Maria Torres-Werlé

Benoît Heinrich

Aline Maisse-François

Stéphane Bellemin- Laponnaz

Synthesis of

Keywords: chiral bisoxazoline, polytopic ligand, zinc, metallopolymer

come

Introduction

Metallo-supramolecular polymers that combine metal ions and polytopic ligands enable constructing unprecedented materials with specific properties and functions that are of interest for fundamental research but also for the development of new technologies. [START_REF] Whittel | Functional soft materials from metallopolymers and metallosupramolecular polymers[END_REF][START_REF] Wojtecki | Using the dynamic bond to access macroscopically responsive structurally dynamic polymers[END_REF] Indeed, the combination of metal ions with polytopic -organic-ligands necessarily imports the properties of the two components (organic and inorganic) into the final polymer as well as additional properties generated by the metal-ligand coordination such as luminescence for example. These metallopolymers are obtained in a straightforward manner by a process of self-assembly of metal ions with a monomer having linked units (i.e. polytopic ligands) and the resulting polymer may exhibit reversible polymerization / depolymerization characteristics due to the strength of the coordinate interactions. [START_REF] Friese | From coordination complexes to coordination polymers through self-assembly[END_REF][START_REF] Lehn | Dynamic Molecular and Supramolecular Polymers[END_REF][START_REF] Yang | Supramolecular Polymers: Historical Development, Preparation, Characterization, and Functions[END_REF][START_REF] Winter | Synthesis and characterization of metallo-supramolecular polymers[END_REF][START_REF] Lehn | Dynamers: dynamic molecular and supramolecular polymers[END_REF] We have recently shown that while most of these polymers consist of a single binding motif, it may be possible to generate alternating metallopolymers with, for example, two different ligands by means of a chiral molecular instruction. [START_REF] Torres | Chirality-Driven Metallo-Copolymer Formation[END_REF][START_REF] Marinova | Chiral Self-Sorting Process with Ditopic Ligands: Alternate or Block Metallopolymer Assembly as a Function of the Metal Ion[END_REF][START_REF] Taupier | Optically Active Sum-Frequency Generation as an Advanced Tool for Chiral Metallopolymer Material[END_REF] The welldefined construction of such copolymers has been possible through heteroleptic coordination under chirality control at the metal center, which contains freely exchangeable ligands under thermodynamic control. We used the complementarity of these bisoxazoline ligands to generate here alternating metallo-copolymers. Therefore, ditopic, tritopic and tetratopic chiral ligands have been synthesized for that purpose and then used to generate metallopolymers. In all cases, the metallopolymers were instantaneously formed upon addition of zinc salt to a mixture of two ligands of opposite configuration. The resulting solids were characterized including powder X-ray diffraction studies.

Materials and Methods

Chemicals and instruments

All manipulations were performed under an inert atmosphere of argon or nitrogen using standard Schlenk line techniques. Valinol was obtained by reduction of valine. [START_REF] Abiko | An improved, convenient procedure for reduction of amino acids to aminoalcohols: Use of NaBH4-H2SO4[END_REF] 1,1-bis[(4S)-4isopropyl-4,5-dihydrooxazol-2-yl]ethane (1) was synthesized according to previous report. [START_REF] Dagorne | Synthesis and Structure of neutral and cationic aluminium complexes incorporating bis-oxazolinato ligands[END_REF][START_REF] Seitz | Bisoxazolines with one and two sidearms: stereodirecting ligands for copper-catalysed asymmetric allylic oxidations of alkenes[END_REF] Solvents were purified and degassed by standard procedures. All other reagents were commercially available and used as received. Metallopolymers were obtained using methanol as solvent (ACS reagent grade). [START_REF] Whittel | Functional soft materials from metallopolymers and metallosupramolecular polymers[END_REF] H and [START_REF] Seitz | Bisoxazolines with one and two sidearms: stereodirecting ligands for copper-catalysed asymmetric allylic oxidations of alkenes[END_REF] C Nuclear Magnetic Resonance (NMR) spectra were recorded on a Bruker AVANCE 300 spectrometer using the residual solvent peak as reference (CDCl3: δH = 7.26 ppm; δC = 77.16 ppm) at 298 K. Chemical shifts are given in ppm (δ) compared to TMS (tetramethylsilane). Infrared (IR) spectra were recorded on a Nicolet 380 FT-IR spectrometer. KBr discs were made for all samples. Elemental analyses were recorded by the 'Institut de Chimie' laboratory, Université de Strasbourg. HRMS ESI analyses were recorded on microTOF, Bruker Daltonics by the 'Institut de Chimie' laboratory, Université de Strasbourg. Specific rotations were recorded at the 'Laboratoire de Stéréochimie', ECPM, Strasbourg. The XRD patterns were obtained with a linear monochromatic Cu-Kα1 beam (λ=1.5405 Å) by using a sealed-tube generator (600W) equipped with a bent quartz monochromator and recorded with a curved Inel CPS120 counter gas-filled detector linked to a data acquisition computer. Periodicities up to 70 Å can be measured, and the sample temperature controlled to within ±0.01 °C from 20 to 200 °C. In all cases, the crude powder was filled in Lindemann capillaries of 1 mm diameter and 10 μm wall-thickness.

Synthesis of polytopic bisoxazolines

(S)-iPr-DiBox (2)

. General procedure. [START_REF] Torres | Highly Recyclable Self-Supported Chiral Catalysts for the Enantioselective α-Hydrazination of β-Ketoesters[END_REF] 1,1'-Bis[(4S)-4,5-dihydro-4-isopropyloxazol-2-yl]ethane (

(S)-iPr-BP-DiBox (3)

The general procedure was followed: 2.00 mmol (51% yield).

[α]D 25 : -0.59. [START_REF] Whittel | Functional soft materials from metallopolymers and metallosupramolecular polymers[END_REF]

(S)-iPr-TriBox (4)

The general procedure was followed to give the product with identical analytical data as already reported by us. [START_REF] Torres | Highly Recyclable Self-Supported Chiral Catalysts for the Enantioselective α-Hydrazination of β-Ketoesters[END_REF]

(S)-iPr-TetraBox (5)

The general procedure was followed to give the product with identical analytical data as already reported by us. [START_REF] Torres | Highly Recyclable Self-Supported Chiral Catalysts for the Enantioselective α-Hydrazination of β-Ketoesters[END_REF] 2.3 Synthesis of metallopolymers.

Metallopolymer (6) [Zn + (R)-iPr-DiBox + (S)-iPr-DiBox]. General procedure.

Ligands (R)-(2) (R)-iPr-DiBox and (S)-(2) (S)-iPr-DiBox (0.12 mmol, 75 mg each) were dissolved in methanol (1.0 mL). The resulting colorless solution was stirred for 10 minutes before adding dropwise solution of Zn(BF4)2 (0.24 mmol, 57 mg) in methanol (1.0 mL). The mixture led to the instantaneous formation of an opaque gel. The polymer was then isolated by addition of diethyl ether followed by filtration, giving a white powder in quantitative yield. IR (KBr): ṽ = 1634 cm-1 (s, C=N); elemental analysis calcd (%) for C72H108B4F16N8O8Zn2: C 51.12, H 6.43, N 6.62; found C 51.53, H 6.68, N 6.75. Since the product is insoluble in most common solvents, the NMR spectrum could not be taken. Thermo-Gravimetric Analysis (TGA) revealed that the decomposition of the samples occurs above 200 °C, without crossing any phase transition.

Metallopolymer (7) [Zn + (S)-iPr-BP-DiBox + (R)-iPr-BP-DiBox]

The general procedure was followed (0.25 mmol of Zn, quantitative yield). Since the product is insoluble in most common solvents, the NMR spectrum could not be taken. Elemental analysis calcd (%) for C84H116B4F16N8O8Zn2: C 54.72, H 6.34, N 6.06; found C 54.29, H 6.43, N 5.87.

Metallopolymer (8) [Zn + (S)-iPr-DiBox + (R)-iPr-BP-DiBox]

The general procedure was followed (0.25 mmol of Zn, quantitative yield). [START_REF] Whittel | Functional soft materials from metallopolymers and metallosupramolecular polymers[END_REF] H NMR (300 MHz, CD3OD): δ (ppm) = 7.51 (d, Jortho= 8.2 Hz, 4H; Harom), 7.25 (d, Jortho= 7.9 Hz, 4H; Harom) 7.10 (bs, 4H, Harom), 4.39-3.94 (m, 24H), 1.86-1.71 (m, 20H), 0.94-0.85 (m, 48H). Elemental analysis calcd (%) for C78H112B4F16N8O8Zn2: C 52.48, H 6.41, N 6.30; found C 52.06, H 6.73, N 6.29.

Metallopolymer (9) [Zn + (R)-iPr-DiBox + (S)-iPr-TriBox]

The general procedure was followed (0.25 mmol of Zn, quantitative yield). [START_REF] Whittel | Functional soft materials from metallopolymers and metallosupramolecular polymers[END_REF] H NMR (300 MHz, CD3OD): δ (ppm) = 7.14-6.99 (m, 7H, Harom), 4.40-3.35 (m, 30H), 2.98-2.92 (m, 10H), 1.96-1.29 (m, 25H), 0.93-0.85 (m, 60H). Elemental analysis calcd (%) for C210H318B12F48N24O24Zn6: C 50.48, H 6.41, N 6.73; found C 50.84, H 6.88, N 6.62.

