol) of the polar solvent (Figure 2f, Supplementary Figure 12) indicative of a cooperative phenomenon. Nothing similar was observed for 2b (Supplementary Figures 15), confirming that changes observed for 2a reflect a conformational transition and not a solvent dependence of chemical shift values. A similar transition is observed for 1a but it is less sharp and requires less DMSO (Supplementary Figures 13,14). Changes in NMR spectra during the transitions support the existence of two equilibria shown in Figure 2g. The first equilibrium (K1) between the folded PP (or MM) helixturn-helix and its unfolded form is solvent dependant and rapid on the NMR timescale: adding DMSO causes chemical shift variations as the proportions between the two species vary (Figure 2f,g). The second equilibrium between unfolded PP/MM and PM conformations is slow on the NMR timescale. Unfolding thus also goes along with the emergence of the signals of the PM conformer whose protons resonate at frequencies that vary little with the proportion of DMSO (Supplementary Figure 12). The sharp transition between folded and unfolded state is to be related to the overall rigidity of the helix backbone which makes it difficult to not establish all inter-helix hydrogen bonds at the same time. Unfolding does not proceed progressively but through the simultaneous disruption of all hydrogen bonds. In addition, the folded helix-turnhelix is somewhat spring-loaded. X-ray structures of 1a and 3a show a certain level of bending of the turn units (Figure 3a,c) whereas it is flat in the unfolded state (Figure 3b,e). This bending had been predicted by modelling (Figure 1g-1i) and reflects that the turn is actually slightly longer than the inter-helix distance in the folded state. Furthermore, helix curvature in the helixturn-helix is slightly reduced with respect to its conformation in a relaxed state (Supplementary Figure 51), suggesting a certain level of strain that is compensated by hydrogen bonding.

Formation of quaternary structures

We then explored the capacity of inter-helix hydrogen bonds to act intermolecularly. In the absence of a turn unit, additional degrees of freedom are allowed. For instance, T imposes hydrogen bonding to occur in register between helices with same handedness and parallel C→N sequence polarity. In the absence of T, out of register, anti-parallel arrangements, and association between P and M helices are allowed. The deca-amide sequence 4a spans four helix turns and possesses two X and two Y units. Despite the possibilities mentioned above, its 1 H NMR spectrum in CDCl3 shows one set of sharp lines (Figure 4a). Three OH resonances are found at 9.90, 10.90, and 11.36 ppm indicating their involvement in hydrogen bonds, but the fourth is not ( = 7.70 ppm). A DOSY spectrum of a 1:1 mixture of 4a and its protected form 4b shows that 4a is larger despite having a lower molecular weight, indicating an aggregated state (Supplementary Figures 18,19). The aggregate does not dissociate upon diluting down to 0.30 mM. Upon adding CD2Cl2, a second set of signals emerges indicating the presence of a distinct structure that becomes major in pure CD2Cl2 (Figure 4b,c). In this structure, all OH groups are involved in hydrogen bonds ( = 10.51, 10.58, 11.07, and at 11.37 ppm). The exchange between the two species is very slow, showing that considerable rearrangements are needed for their interconversion. For instance, evaporating an equilibrated CD2Cl2 sample and dissolving the residue in CDCl3 allows monitoring over 30 min (Supplementary Figures 24-30). This hints at an interconversion mechanism involving the simultaneous disruption of all hydrogen bonds within aggregates, possibly via a disfavored monomeric state. A crystal structure of 4a revealed the formation of a trimeric helix bundle comprised of three helices with the same handedness, parallel helix axes, and identical C→N sequence polarity (Figure 5a-f, Supplementary Video 2). A total of twelve hydrogen bonds are established between X units and between Y units within (4a)3, similar to the eight hydrogen bonds of helix-turn-helix 3a. However, hydrogen bond orientation is different: in both structures O … H-O angles are all close to 180°, but C=O … H angles are also close to 180° in 3a whereas they are closer to 120° in (4a)3. In (4a)3, hydroxyl protons thus point straight to carbonyl lone pairs, an orientation that has been showed to be favoured [START_REF] Lommerse | Hydrogen Bonding of Carbonyl, Ether, and Ester Oxygen Atoms with Alkanol Hydroxyl Groups[END_REF] but that is made impossible in 2a or 3a due restrictions imposed by turn units. as golden yellow and red spheres, respectively. Only the outer rim of the helices is shown as blue tube. In e), only two helices and one array of inter-helix H-bonds are shown. c,f) hydrogenbonded trimeric arrangements of X and Y units. In c), OH groups point toward of the position 5 of the X unit to which they belong, in contrast with helix-turn-helix structures in which OH groups point towards position 3 (Figure 1b). Dashed lines in d) show that X … X and Y … Y hydrogen bonds are almost parallel, in contrast with the structure of 3a (Figure 3d). g) Side view of the tilted dimeric bundle (5a)2 (CPK representation). h) Front view of (5a)2 showing the 60° tilt angle between helix axes. H-bond donors and acceptors are shown as yellow and red spheres, respectively. Only the outer rim of the helices is shown as blue tube. h) Same view as in j, showing exclusively the X and Y units of one helix in grey (left) and each of the hydrogen bonded X and Y units of the other helix in blue. i) Schematic representation of how a given array of six hydrogen bond donors (yellow) and acceptors (red) may allow hydrogen bonding both a in parallel arrangement, and after an 120° rotation.