Metallopolymer (10) [Zn + (R)-iPr-DiBox + (S)-iPr-TetraBox]

The general procedure was followed (0.25 mmol of Zn, quantitative yield). [START_REF] Whittel | Functional soft materials from metallopolymers and metallosupramolecular polymers[END_REF] H NMR (300 MHz, CD3OD): δ = 7.13-6.99 (m, 6H, Harom), 4.87-3.54 (m, 36H), 2.98-2.92 (m, 10H), 1.98-1.24 (m, 30H), 0.96-0.85 (m, 72H). Elemental analysis calcd (%) for C138H210B8F32N16O16Zn4: C 50.15, H 6.40, N 6.78; found C 50.85, H 6.79, N 6.63.

Results and Discussion

The chiral ligands were synthesized according to the general scheme described in Figure 1. They were obtained in one step from methyl-bis(oxazolinyl)methane derivative [START_REF] Dagorne | For a review on bis(oxazolinate), see[END_REF] by deprotonation with n-BuLi followed by reaction with the appropriate electrophile (i.e., α,α'-dibromo-p-xylene, α,α'dibromo-p,p'-bitolyl, 1,3,5-tris (bromomethyl)-benzene or 1,2,4,5-tetrakis (bromomethyl)-benzene). [START_REF] Nano | Ditopic bis(oxazolines): Synthesis and structural studies of zinc(II), copper(II) and nickel(II) complexes[END_REF][START_REF] Torres-Werlé | Asymmetric benzoylation and Henry reaction using reusable polytopic bis(oxazoline) ligands and copper(II)[END_REF] All compounds were characterized by classical analytical methods. For the purpose of our studies both enantiomers of ditopic ligand (2) and (3) were synthesized whereas ligand (4) and ( 5) were synthesized starting from (S) valinol.

FIGURE 1 Synthesis of polytopic ligands in enantiomeric form from (S)-(1) or (R)-(1)

. Conditions: 1) n-BuLi, -78 °C, 15 min in THF; 2) Electrophile (0.5, 0.33 or 0.25 equiv.), R.T. 12 h.

On the basis of chiral molecular assemblies, the successful construction of an alternate metallo-copolymer is only possible by formation of heterochiral complexes, which has been shown to be possible using C2-symmetric bisoxazoline ligands. Indeed, the least sterically hindered [ML2]-type complex in a tetrahedral environment combines two chiral C2-symmetric bisoxazoline ligands of opposite sign, as displayed in Figure 2. [START_REF] Torres | Chirality-Driven Metallo-Copolymer Formation[END_REF][START_REF] Marinova | Chiral Self-Sorting Process with Ditopic Ligands: Alternate or Block Metallopolymer Assembly as a Function of the Metal Ion[END_REF] This strategy allowed us to construct several metallopolymers with ditopic ligands. We then decided to extend this strategy to synthesize various metallopolymers in particular using units with higher topicity. Using ligands (2) to ( 5), we synthesized five different zinc-based metallopolymers as shown in Figure 3. In all cases, they were instantaneously formed upon addition of Zn(BF4)2 to a methanol solution containing two ligands of opposite configuration and in the appropriate stoichiometric proportion. Metallopolymers ( 6) and (7) were generated from the racemic iPr-DiBox ligand (2) and iPr-BP-DiBox ligand (3), respectively. Metallopolymer (8) was obtained from the iPr-DiBox ligand (R)-( 2) and iPr-BP-DiBox ligand (S)-( 3), whereas metallopolymers ( 9) and (10) were obtained by combination of (R)-( 2) with the tritopic ligand (S)-( 4) or the tetratopic ligand (S)-( 5), respectively. In all cases, a white gel-like material was spontaneously formed upon addition of the zinc precursor. All polymers have then been isolated as white solids in close-to-quantitative yield by addition of diethylether and elemental analyses confirmed the purity of the products. Interestingly, all attempts to generate metallopolymers using ligands with the same absolute configuration were unsuccessful thus confirming the need of heterocomplementarity to form the metal complexes. Figure 4 displays the methanol solution containing equimolar amount of ligand (R)-( 2) and (S)-( 3) prior to addition to the metal salt (a) and after addition (b) showing the formation of the polymer (8). Figure 4 (c) displays the 1 H NMR spectra of the resulting polymer that has been isolated by precipitation with diethylether and filtration. Integration of the signals confirmed the stoichiometric composition of resulting metallo-polymer. We then studied our Zn (II) compounds by powder X-ray diffraction (PXRD), to obtain information on the level of crystallinity of our different products. In the case of the metallopolymer (6), the structure turned out to be semicrystalline with nanometer-size crystals leading to broadened reflections, as displayed in Figure 5.

O N N O N O O N i-Pr i-Pr i-Pr i-Pr N O O N i-Pr i-Pr O N N O N O O N N O O N i-Pr i-Pr i-Pr i-Pr i-Pr i-Pr O N N O N O O N N O O N i-Pr i-Pr i-Pr i-Pr i-Pr i-Pr O N N O i-Pr i-Pr O N N O i-Pr i-Pr H Me (1) 1) n-BuLi 2) X N O O N i-Pr i-Pr O N N O i-Pr i-Pr (S)-(2) or (R)-(2) (S)-(3) or (R)- (3)
Zn (S)-(2) + (R)-(2) + 2 Zn(BF 4 ) 2 CH 3 OH (6)
(S)-( 3) Although the overlapping and the poorly defined positions of the reflections precludes the resolution of the average crystalline structure, the information from PXRD pattern is sufficient for a general description of the material state. Specifically, the main diffracted intensity shares in two groups of unresolved reflections: one at small-angles (around the major peak at 9.5° corresponding to 9.3 Å) from the lateral arrangement of polymer chains and counter-ions, and a group at wide-angles (18-25° corresponding to 5-3.5 Å) from periodicities in the range of lateral distances between close-packed molecular segments. Outside these groups, diffracted signals are essentially blurred out by the structural disorder, whereupon the molecular organization could be considered as "soft-crystalline", as meaning in a state intermediate between crystalline and amorphous solids. For metallopolymer (8), also shown in figure 5, crystallinity even totally vanished since the patterns only contains two broad scattering signals from average lateral distances between polymer chains and molecular segments. The similarity of their shapes and positions with those of the unresolved reflection groups of (6) strongly suggests close localrange arrangements for both polymers. They indeed contain the same metal ion and counter-ions, whose arrangement into ionic strata drives the self-assembly. The nature of the organic spacers and their sequence in the backbones allows however to modulate the emerging structure or here to obtain such a structure only for (6). The reason is obviously that (6) exclusively associates enantiomeric units of type (2), while (8) alternates type (2) and type (3) units: complexes from successive strata are hence connected by a phenyl spacer in (6), while they are either connected by a phenyl or a longer biphenyl spacer in (8). Prerequisite for crystallization in this latter case is that polymers self-assemble in a way that all spacers of same nature locate on the same side of ionic strata. Such an unlikely configuration is perhaps not possible for (8), which would then explain that this polymer stays in an amorphous-like state. Confirmation of this view is brought by polymer (7) in figure 5, for which both (2) units were replaced by (3), and thus all phenyl spacers by biphenyl. Constrain from spacer location was therefore removed and crystallization effectively reappeared. Ionic parts being identical for ( 6) and ( 7), structures are also quite similar, with in particular the same major periodicity at 9.3 Å from the lateral arrangement of polymers. There is nevertheless a striking difference in the regularity and correlation length of the structures: for (7), reflections are substantially sharper and appear in the whole scattering range; this material is hence in a fine-powdered crystalline state. It is unclear why the spacer nature has this effect: tentatively, it could be related to the contribution of biphenyl to close-packing or to the higher spacing of ionic strata. Apart from modulating the emerging structure, the organic spacer offers an easy route toward higher topicity and up to 4 coordinating sites could be introduced on phenyl and combined to ditopic ligands, to yield the metallocopolymers ( 9) and ( 10

+ (R)-(3) + 2 Zn(BF 4 ) 2 (7) (R)-(2) + (S)-(3) + 2 Zn(BF 4 ) 2 (8) 3 (R)-(2)

Conclusion

In conclusion, we have shown that using zinc ions alternate metallocopolymers may be easily produced through the formation of heterochiral [ML2]-type complexes. Indeed, chiral C2-symmetric bisoxazoline derivatives exhibit the good spatial arrangement to generate these heterochiral species. When ditopic ligands of opposite configuration were used such as (S)-( 2) and (R)-( 2) or (S)-( 3) and (R)-(3), semi-crystalline solids were obtained. Other combination led to amorphous solids, such as pairing of (S)-( 2) with (R)-( 3) for example. In addition, ligands of higher topicity also led to amorphous materials. Indeed, ramifications may perturb the close-packing and crystallinity disappears as soon as topicity of the ligand is higher than 2.

From these experiments we can further conclude that prerequisite for crystallization of linear metallopolymers from ditopic ligands is that they self-assemble in a way that all spacers of same nature locate on the same side of ionic strata.