The NMR of freshly dissolved crystals of (4a)3 shows that this is the dominant species in CD2Cl2 (4a)3 (Supplementary Figure 23), consistent with the observation that all OH groups are involved in H-bonds (Figure 4c). Dilution causes the dissociation of (4a)3 into the other species which, we have shown above, is also an aggregate, and therefore cannot be anything else than dimeric (Supplementary Figure 32). Ion mobility mass spectrometry (IMS) measurements confirm the coexistence of dimeric and trimeric aggregates (Supplementary Figure 21). The concentration dependence allows to quantify the equilibrium between dimers and trimers: 3x(4a)2 <-> 2x(4a)3 (K4a) with K4a = 2.2 10 5 M -1 in CD2Cl2 and < 0.2 M -1 in CDCl3. We also find that other solvents (CCl4, C2D2Cl4, toluene) affect this equilibrium in various ways (Supplementary Figure 33). The reason for this solvent dependence is unclear.

Pyridine and DMSO cause the dissociation of all aggregates into monomeric helices (Supplementary Figure 34). The effect is more progressive and less cooperative than in the helix turn helix bundle.

The solution behaviour of shorter sequence 5a shed light on the nature of (4a)2. Its 1 H NMR spectrum in CDCl3 shows a major species and two distinct minor species (Figure 4d). A DOSY spectrum of a mixture of 4b and 5a in CDCl3 indicates a larger size for the latter which must therefore be a well-defined aggregate (Supplementary Figures 16,17). Upon adding CD2Cl2, the proportion of one of the two other species increased to become major in pure CD2Cl2, whereas the third species remained minor (Figure 4f). In both major species, all OH groups are hydrogen bonded. Their relative proportion does not depend on concentration indicating that they are composed of the same number of molecules. In contrast, the proportion of the minor species increases with increasing concentration showing that it has a higher molecularity (Supplementary Figure 31). Dissolving crystals of 5a (see below) allowed us to assign the two dominant species to dimers (5a)2. IMS also registered the presence of a trimeric aggregate (Supplementary Figure 20). The minor NMR signals were thus assigned to a trimer (5a)3 which may be similar to (4a)3. The equilibrium constant between the two dimers dimerCDCl3 <-> dimerCD2Cl2 was calculated to be K5a = 2.5 in CD2Cl2 and 0.24 in CDCl3. Again, we found no obvious explanation to the solvent dependence.

The structure in the solid state of the species that prevails in CDCl3 was elucidated (Figure 5g-j, Supplementary Video 3), its identity being established by NMR spectra of freshly dissolved crystals (Supplementary Figure 22). It shows a dimeric arrangement of two helices with the same handedness. The two helix axes are not parallel but tilted by 120° counterclockwise (for P helices) with respect to the expected parallel head-to-head arrangement (or 60° clockwise with respect to a parallel head-to-tail arrangement, Figure 5j). All hydroxyl groups are involved in hydrogen bonds but these are not established between homomeric pairs of X or Y units, but directly between X and Y (Figure 5h). As depicted in Figure 5i (top), the positions of hydrogen bonding sites in one face of a helix can be rotated by 120° and still lead to a matching array of donors and acceptors. The consequence of the 120° tilt in a helix having not 6 but 8 hydrogen bonding sites like 4a is that two hydrogen bonding sites will be unoccupied (Figure 5i, bottom). This is consistent with the 1 H NMR spectrum of (4a)2 in CDCl3 which can reasonably be assigned to a tilted dimer apparently favored in this solvent.

The nature of the (5a)2 dimer that prevails in CD2Cl2 can be proposed based on the partial assignment of NMR spectra and remarkable NOE correlations within the various complexes of 4a and 5a (Supplementary Figures 353637383940414243444546). Head-to-head arrangements of helices of opposite handedness can be ruled out because they lead to X-X and Y-Y mismatches (donors facing donors, acceptors facing acceptors), or X-Y mismatches (different distances between pairs of donors and acceptors). The parallel head-to-head arrangement of helices of identical handedness, as it is found in the structure of 3a, are less favored as the formation of (4a)3 demonstrates. A similar head-to-tail arrangement of helices of opposite handedness (Supplementary Figure 47) is also ruled out based on observed NOEs. Tilted arrangements of helices of opposite handedness (Supplementary Figure 48) can also be ruled out because they would have no symmetry element and give rise to twice the number of 1 H resonances actually observed (i.e. no degeneracy). Finally, the only remaining and most probable option is that the (5a)2 dimer that prevails in CD2Cl2 is also a tilted dimer between helices of identical handedness, but with a tilt direction opposite to that shown in Figure 5j (Supplementary Figures 47484950).

CONCLUSION

Using simple computational considerations, hydrogen bonding has been designed at defined locations on aromatic amide backbones that eventually promote cooperative folding into helixturn-helix motifs in organic solvents. This approach rested on the inherent stability of aromatic amide helices and may be extended to interacting functionalities other than aromatic hydroxyl and amide carbonyl groups. The hydrogen bond-directed aggregation of helices not linked by a turn unit produced several thermodynamically and kinetically stable dimeric and trimeric homochiral bundles having structures distinct from the designed helix-turn-helix. Relative helix orientation within the bundles may be changed from parallel to tilted upon subtle solvent variations such as replacing dichloromethane by chloroform. Altogether, these results prefigure the richness and uniqueness of abiotic tertiary structure behavior. Combining parallel and tilted motifs in the same structure using appropriately designed turn units would yield yet another level of complexity in abiotic tertiary folding. The high stability and rigidity of the helix bundles presented here allow one to speculate that they might accommodate the presence of more flexible backbone features. This would in turn enhance folding entropy and result in melting of the structures upon heating in a tunable fashion. Progress along these lines is being made and will be reported in due course.

Figure 1 .

 1 Figure 1. Foldamer sequence and helix-turn-helix design. a) Structures of units Q, X, Y and

Figure 2 .

 2 Figure 2. Solution studies of helix-turn-helix motifs. Excerpts of 1 H NMR spectra in CDCl3

Figure 3 .

 3 Figure 3. Helix-turn-helix motifs in the solid state. Crystal structures of 1a (a), 2b (b), 3a

Figure 4 .

 4 Figure 4. Solution studies of helix bundles. Excerpts of 1 H NMR spectra showing NH and

Figure 5 .

 5 Figure 5. Helix bundles in the solid state. Crystal structures of (4a)3 (a-f) and (5a)2 (g-j). In

Acknowledgments

. This work was supported by the European Research Council under the European Union's Seventh Framework Programme (Grant Agreement No. ERC-2012-AdG-320892), by the European Union under the People program (FP7 PIIF-2009-254156 postdoctoral fellowship to T.Q.) and by the French-Chinese Foundation for Science and its Applications (postdoctoral fellowship to B.C.). The contribution of Dr. Post in optimizing the synthesis of 4-tBuO protected quinoline monomers is acknowledged. The authors thank Dr. Kauffmann for assistance with x-ray data collection and structure resolution at IECB's X-ray diffraction facility and Dr. Ferrer for beamtime and help during data collection on FIP-BM30A at the European Synchrotron Radiation Facility.

Data Centre (http://www.ccdc.cam.ac.uk/data_request/cif). Reprints and permissions information is available at http://www.nature.com/reprints. The authors declare no competing financial interests. Correspondence and requests for materials should be addressed to I.H. (i.huc@iecb.ubordeaux.fr).

Supplementary


Frédéric Mazenc
email: frederic.mazenc@l2s.centralesupelec.fr

Saeed Ahmed
email: ahmed@ee.bilkent.edu.tr

Finite Time Estimation via Piecewise Constant Measurements ⋆

Keywords: Observer, Estimation, Finite time convergence

We study a broad class of linear continuous-time time-varying systems that contain piecewise continuous disturbances and piecewise constant outputs. Under an observability assumption, we construct a new type of observer to estimate the state of the system in a predetermined finite time in the presence of the disturbances. In contrast to the well-established finite time observer design techniques which estimate the system state using a continuous output, our proposed observer only requires a piecewise constant output. Our simulations illustrate the efficacy of our observer.

INTRODUCTION

The state of the system is not available for measurement in many engineering applications such as automotive systems, bioreactors, communication systems, networked control systems, robotics, and many other fields. Instead, one aims to design an observer to estimate the state using an output that can consist of one or more, but not all, components of the state. Due to this strong motivation, many techniques for state estimation of linear continuoustime systems from output measurements, like Kalman and Luenberger observers, have been proposed in the literature; see, e.g., [START_REF] Kalman | New results in linear filtering and prediction theory[END_REF]; [START_REF] Luenberger | Observing the state of a linear system[END_REF]; [START_REF] Zemouche | Observers for a class of Lipschitz systems with extension to H ∞ performance analysis[END_REF]; [START_REF] Ferrante | An observer with measurementtriggered jumps for linear systems with known input[END_REF].

Most of the above mentioned observer design techniques have the common disadvantage that they guarantee asymptotic convergence of the estimation error to zero, whereas it is often desirable to estimate the exact state of the system in a predetermined finite time for control and supervision purposes. Such finite time observers are of considerable interest in many applications, like in fault detection and state feedback control; see [START_REF] Raff | An impulsive observer that estimates the exact state of a linear continuous time system in predetermined finite time[END_REF]; [START_REF] Sauvage | Design of a nonlinear finite time converging observer for a class of nonlinear systems[END_REF].