  H NMR (300 MHz, CDCl3): δ (ppm) = 7.42 (d, J= 8.2 Hz, 4H; Harom), 7.21 (d, J= 8.3 Hz, 4H; Harom), 4.30-4.23 (m, 4H; -NCH), 4.07-3.91 (m, 8H; -OCH2), 3.33 (q, J= 13.5 Hz, 4H; -CH2), 1.85-1.73 (m, 4H; -CH(CH3)2), 1.47 (s, 6H; -C(CH3), 0.95-0.81 (m, 24H; -CH(CH3)2; 13 C{1H} NMR (300 MHz, CDCl3): δ (ppm) = 167.5 (N=CO), 139.5 (-Carom), 135.6 (-Carom), 130.8 (-Carom), 126.6 (-Carom), 72.0, 71.6 (-NCH), 70.1, 69.8 (-OCH2), 43.4 (-CCH3), 41.8 (-CH2), 32.4, 32.3 (-CH(CH3)2), 21.3 (-C(CH3), 18.8, 18.7, 17.9, 17.5 (-CH(CH3)2); IR (KBr): ṽ = 1655 cm-1 (s, C=N); MS (ESI +): m/z (%): 684.45 [M+H] + ; elemental analysis calcd (%) for C42H58N4O4 (682.93): C 73.86, H 8.56, N 8.20; found C 73.94, H 8.60, N 8.02. (R)-iPr-BP-DiBox was obtained using the same procedure starting from 1,1'-Bis[(4R)-4,5-dihydro-4-isopropyloxazol-2-yl]ethane.

FIGURE 2

 2 FIGURE 2 Shaping coordination spheres with chiral C2-symmetric bisoxazoline derivatives: favorable combination in [Zn(bisox)2] 2+ complexes in a tetrahedral environment, demonstrating the complementarity of opposite enantiomers.

FIGURE 3

 3 FIGURE 3 Synthesis of metallopolymers 6-10, using ditopic, tritopic or tetratopic ligands (2-5). Conditions: MeOH at room temperature.

FIGURE 4

 4 FIGURE 4 Pictures of the resulting solution before (a) and after (b) complexation of Zn(BF4)2 with an equimolar amount of (R)-(2) and (S)-(3); c) 1 H NMR spectra (CD3OD, aromatic region) of (S)-iPr-DiBox (top), (R)-iPr-BP-DiBox (middle) and the corresponding copolymer (bottom) obtained from Zn(BF4)2.

(

FIGURE 5

 5 FIGURE 5 PXRD patterns of metallopolymers (6) (bottom, black), (8) (middle, red) and (7) (top, blue).

  ), whose PXRD patterns are displayed in Figure6. It was expected that ramifications would perturb the close-packing, and effectively crystallinity vanished for both copolymers. The patterns of these amorphous-like polymers display the two scattering signals from the local-range close-packing of entire polymers and molecular segments. The shapes and positions of these signals stay similar to those of the unresolved reflection groups of the linear polymer(6), indicating that ramifications have little impact on the local-range packing of complexes and counter-ions.

FIGURE 6

 6 FIGURE6 PXRD patterns for metallocopolymers (9) and (10) of topicity higher than 2.

Acknowledgements

This work was funded by the French National Research Agency (ANR) through the Programme d'Investissement d'Avenir under contract ANR-11-LABX-0058_NIE within the Investissement d'Avenir program ANR-10-IDEX-0002-02.


Matthew J Cliffe

Joshua A Hill

Claire A Murray

Franc ¸ois-Xavier Coudert

Andrew L Goodwin
email: andrew.goodwin@chem.ox.ac.uk

Defect-dependent colossal negative thermal expansion in UiO-66(Hf) metal-organic framework †

Keywords: Synthesis, experimental methods, and sample characterisation; X-ray powder diffraction refinement details

Thermally-densified hafnium terephthalate UiO-66(Hf) is shown to exhibit the strongest isotropic negative thermal expansion (NTE) effect yet reported for a metal-organic framework (MOF). Incorporation of correlated vacancy defects within the framework affects both the extent of thermal densification and the magnitude of NTE observed in the densified product. We thus demonstrate that defect inclusion can be used to tune systematically the physical behaviour of a MOF.

Introduction

It has long been recognised that defects and their correlations play a central role in determining the physical properties of many functional materials: the mass transport pathways of fast-ion conductors, 1,2 charge localisation in high-temperature superconductors, 3,4 and emergence of polar nanoregions in relaxor ferroelectrics 5 are just three examples. The recent discovery that correlated defects can be systematically introduced into the structures of some canonical metal-organic frameworks (MOFs) 6 suggests the possibility of establishing defect/property relationships in this broad family of materials long-favoured for its chemical versatility. [7][8][9] Perhaps the most obvious benefit of determining such relationships would be the ability to exploit defects and non-stoichiometry to develop MOFs with particularly attractive or otherwise-inaccessible physical properties.

The few recent studies of defective MOFs have focussed either on the nature of defect inclusion itself or on how defects influence chemical properties such as catalytic activity and guest sorption. [10][11][12][13] Here we ask a different question: namely, is it possible that defects can influence the physical properties of MOFs? In attempting to answer this question, we focus in this proof-of-principle study on the thermomechanical behaviour of the canonical hafnium terephthalate MOF UiO-66(Hf). 14 Our choice of this specific pairing of physical property and chemical system is motivated by the following considerations. First, the response of framework geometry to changes in temperature is at once both straightforwardly measured and strongly characteristic of the underlying elastic behaviour of the material in question. 15,16 Second, the empirical propensity for MOFs to exhibit anomalous mechanical effects such as thermal contraction (i.e. negative thermal expansion, NTE; Refs. 17-19) has led to a general expectation that such properties will be especially sensitive to the existence and nature of structural defects. 9 And third, UiO-66(Hf) is the MOF for which there is arguably the greatest and best-understood chemical control over defect incorporation. 6,11,20,21 The idealised structure of non-defective UiO-66(Hf) consists of a face-centred cubic array of oxyhydroxyhafnium(IV) clusters connected via terephthalate (benzene-1,4-dicarboxylate, bdc 2-) linkers [Fig. 1(a)]. 14 The particular defects we study here are introduced by substituting formate for terephthalate during framework synthesis. As a monotopic (capping) ligand, formate reduces the connectivity of the framework. This, in turn, encourages vacancies of entire hafnium clusters. Formate content can be increased at least as far as the situation where, of the 12 carboxylate ligands of each hafnium cluster, four ligands are formates and eight are terephthalate carboxylates [Fig. 1(b)]. 6,11 While this substitution results in substantive structural modification over the unit-cell (nanometre) length scale-indeed one in four hafnium clusters are vacant in this case-the actual coordination environment within the clusters is essentially indistinguishable in the defective and defect-free modifications. 6,22,23 In order to amplify the structural consequences of defect incorporation we exploit the high-temperature ligand elimination reaction known to occur in this system. 11,24,25 Whereas this step involves elimination of H 2 O for defect-free UiO-66, the lability of non-bridging formate ligands is such that it is formic acid that is eliminated in our defective UiO-66, resulting in markedly different cluster geometries for the two cases [Fig. 1(c,d)]. 25,26 Making use of in situ variable-temperature synchrotron Xray diffraction measurements of a series of UiO-66(Hf) samples with a range of defect concentrations, we proceed to demonstrate the following. First, ligand elimination results in framework densification, the magnitude of which is de-pendent on defect concentration. Quantum mechanical calculations help interpret the microscopic origin of this volume reduction. Second, all densified-UiO-66(Hf) samples exhibit isotropic NTE behaviour that is many times stronger than that of any other MOF, and in one case even satisfies the condition of "colossal" NTE (its volumetric coefficient of thermal expansion α V = dV /V dT ≃ -100 MK -1 ). This is the first report of colossal NTE in an isotropic MOF. Third, the magnitude of NTE can also be tuned systematically by defect concentration. Our paper concludes with a discussion of the likely mechanisms by which defect inclusion might influence elastic properties of this particular MOF, the challenges posed for understanding NTE in defective materials, and the implications for exploiting defect/property relationships in other MOF systems.

Results and discussion

Having used the method of Ref. 6 to prepare a series of six UiO-66(Hf) samples with a range of defect concentrations, we sought first to establish that our samples included defects of the type discussed above. X-ray powder diffraction measurements revealed the existence of structured diffuse scattering in the form of broad superlattice reflections that are symmetryforbidden in the defect-free system but are associated with nanodomains of correlated cluster vacancies in defective UiO-66 [Fig. 2(a)]. The ratio of the intensity of this diffuse scattering to that of the parent reflections is a measurable quantity that is sensitive to defect concentration. We find an approximately linear relationship between this ratio and the quantity of terephthalic acid used during synthesis, suggesting that our ensemble of defective UiO-66(Hf) samples spans a suitable range of defect concentrations [Fig. 2(a,b)].

The onset temperature of ligand elimination was determined using thermogravimetric analysis (TGA). The TGA traces of all our samples are qualitatively similar and can be interpreted on the basis of previous studies of this system: 11,25,27 tion and TGA measurements point to a systematic variation in defect concentration across our series of UiO-66(Hf) samples. In order to determine the effect of defect concentration on thermomechanical behaviour, we measured variabletemperature X-ray powder diffraction patterns for all six samples. Because we were interested in determining the thermal expansivity of the phases produced by ligand elimination, we paid particular care to ensuring that our samples were heated sufficiently to drive this elimination step but not so far as to initiate framework decomposition. In practice this dictated an experimental protocol of heating each sample to a temperature of 340 • C and then measuring thermal expansivity on subsequent cooling to 100 • C. So the temperature range probed in these experiments corresponds to the first two stages observed in the TGA measurements described above.

For all samples the same qualitative behaviour was observed (although, as will form the focus of subsequent discussion, the magnitudes of these changes varied from sample to sample). On heating through the first stage of mass loss (i.e. up to 250 • C) the most noticeable changes to the diffraction pattern were in the relative intensities of reflections sensitive to solvent occupation within the framework pores. There was only a very small change (∼0.1%) in the size of the unit cell within this regime. These observations are consistent with our interpretation of the TGA trace and with previous investigations. 11,25 In contrast, the ligand elimination process that occurs between 250 and 350 • C resulted in an orderof-magnitude larger decrease in the cubic unit cell parameter that has not previously been observed in defect-free samples [Fig. 3(a)]. The diffraction pattern of the densified framework formed at this higher temperature can still be accounted for by a model based on the ambient-temperature structure, adjusted for the change in lattice dimensions; however the scattering intensity in high-angle Bragg peaks was much reduced such that detailed structural refinement did not prove possible. On subsequent cooling from 340 to 100 • C all samples showed NTE [Fig. 3(a,b)]. The two processes of densification and NTE, together with their dependence on defect concentration, are discussed in turn below.

Thermal densification

In the absence of a diffraction-based structural model for the densified framework, we turned to quantum mechanical calculations to verify that the structural changes associated with formate elimination were consistent with the magnitude of volume collapse observed experimentally. From a computational viewpoint it would have been unfeasible to use ab initio methods to relax large atomistic configurations describing In contrast, it is already known that elimination of water from defect-free UiO-66 does not result in any significant volume change. 26 By inspection of the relaxed structural model for Hf 6 O 8 (bdc) 4 , we conclude that distortions of the Hf 6 octahedra are responsible for this framework densification. Elimination of HCOOH amplifies the tetragonal distortion of Hf 6 octahedra, reducing the effective octahedral volume by 4.2% and bending the cluster-bdc-cluster connection so as to couple local distortion to macroscopic volume contraction [ESI Fig. 4 †]. The relaxed structural models are available as crystallographic information files as part of the ESI. † If, as this analysis would suggest, the process of framework densification depends on the presence of structural defects, then we should expect stronger densification in UiO-66(Hf) samples with larger defect concentrations. By heating our samples to sufficiently-high temperatures to ensure complete densification (and working within the constraints of available synchrotron beam time), we found this expected property/defect relationship to be borne out in practice [Fig. 4(a,b)]. Our most defective samples showed densifications that were larger by ∼1% (of the total lattice parameter) than for the least defective samples; this result is consistent not only with the results of our ab initio calculations but also with previous experimental reports. 26

Negative thermal expansion

Perhaps the most intriguing physical property of densified-UiO-66(Hf) is its NTE response, which is evident in the increase of the cubic unit cell parameter on cooling from 340 • C [Fig. 3(a)] and in the anomalous thermal shifts of Bragg peaks in the raw diffraction data themselves [Fig. 3(b)]. While NTE is increasingly frequently observed amongst MOFs, there are two particular aspects of the behaviour we observe here that ‡ The tendency for calculations to overestimate lattice parameters for organic and organic-containing materials is well-known phenomenon and arises due to the difficulty of correctly accounting for dispersive interactions. It does not impact the analysis of structural and lattice parameter trends in families of materials of similar composition and porosity. 28 MOF-14 are especially attractive: on the one hand, the cubic crystal symmetry of defective UiO-66 means that its NTE is isotropic-i.e., it occurs with constant magnitude in all crystal directions; and, on the other hand, the magnitude itself is really very large. Thermal expansion effects in different materials can be compared via the lattice expansivities

α V = 1 V ∂V ∂T p , (1)
with typical (positive) thermal expansion of engineering materials corresponding to values of α V in the range 10-30 MK -1 . [START_REF] Krishnan | Thermal Expansion of Crystals[END_REF] Here, the lattice parameter data of Fig. 3(a) correspond to a thermal expansivity of α V = -97(4) MK -1 over the temperature range 160-340 • C, which is comparable in magnitude to that of "colossal" thermal expansion materials (|α| ≃ 100 MK -1 ) such as Ag 3 [Co(CN) 6 ]. [START_REF] Goodwin | [END_REF] Although we observe an even more rapid increase in lattice parameter on further cooling from 160-100 • C, we cannot rule out the possibility that this feature may arise from re-uptake of guest molecules at these lower temperatures, and hence do not attach particular weight to that observation here. We have also carried out separate variable temperature diffraction measurements confirming the reversibility of the NTE behaviour [ESI Fig. 4].

Isotropic NTE behaviour of this magnitude is rare indeed. First discovered in ZrW 2 O 8 in the late 1960s, 31 isotropic NTE is now known to occur in a handful of materials, such as ZrW 2 O 8 itself and related substitutional variants, [31][32][33] the perovskite analogues ScF 3 and ReO 3 , [34][35][36] the cuprite-structured oxides Ag 2 O and Cu 2 O, 37 the Zn x Cd 1-x (CN) 2 family and anhydrous Prussian Blue analogues, [38][39][40][41] and the three MOFs HKUST-1, MOF-5, and MOF-14. [17][18][19] Recent theoretical calculations suggest that isotropic NTE may occur in a wider range of MOF materials that currently experimentally reported and re-emphasise the important role of topology. 42 Of these various systems, single-network Cd(CN) 2 is the only colossal NTE material (α V = -100.5(15) MK -1 ). 41 Despite decades of theoretical and experimental studies there is only general consensus that low-energy transverse vibrational modes play at least some role in the NTE mechanisms for most of these materials. The details of NTE mechanisms remain controversial, with new studies frequently demanding reinterpretation of long-held assumptions as and when they appear. 17,[43][44][45] Here we avoid speculating on a detailed NTE mechanism and focus instead on the measurement and defect dependence of the phenomenon. We note simply that, with its open framework structure of high-nuclearity clusters connected by long, light, and flexible organic linkers, densified-UiO-66(Hf) certainly shares many of the basic design features thought to favour low-energy NTE modes. Fig. 4(d) places the isotropic NTE behaviour of UiO-66(Hf) in the context of the three other MOFs known to show the effect. [17][18][19] As anomalous as the thermal expansion behaviour of densified-UiO-66(Hf) is, our key interest is in the possibility that the magnitude of NTE might be sensitive to defect concentration. What sensitivity, if any, do we actually expect? An earlier study of the geometric contribution to NTE in molecular frameworks suggested that defects of the type thought to occur in UiO-66(Hf) should actually encourage NTE because they act to increase the relative contribution of NTE vibrational modes to the overall framework dynamics. 46 Likewise, even simple Maxwellian counting suggests that the balance of degrees of freedom and geometric constraints would be tipped increasingly in favour of the former (and hence NTE) as linkers are removed from a three-dimensional network. [START_REF] Thorpe | Encyclopedia of Complexity and Systems Science[END_REF] Not taken into account in either analysis, however, is the observation we make here that defects are also linked to framework densification. Empirically, one finds that higher densities usually result in weaker NTE-this is exactly what is observed in zeolites, Prussian blues, and the Zn x Cd 1-x (CN) 2 family alike. 38,40,[START_REF] Lightfoot | [END_REF] To complicate further this balance of competing effects, we remark that the ligand elimination process alters the point symmetry of the Hf 6 clusters in a way that may itself affect the dynamics-and hence the NTE-of defective UiO-66(Hf). The point symmetries of both defect-free and defective Hf 6 clusters (D 3d and D 4h , respectively) are incompatible with one another, as they are with the point symmetry of the crystallographic site on which they both sit (O h ) [Fig. 1(c,d)]. Consequently, the effective energy potential governing transverse vibrational motion-which, by analogy to better-understood NTE materials, is likely to be implicated in NTE-will not be constrained by symmetry and hence will vary from site to site throughout the defective lattice. Despite the differences in local symmetry between the reo nanodomains and fcu matrix, both phases retain their cubic symmetry on the unit cell level. The thermal expansion within each domain will therefore remain isotropic. Differences in thermal expansion between phases will lead to the development of thermal strains at the domain walls, which would in principle manifest themselves as differential peak broadening between primitive and face-centered reflections on heating. The large strains produced by the densification precluded our measurement of this comparatively subtle effect. The effect of defect concentration on the average effective potential, its asymmetry, and its variance, collectively poses an interesting challenge for computational and theoretical methods of understanding NTE lattice dynamics in this system. What we observe in practice is that our UiO-66(Hf) samples with higher defect concentrations actually show reduced NTE effects relative to defect-poor samples. The coefficients of thermal expansion determined from our variable-temperature X-ray powder diffraction measurements range systematically from -70(2) to -97(4) MK -1 as defect concentrations are reduced [Fig. 4(c)]. So any effect of increased flexibility arising from the reduced framework connectivity at high defect concentrations must be outweighed by considerations of densification and/or local symmetry. Whatever the microscopic origin for this defect dependence, we note that the chemical control over NTE behaviour we observe here would usually be associated with aliovalent substitution, as exploited in conventional solid-state NTE compounds (e.g. the ZrV x P 2-x O 7 family 32 ). That we can achieve similar control over the physical properties of MOFs through the use of defect chemistry is precisely the proof-of-principle needed to demonstrate that defects might be used as a design element in their own right

1-8 | 5
when engineering MOFs with targeted properties.