Moreover, most of the observers discussed in the literature require continuous measurements. However, in many engineering applications, the measurements are piecewise constant. These systems are called continuous-discrete systems where the system dynamics are continuous while the measurements are only available at discrete instants; see [START_REF] Jazwinski | Stochastic Processes and Filtering Theory[END_REF] and [START_REF] Ahmed-Ali | Continuous-discrete adaptive observers for state affine systems[END_REF] for the notion of a continuous-discrete system. This motivates the problem of constructing finite time converging observers for systems with piecewise constant outputs. There are several works on finite time observer design for cases where the measurements are continuous instead of being piecewise constant; see, e.g., [START_REF] Engel | A continuous time observer which converges in finite time[END_REF]; [START_REF] Raff | An impulsive observer that estimates the exact state of a linear continuous time system in predetermined finite time[END_REF]; [START_REF] Raff | An observer that converges in finite time due to measurement-based state updates[END_REF]; [START_REF] Li | A finite time convergent observer with robustness to piecewise constant measurement noise[END_REF]; [START_REF] Mazenc | Estimation of solutions of observable nonlinear systems with disturbances[END_REF]; [START_REF] Mazenc | Finite Time Estimation Through a Continuous-Discrete Observer[END_REF]. However, to the best of our knowledge, the finite time estimation we study in this work via piecewise constant measurements has remained unsolved, even in the case of linear systems, due to the challenges of quantifying the effects of piecewise continuous disturbances on the observer performance. By contrast, for simpler cases where there are no such disturbances in the system, notable works on finite time observers include [START_REF] Qayyum | Finite time estimation of a linear system based on sampled measurement through impulsive observer[END_REF], which uses periodic sampling times in the outputs and an observability assumption that is similar to the one we use in this work.

In the present paper, we propose a solution to the preceding problem for a family of linear continuous-time systems. We construct an observer to estimate the exact state of the system from synchronously sampled outputs. We consider a sequence of real numbers {t i } and a constant ν > 0 such that t 0 = 0 and t i+1 -t i = ν for all integers i ≥ 0. Then the t i 's will serve as the measurement instants for the output and ν will be a tuning parameter that will govern the estimation error. We will show that the smaller the tuning parameter ν, the better the estimation. We also provide an approximate estimate of the system's state that overcomes the problem of determining explicit formulas for fundamental solutions. Our strategy has several steps. We use a classical prediction result, the finite time observer design technique of [START_REF] Mazenc | Estimation of solutions of observable nonlinear systems with disturbances[END_REF], [START_REF] Mazenc | Finite Time Estimation Through a Continuous-Discrete Observer[END_REF], and finally a novel construction of continuous-discrete observers to complete the observer de-sign; see, e.g., [START_REF] Mazenc | Design of continuous-discrete observers for time-varying nonlinear systems[END_REF] for the notion of continuous-discrete observer. We establish robust stability of the observer with respect to disturbances in the system dynamics.

Our paper shares fundamental features with the significant work of [START_REF] Shim | Asymptotic controllability and observability imply semiglobal practical asymptotic stabilizability by sampled-data output feedback[END_REF]. The idea of repeatedly reconstructing the state values in a short amount of time is already present in [START_REF] Shim | Asymptotic controllability and observability imply semiglobal practical asymptotic stabilizability by sampled-data output feedback[END_REF], where a semiglobally stabilizing sampled output feedback for a nonlinear system is proposed. However, there are three key differences between the present paper and [START_REF] Shim | Asymptotic controllability and observability imply semiglobal practical asymptotic stabilizability by sampled-data output feedback[END_REF]. First, in [START_REF] Shim | Asymptotic controllability and observability imply semiglobal practical asymptotic stabilizability by sampled-data output feedback[END_REF], the output is assumed to be known at any instant. Second, high gain observers are used in [START_REF] Shim | Asymptotic controllability and observability imply semiglobal practical asymptotic stabilizability by sampled-data output feedback[END_REF] to obtain approximate values of the state variable, while here we adopt a finite time reconstruction strategy. Third, although [START_REF] Shim | Asymptotic controllability and observability imply semiglobal practical asymptotic stabilizability by sampled-data output feedback[END_REF] covers nonlinear systems and the present paper is confined to linear systems, [START_REF] Shim | Asymptotic controllability and observability imply semiglobal practical asymptotic stabilizability by sampled-data output feedback[END_REF] imposes a limitation on the size of the sampling period of the feedback, while none of our results here rely on a restriction of this type. In particular, the piecewise continuous disturbances in our systems can capture the effects of sampled feedbacks with arbitrarily large sampling periods.

In Section 2 we describe our objectives in detail and present two lemmas that we will use to prove our main result in Section 3. Our illustration in Section 4 includes numerical simulations and demonstrates the utility of our theory, and in Section 5, we summarize the value added by our paper and suggest future research directions.

Throughout the sequel, the notation will be simplified whenever no confusion can arise from the context. The dimensions of our Euclidean spaces are arbitrary unless otherwise noted. The Euclidean norm in R a in any dimension a, and the induced norm of matrices, are denoted by | • |. Let I denote the identity matrix of any dimension. Let | • | ∞ denote the sup norm of any matrix valued function over its entire domain, and exp(f ) denotes the real valued function e f for any real valued function f . For any matrix valued function Ω : R → R n×n , let Φ Ω : R × R → R n×n denote the function such that ∂ΦΩ ∂t (t, t 0 ) = -Φ Ω (t, t 0 )Ω(t) and Φ Ω (t 0 , t 0 ) = I for all t ∈ R and t 0 ∈ R, where I is the identity matrix. Then Φ Ω is the inverse of the fundamental solution for the time-varying linear system q = Ω(t)q.

PROBLEM STATEMENT AND PRELIMINARIES

Our objective in this section is to construct an observer for a linear continuous-time system with a piecewise constant output such that the observer converges in predetermined finite time in the presence of a disturbance in the dynamics of the system. The observer is expressed in terms of the fundamental solution of suitable time-varying system. Since the disturbance is a general piecewise continuous function, this allows systems with a discontinuous right side which were beyond the scope of [START_REF] Qayyum | Finite time estimation of a linear system based on sampled measurement through impulsive observer[END_REF] and other works. Then in the next section, we use ideas from this section to obtain more explicit formulas for finite time observers that do not contain the fundamental matrix and therefore may be better suited to implementations where the fundamental matrix is not available in explicit closed form.

Our systems have the form

ẋ(t) = Ax(t) + δ(t) y(t) = Cx(t i ) (1)
with x valued in R n for any n ∈ N, y valued in R q for any q ∈ N, the sampling times t i being nonnegative values for all integers i ≥ 0, and δ : [0, +∞) → R n being a known bounded and piecewise continuous disturbance. We assume that A and C are known matrices of appropriate dimensions and the following assumption throughout this paper: Assumption 1. There is a constant ν > 0 such that t i+1t i = ν for all i ≥ 0. Also, the pair (A, C) is observable.

When Assumption 1 is satisfied, we can use [START_REF] Mazenc | Estimation of solutions of observable nonlinear systems with disturbances[END_REF], Lemma 1) to find a constant T > 0 and a constant matrix L such that with the choice

F = A + LC, the matrix M T = e -AT -e -F T
(2) is invertible and such that T /ν is an integer.

To prove our main results, we use the following two lemmas, which we prove in the appendices. The first of these lemmas is from [START_REF] Mazenc | Finite Time Estimation Through a Continuous-Discrete Observer[END_REF]. Lemma 1. Let M ∈ R n×n be an invertible matrix. Let N ∈ R n×n be a matrix. Let n and m be two constants such that |M -1 | ≤ m and |N | ≤ n. Assume that mn < 1 .

(3) Then the matrix M + N is invertible and

(M + N ) -1 -M -1 ≤ m2 n 1 -mn (4) is satisfied. Lemma 2. Let A ∈ R n×n be a constant matrix. Consider the system ζ(t) = [A + E(t)] ζ(t)
( 5 ) where ζ is valued in R n and E : [0, +∞) → R n×n is a bounded piecewise continuous function. Let φ denote the fundamental solution of the system (5). Then for all t 1 ∈ R and t 2 ∈ R such that t 2 ≥ t 1 , the inequality

φ(t 2 , t 1 ) -e (t2-t1)A ≤ |E| ∞ e (t2-t1)|A| e 2|A|(t2-t1) -1 2|A| exp |E| ∞ e 2|A|(t2-t1) -1 2|A| is satisfied.

FINITE TIME OBSERVER DESIGN

Throughout this section, we consider the system (1) and assume that Assumption 1 is satisfied.

Exact Estimate

We provide an exact estimate of the system's state that converges in a predetermined finite time, using the piecewise constant function ϕ(t) = t i when t ∈ [t i , t i+1 ) and i ≥ 0. Here and in what follows, all equalities and inequalities are for all t ≥ 0, unless otherwise indicated. We have

ẋ(t) = F x(t) + δ(t) -Ly(t) + LC[x(ϕ(t)) -x(t)] . We also have x(ϕ(t)) = e A(ϕ(t)-t) x(t) + ϕ(t) t e A(ϕ(t)-m) δ(m)dm .
As an immediate consequence,

ẋ(t) = [F + µ(t)] x(t) + δ(t) -Ly(t) +LC ϕ(t) t e A(ϕ(t)-m) δ(m)dm (6)
where µ(t) = LC e A(ϕ(t)-t) -I .

Let ξ(t) = Φ F +µ (t, 0)x(t). Then

ξ(t) = -Φ F +µ (t, 0)[F + µ(t)]x(t) + Φ F +µ (t, 0) ẋ(t). (7)
Using ( 6) and ( 7), we obtain

ξ(t) = Φ F +µ (t, 0) δ(t) -Ly(t) +LC ϕ(t) t e A(ϕ(t)-m) δ(m)dm . (8)
For any T > 0 and t ≥ T , we can integrate ( 8) over [t-T, t] to obtain

ξ(t) = ξ(t -T ) + t t-T Φ F +µ (m, 0) δ(m) -Ly(m) +LC ϕ(m) m e A(ϕ(m)-s) δ(s)ds dm .
From the definition of ξ, and from the semigroup property of flow maps applied to the flow map Ψ -1 F +µ of the system q = (F + µ(t))q, we deduce that

x(t) = Φ F +µ (t, 0) -1 Φ F +µ (t -T, 0)x(t -T ) + t t-T Φ F +µ (t, 0) -1 Φ F +µ (m, 0) × δ(m) -Ly(m) +LC ϕ(m) m e A(ϕ(m)-s) δ(s)ds dm = Φ -1 F +µ (t, t -T )x(t -T ) + t t-T Φ -1 F +µ (t, m) δ(m) -Ly(m) +LC ϕ(m) m e A(ϕ(m)-s) δ(s)ds dm . (9)
Notice that (9) gives the exact value of the solution of the system (1) in a predetermined finite time T . In other words, the right hand side of (9) provides a finite time observer. However, finding an explicit expression for Φ F +µ may be difficult, which motivates our work in the next section.