Concluding remarks

One of the key motivations for exploring NTE behaviour in defective MOFs was always that NTE is very often diagnostic of other lattice-dynamical anomalies, such as pressure-induced amorphisation and/or mechanical softening. Consequently the variation in magnitude of NTE documented in this study implies that changes in defect concentrations are likely also to affect the mechanical stability of MOFs in high pressure environments and their resistance to amorphisation. We find that this relationship is likely to be entirely counterintuitive for defective UiOs (at the very least): reduced NTE nearly always translates to greater pressure stability, and so higher defect concentrations might now be expected actually to frustrate amorphisation rather than to encourage it. In turn this may provide a mechanism to improve the mechanical properties of MOFs, such as their machinability and durability under operating conditions. Whether this bizarre relationship of defect-driven toughening actually holds in practice remains to be shown, but what is clear even at this stage is that further studies of defect/property relationships in MOFs are set to challenge our collective intuition of the physical consequences of structural disorder.

While we have focussed heavily on the implications of variable NTE in the UiO-66 system, we note for completeness that the control over framework densification evident in Fig. 4(b) may highlight another means of exploiting defects in MOFs. It has recently been shown that densification of guest-containing MOFs through pressure or mechanical grinding can be an effective method of immobilising dangerous species (e.g. radioactive iodine). 49,50 Perhaps the incorporation of defects might simultaneously increase capacity for adsorption of these guest molecules while also trapping these molecules more efficiently as a result of an increased density change on framework collapse.

In summary, we have shown for the first time how the thermomechanical properties of a MOF can be systematically varied via controlled incorporation of defects. The thermomechanical properties we observe in defective UiO-66(Hf) are extremely unusual, and include the strongest isotropic NTE effect ever observed for a MOF. This property in itself may have application in counteracting the positive thermal expansion behaviour of other engineering materials. Yet the most important result remains a conceptual one: namely that defects might now realistically play a key role in the design of new classes of functional MOFs.

Methods Preparation of defective UiO-66(Hf)

We synthesised defective UiO-66(Hf) using the method described for large scale synthesis in Ref. 6. The most defective sample was synthesised as follows. HfCl 4 (3 mmol, 99.9% (metals basis, < 0.5% Zr), Alfa Aesar) and terephthalic acid (2.5 mmol, Sigma Aldrich) were added to a 250 ml Schott bottle and dissolved in N,N dimethylformamide (DMF) (40 ml, Sigma Aldrich). This mixture was sonicated until the metal and ligand had dissolved and then formic acid (20 ml, 95%, Sigma Aldrich) was added. The bottle was sealed then placed in an oven at 120 • C for 24 h. The product was isolated by filtration, washed (DMF) and then heated at 60 • C with DMF for 3 days to ensure the removal any residual ligand. Further impurities were removed from the filtered product by Soxhlet extraction (ethanol). The purified product was then dried at 150 • for 24 h in vacuo. Less defective samples were prepared using larger concentrations of terephthalic acid: 3.0, 4.0, 4.5 and 5.0 mmol. All samples were loaded in 0.5 mm quartz glass capillaries with Al 2 O 3 as an internal temperature standard (ca 50% by mass) and sealed with cotton wool.

Thermogravimetric Analysis

Thermogravimetric analysis was carried using a Mettler Toledo TGA/DSC 1 System, heating from 30 • C to 650 • C at a rate of 20 • C min -1 before cooling down to room temperature.

Variable-temperature synchrotron X-ray diffraction measurements

Variable temperature X-ray powder diffraction measurements were carried out on the I11 beamline at the Diamond Light Source (λ = 0.82562 Å) using a position sensitive detector with a Cyberstar hot air blower. 51,52 Analysis of the data was carried out using Topas Academic version 4.1. [START_REF] Coelho | TOPAS -Academic: General Profile and Structure Analysis Software for Powder Diffraction Data[END_REF] All samples' purity were checked using a Rietveld fit to the reported structure for UiO-66(Zr), [START_REF] Cavka | [END_REF] with Hf substituted for Zr. The additional primitive peaks were fitted using a secondary Pawley phase with the same lattice parameter but with additional size broadening as described in Ref. 6. Lattice parameters were determined using Pawley refinement in space group Fm 3m, omitting the low angle region where the broad primitive reflections make an important contribution. The Pawley refinements were carried out using the previous temperature's refined parameters as the input for the next refinement.

Quantum Mechanical Calculations

All structures were fully relaxed by optimizing both atomic positions and unit cell parameters. We performed quantum mechanical calculations in the density functional theory approach with localized basis sets (CRYSTAL14 code 55 ). We used the B3LYP hybrid exchange-correlation functional 56 and all-electron basis sets for all atoms, 25 except Hf for which we used a small relativistic effective core potential with 12 outer electrons considered explicitly. 57 Full methodological details can be found in in Refs. 6 and 57. The optimised structures are included as crystallographic information files as part of the ESI †.

Fig. 1

 1 Fig. 1 The ideal non-defective UiO-66(Hf) consists of oxyhydroxyhafnium(IV) clusters connected via terephthalate linkers into a face-centred cubic fcu topology network. A unit cell is shown in panel (a), with the Hf clusters represented as blue truncated octahedra and linkers as red rectangles. The atomic-scale structure of a single cluster is shown in detail in the top-left of the panel. (b) at high concentrations of formic acid promotes the formation of nanodomains of the primitive reo topology, which contains correlated linker and hafnium cluster absences. Absent hafnium clusters are shown in skeletal form. In spite of the differences in framework topology, the cluster coordination environments of fcu (c) and reo (d) UiO-66(Hf) are very similar. 6,23 After ligand elimination (right-hand panels of (c),(d)) the two cluster types are distinct in both coordination and symmetry. Atoms are coloured as follows: Hf, blue; O, red; and C, grey. Atoms lost during thermal elimination are coloured black.

Fig. 2

 2 Fig.2 (a)A diffraction pattern of the most defective sample of UiO-66(Hf). Structure diffuse scattering located at primitive superlattice positions is a signature of the presence of correlated defects in UiO-66(Hf). The most intense of these broad reflections are the (100) and (110). On increasing the quantity of H 2 bdc in the reaction mixture, the concentrations of defects decreases, and so the relative intensity of these primitive reflections decreases ((a) inset and (b)). (c) The TGA trace of UiO-66(Hf) shows three steps corresponding to, in turn, guest volatilisation, ligand elimination and framework decomposition. (d) The decrease in defect concentration is evident in the variation with bdc concentration of normalised mass losses both on ligand elimination (∆m elim ), which decreases, and framework decomposition (∆m decamp ). Fitted curves are included as guides to the eye.

Fig. 3 (

 3 Fig. 3 (a) Evolution of the lattice parameter of the UiO-66(Hf) sample synthesised with the largest quantity of H 2 bdc (5.0 mmol)i.e. the lowest defect concentration -on heating from room temperature to 340 • C then cooling to 100 • C. A linear fit to the lattice parameter over the temperature range 340-160 • C is shown, for the determination of α V . (b) The evolution of the most intense peak, the (111) reflection, throughout this process.

Fig. 4 (

 4 Fig.4(a) The evolution of the lattice parameter through the ligand elimination process for a series of samples with different defect concentrations. To correct for minor variations in the onset temperature of ligand elimination between sample, the curves are offset in temperature such that the maximum change in volume corresponds to 300 • C. (b) The densification (percentage linear decrease) after ligand elimination, measured at 380 • C, highlighted in (a) by a dashed grey line. (c) Coefficients of thermal expansion for isotropic NTE MOFs. The two values given for densified-UiO-66(Hf) correspond to upper and lower defect concentrations.17-19 (d) Variation in NTE behaviour as a function of decreasing defect concentration (increasing quantity of H 2 bdc in the reaction mixture).

  the nanodomain defect structure reported in Ref.6. Instead we focussed on an ordered defect structure corresponding to substitution of four equatorial carboxylate sites by formate on each hafnium cluster; this is the 'ordered-reo' model of Ref. 6. For this model, ligand elimination corresponds to the chemical process Hf 6 O 4 (OH) 4 (bdc) 4 (OOCH) 4 → Hf 6 O 8 (bdc) 4 + 4HCOOH↑. In silico relaxation of structural models for the framework compositions represented on either side of this reaction equation suggested that a 1.0% change in lattice parameters is expected upon HCOOH elimination: a = 20.862 Å for Hf 6 O 4 (OH) 4 (bdc) 4 (OOCH) 4 and 20.652 Å for Hf 6 O 8 (bdc) 4 . ‡

1-8 | 3

|1-8

Acknowledgments M.J.C., J.A.H., and A.L.G. gratefully acknowledge financial support from the ERC (Grant No. 279705) and EPSRC (Grant No. EP/G004528/2). This work was performed using HPC resources from GENCI-IDRIS (grant x2015087069). We thank Diamond Light Source for access to beamline I11 (Proposal Numbers EE9940-1).


Amandine Gamble
email: amandine.gamble@gmail.com.