Approximate Estimate

It is often difficult to determine explicit expressions for fundamental solutions in order to estimate the system's state using (9). Our next objective is to provide an approximate estimate of the system's state that overcomes the problem of determining explicit formulas for the fundamental solutions, under our standing Assumption 1. In terms of the functions

Σ(T, ν) = G T, |LC|(e ν|A| -1) , G(T, s) = se T |F | e 2|F |T -1 2|F | exp s e 2|F |T -1 2|F | , Ḡ(T, ν) = |e -F T | 2 Σ(T, ν) 1 -|e -F T |Σ(T, ν) , ᾱ(T, ν) = |M -1 T | 2 Ḡ(T, ν) 1 -|M -1 T | Ḡ(T, ν) , (10)
β(T, ν) = e -F T ᾱ(T, ν)

+ |M -1 T | + ᾱ(T, ν) Ḡ(T, ν), (11)
and γ(T, ν) = |M -1 T | + ᾱ(T, ν) e -F T + Ḡ(T, ν) , (12) we prove the following result: Theorem 1. Let the system (1) satisfy Assumption 1, where A, B, and C are known constant matrices. Let F and T be such that M T as defined in ( 2) is invertible and such that T /ν is an integer, where the constant

ν > 0 is such that max |e -F T |Σ(T, ν), |M -1 T | Ḡ(T, ν) < 1. (13) Let x(t i ) = M -1 T ti ti-T e A(ti-m-T ) δ(m)dm -M -1 T e -F T T (t i , δ, y) (14)
where

T (t i , δ, y) = ti ti-T e F (ti-m) δ(m) -Ly(m) + LC ϕ(m) m
e A(ϕ(m)-s) δ(s)ds dm .

Then

|x(t i ) -x(t i )| ≤ ᾱ(T, ν) ti ti-T e A(t-m-T ) δ(m)dm + β(T, ν)| T (t i , δ, y)| + γ(T, ν)Σ(T, ν)T △ (t i , δ, y) holds with the choice T △ (t i , δ, y) = ti ti-T δ(m) -Ly(m) + LC ϕ(m) m e A(ϕ(m)-s) δ(s)ds dm (15)
for all integers i ∈ N such that t i > T .

Proof: Set k = T /ν, which is a positive integer, by our assumptions. By integrating (1), we obtain

e -AT x(t i ) = x(t i-k ) + ti ti-T e A(ti-m-T ) δ(m)dm . (16)
Using (9) and Lemma 2 (applied with A = F and E = µ), we obtain

x(t i ) = e F T + κ(t) x(t i-k ) + T (t i , δ, y) (17) with T (t i , δ, y) = ti ti-T Φ -1 F +µ (t i , m) δ(m) -Ly(m) + LC ϕ(m) m e A(ϕ(m)-s) δ(s)ds dm and |κ(t)| ≤ G(T, |µ| ∞ ).
(18) Here and in the sequel, all equalities and inequalities should be understood to hold for all t ≥ t i and all i such that t i > T .

Our formula µ(t) = LC e A(ϕ(t)-t) -I gives

|µ| ∞ = LC e A(ϕ(t)-t) -I ∞ ≤ |LC| ∞ k=1 A k (ϕ(t) -t) k k! ∞ ≤ |LC| ∞ k=1 |A| k |ϕ(t) -t| k k! . Since |t -ϕ(t)| ∞ ≤ ν, we deduce that |µ| ∞ ≤ |LC| (e ν|A| -1) . ( 19
) Using ( 18) and ( 19), we have

|κ(t)| ≤ G T, |LC| e ν|A| -1 = Σ(T, ν) (20)
for all t ≥ 0. Since our condition ( 13) on ν gives |e -|F |T |Σ(T, ν) < 1, we can use the inequality (20) and Lemma 1 (applied with M = e F T and N = κ(t)) to deduce that e F T + κ(t) is invertible for all t. Then ( 17) gives

e F T + κ(t) -1 x(t i ) = x(t i-k ) + e F T + κ(t) -1 T (t i , δ, y) . (21)
Combining ( 16) and ( 21), we obtain

e -AT -e F T + κ(t) -1 x(t i ) = ti ti-T e A(ti-m-T ) δ(m)dm -e F T + κ(t) -1 T (t i , δ, y) .
Using the definition of M T , we have

[M T + G(t, T )] x(t i ) = ti ti-T e A(ti-m-T ) δ(m)dm -e F T + κ(t) -1 T (t i , δ, y) (22)
where G(t, T ) = e -F T -e F T + κ(t) -1 . Lemma 1 (applied with M = e F T and N = κ(t)) also ensures that |G(t, T )| ≤ Ḡ(T, ν) ( 2 3 ) where Ḡ is from (10). Since M T is invertible, it follows from our condition (13) and the inequality ( 23) and Lemma 1 (applied with M = M T , N = G(t, T ), and n = Ḡ(T, ν)) that M T + G(t, T ) is invertible and from ( 22), we have

x(t i ) = [M T + G(t, T )] -1 × ti ti-T e A(t-m-T ) δ(m)dm -[M T + G(t, T )] -1 × e F T + κ(t) -1 T (t i , δ, y) . ( 24
)
From ( 14) and ( 24), we deduce that

|x(t i ) -x(t i )| ≤ β(t, T )| T (t i , δ, y)| +α(t, T ) ti ti-T e A(t-m-T ) δ(m)dm +γ(t, T )|T (t i , δ, y) -T (t i , δ, y)| (25)
where

α(t, T ) = |[M T + G(t, T )] -1 -M -1 T |, β(t, T ) = M -1 T e -F T -[M T + G(t, T )] -1 e F T + κ(t) -1 , and γ(t, T ) = [M T + G(t, T )] -1 e F T + κ(t) -1 .
Lemma 1 (applied with M = M T and N = G(t, T )) ensures that α(t, T ) ≤ ᾱ(T, ν) ( 2 6 ) where ᾱ was defined in (10). We have

β(t, T ) = M -1 T -[M T + G(t, T )] -1 e -F T + [M T + G(t, T )] -1 × e -F T -e F T + κ(t) -1 ≤ M -1 T -[M T + G(t, T )] -1 e -F T + [M T + G(t, T )] -1 × e -F T -e F T + κ(t) -1 ≤ ᾱ(T, ν) e -F T + |M -1 T | + ᾱ(T, ν)ν Ḡ(T, ν) = β(T, ν)
with β also as defined in (10). We also have

γ(t, T ) = [M T + G(t, T )] -1 e F T + κ(t) -1 ≤ [M T + G(t, T )] -1 e F T + κ(t) -1 ≤ γ(T, ν)
where γ is also from (10). Observe that Lemma 2 gives

|T (t i , δ, y) -T (t i , δ, y)| ≤ ti ti-T Φ -1 F +µ (t i , m) -e F (ti-m) × δ(m) -Ly(m) + LC ϕ(m) m e A(ϕ(m)-s) δ(s)ds dm ≤ Σ(T, ν) ti ti-T δ(m) -Ly(m) + LC ϕ(m) m e A(ϕ(m)-s) δ(s)ds dm = Σ(T, ν)|T △ (t i , δ, y)|
with T △ as defined in (15). It follows from ( 25)-(26) that

|x(t i ) -x(t i )| ≤ β(T, ν)| T (t i , δ, y)| +ᾱ(T, ν) ti ti-T e A(t-m-T ) δ(m)dm +γ(T, ν)Σ(T, ν)T △ (t i , δ, y), (27)
which is our desired estimate. This concludes the proof.

ILLUSTRATION

We illustrate Theorem 1 with the system ẋ(t) = 0 0.15 -0.15 0

x(t) + d(t) 0 (28)
where x = (x 1 , x 2 ) is valued in R 2 , d is scalar valued and represents a perturbation, and the measurement is

y(t) = [ 0.3 0 ] x (t i ) ( 2 9
) where t i = iν for all i ∈ N. One can easily check that Assumption 1 is satisfied with C = [0.3 0], T = 6, and that with the choice (28), we have e At = cos(0.15t) sin(0.15t) -sin(0.15t) cos(0.15t)

where

A = 0 0.15 -0.15 0 . (31)
Hence, choosing L = 0 0.1 and F = A + LC = 0 0.15 -0.12 0 ,

we obtain

e -F T = ⎡ ⎣ cos √ 35T 50 - √ 5 2 sin √ 35T 50 sin √ 35T 50 cos √ 35T 50 ⎤ ⎦ , (32)
e.g., by checking that (32) has derivative -F e -F T with respect to T . Choosing T = 6, we have M T = e -AT -e -F T = -0.0715 0.0226 0.1386 -0.0715 which has a nonzero determinant equal to 0.0020. Then M T is invertible and -36.0244 -11.3718 -69.8228 -36.0244 .

M T -1 =
Now choosing the sampling rate to be ν = 0.05, one can corroborate that ( 13) is satisfied with |e -F T | = 1.0838, Σ(T, ν) = 0.0094, |M T -1 | = 86.9858, and Ḡ(T, ν) = 0.0111. Therefore, we can use (33) in the formula ( 14) for the continuous-discrete observer from Theorem 1 for the system (28) with t i = 0.05i for all i ∈ N.

To illustrate our result, Fig. 1 shows MATLAB simulation of our observer (14) for the system (28) under a piecewise continuous perturbation d(t) = 0.5u(t) with initial conditions x 1 (0) = x1 (0) = x2 (0) = 0, and x 2 (0) = 2. We have also include a zoomed plot in Fig. 1 to depict that we have used a zero-order hold with ν = 0.05 to construct the piecewise continuous estimate x2 from its discrete samples. The fundamental sampling rate of our simulation is 0.1 kHz. The simulation results corroborate convergence of our estimate after T = 6 seconds. Since our simulations show good tracking performance, they help illustrate our general theory in the special case of the system (28) with the measurement (29).