Romain Garnier

Thierry Chambert

Olivier Gimenez

Thierry Boulinier

Next generation serology: integrating cross-sectional

Keywords: eco-epidemiology, detectability, immunity persistence, sampling strategy, study design, wildlife

come

INTRODUCTION

Understanding the ecology and evolution of infectious diseases in wildlife has been highlighted as critical for public health [START_REF] Jones | Global trends in emerging infectious diseases[END_REF]) and biodiversity conservation [START_REF] Smith | Evidence for the role of infectious disease in species extinction and endangerment[END_REF]. Natural host-parasite systems also offer useful models to obtain valuable insights on evolutionary ecology processes such as coevolution and local adaptation [START_REF] Gandon | Local adaptation and the geometry of host-parasite coevolution[END_REF] or host and vector movements [START_REF] Boulinier | Migration, prospecting, dispersal? What host movement matters for infectious agent circulation?[END_REF]). However, investigations in the wild have been hampered by the difficulty of collecting data allowing efficient inference of eco-epidemiological dynamics [START_REF] Plowright | Sampling to elucidate the dynamics of infections in reservoir hosts[END_REF]. For instance, the force of infection (i.e., the rate at which susceptible individuals acquire an infectious disease), a key ecoepidemiological parameter [START_REF] Hens | Modeling infectious disease parameters based on serological and social contact data: a modern statistical perspective[END_REF], is difficult to estimate from field data as it requires assessing how many individuals went from susceptible (e.g., non-infected and nonimmunized) to infected in a given time period, which is rarely observable. Estimating these parameters is however a critical step in the characterization of epidemiological dynamics and factors impacting them. Methods allowing their estimation from field data are thus needed.

The benefits of longitudinal setups, defined here as the repeated sampling of the same individuals across time, notably using capture-recapture designs, are increasingly acknowledged in the disease ecology community (e.g., Jenelle et al. 2007[START_REF] Lachish | The impact of disease on the survival and population growth rate of the Tasmanian devil[END_REF], Chambert et al. 2012[START_REF] Buzdugan | Inference of the infection status of individuals using longitudinal testing data from cryptic populations: towards a probabilistic approach to diagnosis[END_REF][START_REF] Marescot | Social status mediates the fitness costs of infection with canine distemper virus in Serengeti spotted hyenas[END_REF]). However, cross-sectional data, defined here as the sampling of unmarked individuals at one or more points in time, remain largely over-represented in the literature, probably because of the inherent costs of longitudinal surveys. It requires much more time and skills to spot marked individuals and to recapture them than to capture a random sample of individuals in a target population (e.g., if a marked fur seal is spotted in the middle of a harem, field workers may have to postpone the capture to limit disturbance and biting risks, while in a cross-sectional sampling design, the capture of another, more peripheral, individual would be much easier).

Recent advances in population ecology, such as the advent of integrated modeling, may open new perspectives for the estimation of eco-epidemiological parameters. Indeed, Integrated Population Modelling (IPM) has proven effective to improve demographic parameter estimations by integrating datasets of different natures (e.g., capture-recapture and counts) on the condition that they depend partly on the same set of (demographic) parameters [START_REF] Besbeas | Integrating Mark-Recapture-Recovery and Census Data to Estimate Animal Abundance and Demographic Parameters[END_REF][START_REF] Schaub | Use of Integrated Modeling to Enhance Estimates of Population Dynamics Obtained from Limited Data[END_REF][START_REF] Abadi | An assessment of integrated population models: bias, accuracy, and violation of the assumption of independence[END_REF][START_REF] Fletcher | A practical guide for combining data to model species distributions[END_REF]. In disease ecology, a similar approach could thus be used to integrate low cost cross-sectional data with longitudinal data that provide key elements about processes underlying the dynamics of the considered variables (e.g., the kinetics of the immune response). IPM has been recently applied in an epidemiological context [START_REF] Mcdonald | Demographic buffering and compensatory recruitment promotes the persistence of disease in a wildlife population[END_REF]), but to our knowledge approaches integrating cross-sectional and capture-recapture epidemiological data have never been explicitly used to estimate epidemiological parameters.

In some species, individuals can be marked and repeatedly (re)captured across time, allowing longitudinal sampling. This is particularly true for long-lived vertebrates showing seasonal and colonial breeding (such as seabirds, pinnipeds, and chiropterans) and which are often faithful to their breeding or roosting site (e.g., Chambert et al. 2012b[START_REF] Robardet | Longitudinal survey of two serotine bat (Eptesicus serotinus) maternity colonies exposed to EBLV-1 (European Bat Lyssavirus type 1): Assessment of survival and serological status variations using capturerecapture models[END_REF], Gamble et al. 2019a). In these systems, capture-recapture approaches have started to be used to estimate epidemiological state transition probabilities (e.g., from healthy to symptomatic) while accounting for recapture probabilities below unity, which are unavoidable in wild settings (Jennelle et al. 2007, Conn and[START_REF] Conn | Multistate capture-recapture analysis under imperfect state observation: an application to disease models[END_REF]. However, longitudinal studies are usually based on relatively small sample sizes because field efforts needed to resight and recapture marked individuals tend to be intensive. In contrast, cross-sectional studies are usually less costly and may also allow the estimation of epidemiological state transition probabilities. This type of data can generally be used to monitor variations of prevalences (i.e., the proportion of infected individuals) or seroprevalences (i.e., proportions of seropositive individuals). However, linking variations of prevalences or seroprevalences to epidemiological dynamics often requires additional data seldom available in wild populations, such as knowledge on the infectious period (e.g., [START_REF] Hénaux | Model-based evaluation of highly and low pathogenic avian influenza dynamics in wild birds[END_REF]) and/or refined antibody kinetic curves (e.g., [START_REF] Borremans | Estimating time of infection using prior serological and individual information can greatly improve incidence estimation of human and wildlife infections[END_REF][START_REF] Pepin | Inferring infection hazard in wildlife populations by linking data across individual and population scales[END_REF], or strong assumptions on the host demography (e.g., [START_REF] Samuel | The dynamics of avian influenza in lesser snow geese: implications for annual and migratory infection patterns[END_REF]. Both approaches (longitudinal and crosssectional) thus present relative pros and cons. Because cross-sectional and longitudinal data are outcomes of the same eco-epidemiological processes based on the same demographic and epidemiological parameters (notably survival, force of infection, and antibody level persistence), their combination into an integrated model should improve the estimation of these parameters.

Serology has proven effective to detect patterns of exposure to many infectious agents and infer eco-epidemiological processes [START_REF] Gilbert | Deciphering serology to understand the ecology of infectious diseases in wildlife[END_REF][START_REF] Metcalf | Use of serological surveys to generate key insights into the changing global landscape of infectious disease[END_REF]. Moreover, a wide range of approaches are now available to apply serology to wild settings (e.g., [START_REF] Garnier | Interpreting ELISA analyses from wild animal samples: some recurrent issues and solutions[END_REF]. However, the interpretation of serological data is not straightforward as they do not directly inform on the timing of infection. The reliability of the inference that can be made from serological data is thus dependent on the ecological and epidemiological characteristics of the considered system. Sampling schemes may need to be adjusted to reflect both these characteristics and what is possible in terms of field efforts. For instance, in some hostparasite systems, detectable antibody levels persist for many years after exposure (e.g., antibody level against the Newcastle disease virus vaccine in [START_REF] Ramos | Long antibody persistence and transgenerational transfer of immunity in a long-lived vertebrate[END_REF], while in other cases, they wane within a few weeks (e.g., antibody level against the avian cholera agent in [START_REF] Samuel | Comparison of methods to detect Pasteurella multocida in carrier waterfowl[END_REF], complicating interpretation of serological data. Methods allowing the estimation of the force of infection from serological data when the kinetics of the immune response is not known are needed to better characterize the factors driving epidemiological dynamics.

In the present study, we use a simulation approach to compare the performances of different sampling designs to estimate the seroconversion probability, a proxy of the force of infection, when the kinetics of the immune response after exposure is not known. This parameter can be estimated either from the temporal variations of the seroprevalence based on cross-sectional data (e.g., [START_REF] Samuel | The dynamics of avian influenza in lesser snow geese: implications for annual and migratory infection patterns[END_REF] or as the transition probability from seronegative to seropositive states in a capture-recapture model based on longitudinal data (e.g., [START_REF] Conn | Multistate capture-recapture analysis under imperfect state observation: an application to disease models[END_REF]. We moreover consider the possibility of integrating both sources of data in an integrated framework inspired from IPM. Based on data simulated under different scenarios, we notably account for several key parameters expected to have a strong impact on the observation process and the inference that can be made from serological data: host lifespan, temporal persistence of antibody levels, and detection and recapture probabilities. For instance, low annual survival will increase the turnover of individuals in the host population, which is expected to lower the benefit of longitudinal sampling designs, which rely on the repeated sampling of individuals. Finally, we illustrate how this method could be used on empirical data by considering the case of a serotine bat (Eptesicus serotinus) colony exposed to a rabies virus.

The results of the present study could have important implications regarding current practices in eco-epidemiology by (1) highlighting the benefits of longitudinal sampling designs compared to cross-sectional sampling designs, and (2) opening to possibility of integrating the two types of approaches to design cost-efficient sampling protocols in study

systems not yet subject to longitudinal monitoring programs.

MATERIALS AND METHODS

Eco-epidemiological model

Individual data resulting from an eco-epidemiological inter-annual process were simulated with a set of parameters fixed to different values in order to represent different demographic and epidemiological situations (Fig. 1 a): survival (ϕ), seroconversion (λ; i.e., the probability for a seronegative individual to become seropositive, which usually corresponds to the mounting of an antibody response after exposure to an infectious agent) and seroreversion (ω;

i.e., the probability for a seropositive individual to become seronegative, which corresponds to the waning of the antibody response) probabilities. To illustrate how eco-epidemiological parameters could be quantified from serological data, we have chosen the simple situation of populations at the demographic and endemic equilibria with all individuals recruiting as seronegative and exposure having no impact on survival or detectability. Additional details and illustrations are given in Appendix S1-A.