CONCLUSION

For linear continuous-time systems with a piecewise constant output, we proposed an observer of a new type, estimating the system state in a predetermined finite time in the presence of a disturbance in the dynamics of the system. It provides an exact estimate which in general is not given by an explicit formula. This led us to propose an approximate formula, which is given by an explicit formula and whose accuracy is proportional to the size of the sampling interval. We also provided an approximate estimate to overcome the problem of computing the explicit To prove that the matrix M +N is invertible, let us proceed by contradiction. We suppose that it is not invertible. Then there is a nonzero vector V ∈ R n such that V ⊤ (M + N ) = 0, so invertibility of M gives V ⊤ = -V ⊤ N M -1 , and so also |V | ≤ |V | mn. Since V ̸ = 0, we conclude that 1 ≤ mn, which contradicts (3). We deduce that M + N is invertible. To prove the inequality (4), we first set

R = (M + N ) -1 -M -1 . By multiplying R by M + N and M , we obtain (M + N )RM = M -(M + N ) = -N , and so also M RM = -N -N RM . We deduce that R = -M -1 N M -1 -M -1 N R.
As an immediate consequence, we obtain |R| ≤ m2 n+ mn|R|, which allows us to conclude the proof of Lemma 1.

A.2. Proof of Lemma 2

Let φ be the fundamental solution of the system ∂φ ∂t (t, t 0 ) = [A + µ(t)]φ(t, t 0 ) .

Here and in the sequel, t 0 ≥ 0 and t ≥ t 0 are arbitrary. Let ψ(t, t 0 ) = e -A(t-t0) φ(t, t 0 ). Then ∂ψ ∂t (t, t 0 ) = ω(t, t 0 )ψ(t, t 0 ) ( 3 5 ) holds with ω(t, t 0 ) = e -A(t-t0) E(t)e A(t-t0) .

(36) For any vector V ∈ R n , we have ∂ ∂t (ψ(t, t 0 )V ) ⊤ ψ(t, t 0 )V = V ⊤ ψ(t, t 0 ) ⊤ ω(t, t 0 )ψ(t, t 0 )V .

( |E| ∞ e 2|A|(s-t0) exp |E| ∞ e 2|A|(s-t0) -1 2|A| ds (46) where the last inequality is a consequence of ( 43) and (40). We deduce that

|ψ(t, t 0 ) -I| ≤ |E| ∞ t t0 e 2|A|(s-t0) ds exp |E| ∞ e 2|A|(t-t0) -1 2|A| = |E| ∞ e 2|A|(t-t0) -1 2|A| exp |E| ∞ e 2|A|(t-t0) -1 2|A| . ( 47
)
We also have φ(t, t 0 ) -e (t-t0)A = e (t-t0)A e -(t-t0)A φ(t, t 0 ) -I ≤ e (t-t0)|A| |ψ(t, t 0 ) -I| .

(48)

The inequality in conjunction with (47) gives φ(t, t 0 ) -e (t-t0)A ≤

|E| ∞ e (t-t0)|A| e 2|A|(t-t0) -1 2|A| exp |E| ∞ e 2|A|(t-t0) -1 2|A| which is the desired conclusion.

  Since this inequality is valid for all V ∈ R n , we have|ψ(t, t 0 )| ≤ exp |E| ∞

						37)
	Consequently,
		∂(|ψ(t, t 0 )V | 2 ) ∂t	≤ |ω(t, t 0 )||ψ(t, t 0 )V | 2 .	(38)
	Through a simple integration, we obtain
						t
			|ψ(t, t 0 )V | ≤ e	t 0	|ω(m,t0)|dm	|V | .	(39)
	One can check readily that
	Consequently,	|ω(t, t 0 )| ≤ |E| ∞ e 2|A|(t-t0) .	(40)
	t					t
	t0	|ω(m, t 0 )|dm ≤ |E| ∞ = |E| ∞	e 2|A|(m-t0) dm 2|A| . e 2|A|(t-t0) -1 t0	(41)
	Combining (39) and (41), we obtain
	|ψ(t, t 0 )V | ≤ exp |E| ∞	e 2|A|(t-t0) -1 2|A|	|V | .	(42)
						e 2|A|(t-t0) -1 2|A|	.	(43)
	Again using (35), we deduce that
		t t0	∂ψ ∂t	(s, t 0 )ds =

t t0 ω(s, t 0 )ψ(s, t 0 )ds . (44) It follows from the Fundamental Theorem of Calculus that ψ(t, t 0 ) -I = t t0 ω(s, t 0 )ψ(s, t 0 )ds . (45) We deduce that |ψ(t, t 0 ) -I| ≤ t t0 |ω(s, t 0 )||ψ(s, t 0 )|ds ≤ t t0

⋆ This work is supported by the PHC Bosphore 2016 France-Turkey Project under project numbers 35634QM (France) and EEEAG-115E820 (T ÜB İTAK-The Scientific and Technological Research Council of Turkey).