Cross-sectional sampling

Each year, nCS individuals are randomly captured and sampled for serological analyses.

Seroprevalence at time t (πt) is calculated as the proportion of seropositive individuals among the tested individuals. πt thus corresponds to the probability for a sample randomly collected in a population to be seropositive at time t. Seroprevalences at times t and t+1 are linked by a function of survival, seroreversion, and seroconversion probabilities. Such approaches have previously been used to estimate seroconversion probabilities in wild populations (e.g., Hénaux et al. 2013[START_REF] Samuel | The dynamics of avian influenza in lesser snow geese: implications for annual and migratory infection patterns[END_REF]. Under the eco-epidemiological model assumptions (see above), this relation is given by equation 1:

πt+1 = πt ϕ (1 -ω) + πt ϕ ω λ + (1 -πt) ϕ λ + r λ (1)
In equation 1, the first additive term [πt ϕ (1 -ω)] corresponds to seropositive individuals at time t that survive and maintain detectable antibody levels between time t and t+1; the second [πt ϕ ω λ] to seropositive individuals at time t that survive, lose their antibodies and seroconvert between t and t+1; the third [(1 -πt) ϕ λ] to seronegative individuals at time t that survive and seroconvert between t and t+1; and the last [r λ] to individuals that recruit (here with a probability r) and seroconvert between t and t+1.

Under the assumption of demographic equilibrium, recruitment exactly compensates for mortality and r can be written as (1 -ϕ); and under the assumption of endemic equilibrium, seroprevalence (π*) is stable over time (equation 2; intermediary steps are clarified in Appendix S1, equations S1-3). Serological states of the samples thus follow the binomial distribution given in equation 3.

π* = - λ ϕ (1-ω + ω λ-λ)-1 (2) y ~ B (𝑛 𝐶𝑆 , - λ ϕ (1-ω + ω λ-λ)-1 ) (3)
The estimation of unknown parameters will be facilitated if some of these parameters are known a priori. In this study, we thus notably considered the case when the model was informed with some values for the survival and the seroreversion probabilities (true or erroneous, e.g., based on the literature). Additional details are given in Appendix S1-A.

Longitudinal sampling

On the first year of the observation process, nLG random individuals are captured and marked with a tag allowing individuals to be identified without recapture (e.g., rings or PIT tags). Each of the following years, each alive marked individual is resighted with a probability p and its serological state is ascertained with a probability δ corresponding to the recapture probability after resighting (the serological state being ascertained at the same time from a blood sample). A fixed number of individuals is captured each year, with a priority on marked individuals and some newly marked individuals if necessary to complete the sample size to

nLG. An observation event is then attributed each year to each marked individual of the study and recorded in the matrix m: 0 if not seen (for an individual either dead, alive but not present in the study site, or present but not detected), 1 if captured and ascertained as seronegative, 2

if captured and ascertained as seropositive or 3 if seen but not captured (uncertain serological state; Appendix S1-A). Note that we considered no state misclassification (i.e., test sensitivity and specificity are equal to one). These assumptions are discussed in Appendix S1-A.

Multievent models allowing for state uncertainty (corresponding to event 3) were then fitted on the individual histories [START_REF] Pradel | Multievent: an extension of multistate capture-recapture models to uncertain states[END_REF], similarly to classical applications to demographic studies [START_REF] Gimenez | Estimating demographic parameters using hidden process dynamic models[END_REF]. Such models are increasingly used in population ecology and in eco-epidemiology (e.g., [START_REF] Conn | Multistate capture-recapture analysis under imperfect state observation: an application to disease models[END_REF][START_REF] Robardet | Longitudinal survey of two serotine bat (Eptesicus serotinus) maternity colonies exposed to EBLV-1 (European Bat Lyssavirus type 1): Assessment of survival and serological status variations using capturerecapture models[END_REF][START_REF] Buzdugan | Inference of the infection status of individuals using longitudinal testing data from cryptic populations: towards a probabilistic approach to diagnosis[END_REF][START_REF] Marescot | Social status mediates the fitness costs of infection with canine distemper virus in Serengeti spotted hyenas[END_REF].

Integrated modelling

For a given simulated population, the cross-sectional and the longitudinal datasets (y and m respectively) can be integrated together (Fig. 1 b; [START_REF] Schaub | Use of Integrated Modeling to Enhance Estimates of Population Dynamics Obtained from Limited Data[END_REF]. Under the assumption of independence of the two datasets (only data from unmarked individuals are included in the cross-sectional dataset), the combined likelihood function (LIPM) can thus be expressed as the product of the likelihood function of the cross-sectional (LCS) and longitudinal (LLG) models:

LIPM (y, m | ϕ, λ, ω, p, δ) = LCS (y | ϕ, λ, ω) × LLG (m | ϕ, λ, ω, p, δ) (4)
These parameters can thus conjointly be estimated based on the cross-sectional and longitudinal datasets (y and m). As both datasets result from processes sharing some similar eco-epidemiological parameters, the integrated estimator of these parameters is expected to be less biased and more precise [START_REF] Schaub | Use of Integrated Modeling to Enhance Estimates of Population Dynamics Obtained from Limited Data[END_REF][START_REF] Abadi | An assessment of integrated population models: bias, accuracy, and violation of the assumption of independence[END_REF]). As we considered situations in which a small proportion of the population is sampled (≤ 10 % unmarked individuals and ≤ 10 % marked individuals) and cross-sectional and longitudinal samples were chosen randomly, leading to only a potentially small overlap of the two datasets, we made the assumption that our cross-sectional and longitudinal datasets were independent. In addition to the assumption of independence of the two datasets typical to integrated models, the main assumptions are the ones made by the multievent capture-recapture model (see for instance [START_REF] Riecke | Integrated population models: model assumptions and inference[END_REF]) and when formalizing the temporal variations of the seroprevalence. These assumptions are discussed in more details in Appendix S1-A.

Simulations and model fitting

For each set of parameters, 1000 populations with a size of 600 individuals were simulated

Illustrative example

The integrated estimator was then applied to a real case study of serotine bats (Eptesicus serotinus) exposed to a bat rabies virus (European Bat Lyssavirus type 1; EBLV-1) in Pagnysur-Moselle, France [START_REF] Robardet | Longitudinal survey of two serotine bat (Eptesicus serotinus) maternity colonies exposed to EBLV-1 (European Bat Lyssavirus type 1): Assessment of survival and serological status variations using capturerecapture models[END_REF]). Because we were unable to find a dataset combining cross-sectional and capture-recapture setups in the literature, we chose to use this capture-recapture dataset and to simulate additional cross-sectional data using the simulation model presented above and parameterized based on the demographic and epidemiological parameters estimated using a multievent model. Juvenile serotine bats from the study site are known to be exposed to EBLV-1 [START_REF] Robardet | Longitudinal survey of two serotine bat (Eptesicus serotinus) maternity colonies exposed to EBLV-1 (European Bat Lyssavirus type 1): Assessment of survival and serological status variations using capturerecapture models[END_REF]). Thus, instead of making the assumption that individuals recruit as seronegative (as in equation 1), we made the assumption that females recruit in the breeder pool with the same probability of being seropositive as former breeders (i.e., seroprevalence is similar in the new recruit and former breeder pools), leading to equation 5 in which new recruits and former breeders are not distinguished: πt+1 bats, EBLV-1 = πt bats, EBLV-1 ϕ (1 -ω) + πt bats, EBLV-1 ϕ ω λ + (1 -πt bats, EBLV-1) ϕ λ ( 5)

And seroprevalence at the equilibrium can be written:

π* bats, EBLV-1 = λ (ω -ω λ + λ) (6)
The simulation of the cross-sectional data was also modified to reflect this assumption.

We considered 102 marked individuals captured between one and five times over eight capture occasions (corresponding to the empirical longitudinal data). In parallel, during each of the eight capture occasions, ncs (20, 40 or 60) unmarked individuals were randomly captured and used to calculate the seroprevalence at each occasion (corresponding to the simulated cross-sectional data). We then estimated the survival, seroconversion and seroreversion probabilities using the integrated estimator based on the best model retained in [START_REF] Robardet | Longitudinal survey of two serotine bat (Eptesicus serotinus) maternity colonies exposed to EBLV-1 (European Bat Lyssavirus type 1): Assessment of survival and serological status variations using capturerecapture models[END_REF], in which the resighting probability varies over time: ϕ(.), λ(.), ω(.), p(t), δ(.). Additional details are given in Appendix S1-B and codes in Appendix S3.

RESULTS

Cross-sectional estimator. The seroconversion probability (λ) was estimated without bias (i.e. absolute difference between the true and estimated value close to zero) when using the cross-sectional estimator informed with the true values of the survival (ϕ) and seroreversion (ω) probabilities (Fig. 2 a andb). In contrast, informing the cross-sectional estimator with slightly erroneous values for these parameters led to biases when lifespan and antibody persistence were long (when ϕ tends to one and ω tends to zero). The bias was smaller when lifespan or antibody persistence were short, which can easily be explained by the fact that when ϕ tends to zero and/or ω tends to one, π* tends to λ (equation 2) and the seroconversion probability can thus be directly deducted from the observed seroprevalence. Hence, the crosssectional estimator overall performed better (lower MSE independently of the a priori knowledge) when ϕ was low (i.e., short-lived host species) and/or ω was high (i.e., short-lived immune response; Fig. 2 a and b and sensitivity analyses presented in Fig. S4).

Longitudinal estimator. When using the longitudinal estimator, the seroconversion probability (λ) was estimated without bias without any a priori knowledge of the true survival (ϕ) and seroreversion (ω) probabilities, except when the survival probability was close to zero (Fig. 2 a). In addition to the higher bias, precision was also lower at low survival probabilities.

The lower performances observed for low survival probabilities are expected when using capture-recapture models as fewer individuals can be recaptured over the years, reducing the effective sample size. Precision was also slightly decreased when antibody level persistence was longer (low ω). This could be explained by the model not being able to distinguish individuals that maintained their antibody levels (at a probability 1 -ω) from individuals that were observed seropositive once and then seroreverted and got exposed again (at a probability ω × λ) as both situations fit with the observation of the individuals as seropositive during two consecutive occasions. This is supported by the fact that the precision was lower for higher seroconversion probabilities when the seroreversion was low but not when it was high (Fig. 2 c and d). Hence, the longitudinal estimator overall performed better when ϕ was high (i.e., long-lived host species) and/or ω was high (i.e., short-lived immune response; sensitivity analyses presented in Fig. S4).

Integrated estimator.

Similarly to the longitudinal estimator, the integrated estimator of the seroconversion probability (λ) was unbiased without having to rely on any a priori knowledge on the survival (ϕ) and/or seroreversion (ω) probabilities (Fig. 3). In addition, integrating gives a more accurate estimation than sampling longitudinally 40 marked individuals (Fig. 3 b), while this is not the case for persisting antibody levels (Fig. 3 a). In such comparisons, one need to keep in mind the relative field costs (in time spent and skills required) associated with (re)capturing marked versus unmarked individuals (see Fig. S9 for illustrative examples).

Additional results are presented in the Appendix S1-C, notably considering the effects of various biological (host survival, antibody persistence; Fig. S5) and observations parameters (resighting and recapture probabilities, study duration, sample sizes; Fig. S6-8).

Illustrative example. The estimation of seroconversion probability was improved (smaller confidence interval) when longitudinal and cross-sectional data were integrated together (compared to using only longitudinal data; Table 1). For instance, the seroconversion probability [95% confidence interval] was estimated at 0.085 [0.033; 0.201] using the longitudinal design and 0.079 [0.043; 0.139] using the integrated design including data from 60 unmarked individuals each year. The estimates of survival, resighting and recapture probabilities were unchanged, as expected considering that these parameters were not expected to impact seroprevalence (equation 5).

DISCUSSION

Based on an eco-epidemiological model and simulations under different sampling scenarios, our results suggest that longitudinal data analyzed in capture-recapture frameworks are preferable to cross-sectional data when poor a priori knowledge (for instance on the survival and seroreversion probabilities) is available on the system, which is the case with most wildlife-parasites systems. The cross-sectional estimator can nonetheless be accurate for hosts with short lifespan and/or short antibody level persistence or when informed with reliable a priori knowledge on these parameters. In contrast, the longitudinal approach provided accurate estimates and also allowed survival and seroreversion probabilities to be estimated along with observation parameters (resighting and recapture probabilities; e.g., the serotine bat example). Finally, the integrated estimator benefited from the performances of longitudinal designs, notably it did not rely on any a priori known parameters, even with relatively small longitudinal sample sizes. Based on these results, we hope to encourage researchers to think about the benefits of implementing longitudinal setups, potentially of relatively small scope, in parallel to already existing cross-sectional studies. We also propose a method to integrate these two types of data, which we believe could be useful in the future to motivate researchers to switch from cross-sectional to integrated designs. The method we present here also offers the possibility to integrate datasets that were previously analyzed independently, and thus to improve the inference of eco-epidemiological processes made from these data. For instance, multi-site cross-sectional data could be integrated with single-site longitudinal data (e.g., Picard-Meyer et al. 2011 and[START_REF] Robardet | Longitudinal survey of two serotine bat (Eptesicus serotinus) maternity colonies exposed to EBLV-1 (European Bat Lyssavirus type 1): Assessment of survival and serological status variations using capturerecapture models[END_REF] to overcome the need of a priori knowledge on the host kinetics of the immune response, which is likely conserved within a species sampled across sites.

Although the benefits of longitudinal setups are increasingly acknowledged in the disease ecology community, our study is the first to our knowledge to explore the conditions in which these benefits are actually found. Overall, the results highlight that the key elements to determine an optimal sampling design are: (1) host species life history, (2) the degree of antibody persistence and (3) the degree of a priori knowledge and uncertainty on demographic and epidemiologic parameters. This work also stresses the potential benefits of incorporating data from capture-recapture sampling designs in eco-epidemiological analyses, often largely based on cross-sectional field surveys. In practice, this integrated approach would be particularly beneficial in systems in which (1) individuals can be recaptured over several years (relatively long lifespan and high site faithfulness) and ( 2) large numbers of unmarked individuals can be sampled without increasing too much the cost of the study. This is for instance the case when samples can be collected when accidental capture is frequent (e.g., when using non-targeted capture methods such as mist nets, harp or Sherman traps: e.g., [START_REF] Robardet | Longitudinal survey of two serotine bat (Eptesicus serotinus) maternity colonies exposed to EBLV-1 (European Bat Lyssavirus type 1): Assessment of survival and serological status variations using capturerecapture models[END_REF][START_REF] Mariën | Arenavirus infection correlates with lower survival of its natural rodent host in a long-term capture-mark-recapture study[END_REF], or as part of harvesting practices (e.g., [START_REF] Rossi | Long-term monitoring of classical swine fever in wild boar (Sus scrofa sp.) using serological data[END_REF], or from the offspring of colonial breeders (e.g., Chambert et al. 2012b). In such cases, seroprevalence data from unmarked individuals may be collected with minimal additional effort in parallel to capture-recapture setups. For instance, particularly efficient cross-sectional sampling designs may not even require the capture of adults if the sampling of offspring, or eggs, can be used as a reliable alternative to adult blood sampling [START_REF] Alekseeev | Antibodies to Newcastle Disease Virus in egg yolks of great cormorant (Phalacrocorax carbo) at Qinghai Lake[END_REF][START_REF] Hammouda | Egg sampling as a possible alternative to blood sampling when monitoring the exposure of yellow-legged gulls (Larus michahellis) to avian influenza viruses[END_REF], Gamble et al. 2019b; discussed in Appendix S1-D). Further simulation work could aim at optimazing designs (e.g., sample sizes, sampling frequencies, study duration…) for various scenarios, similar to work performed for occupancy models (Mackenzie andRoyle 2005, Guillera-Arroita and[START_REF] Guillera-Arroita | Designing studies to detect differences in species occupancy: power analysis under imperfect detection[END_REF].

The present study illustrates that setting up a capture-recapture program, potentially in parallel to extensive cross-sectional sampling, to estimate epidemiological parameters may be particularly rewarding in long-lived host species and when specific antibody level persistence is unknown, which is often the case for non-model species (e.g., seabirds, Chambert et al. 2012b; or marine mammals, Chambert et al. 2012a). Conversely, in a species expected to be subjected to high yearly mortality probabilities (e.g., small passerines, [START_REF] Grosbois | Climate impacts on Mediterranean blue tit survival: an investigation across seasons and spatial scales[END_REF]or rodents, Mariën et al. 2018), cross-sectional surveys may be the most efficient way to explore inter-annual processes. Nevertheless, implementing longitudinal, or integrated, setups can still be valuable in short-lived species to study processes occurring at smaller time scales (e.g., monthly; [START_REF] Mariën | Arenavirus infection correlates with lower survival of its natural rodent host in a long-term capture-mark-recapture study[END_REF]. In case of doubt about annual survival and/or the temporal persistence of antibody levels, it is always advisable to implement a capture-recapture program at a time scale adapted to the host species phenology. The inter-annual time scale we considered here may be particularly suited to the long-term monitoring of seasonally breeding species or to investigate the potential impact of diseases on long-lived populations (e.g., [START_REF] Lachish | The impact of disease on the survival and population growth rate of the Tasmanian devil[END_REF][START_REF] Robardet | Longitudinal survey of two serotine bat (Eptesicus serotinus) maternity colonies exposed to EBLV-1 (European Bat Lyssavirus type 1): Assessment of survival and serological status variations using capturerecapture models[END_REF]. In disease systems with strong expected within-and between-year dynamics, the approach would need to incorporate some temporal hierarchy in considered eco-epidemiological parameters and in the corresponding timing of sampling.

Overall, given the relatively realistic situations we considered and the possibility to tailor the approach to more specific cases, the present study could have important implications regarding current practices in eco-epidemiology. For instance, the presented approach could be adapted to consider the time variations of the force of infection to account for epidemic cases or to incorporating parameters to account for a potential disease-induced mortality (discussed in Appendix S1-A). Our study continues to expand the currently proposed framework to improve inference of the circulation of infectious agents in wild populations using serological data (see Appendix S1-E). The sampling design will of course have to be adapted to the main objective of the survey [START_REF] Yoccoz | Monitoring of biological diversity in space and time[END_REF]. For instance, if the main objective of the study is to estimate the seroconversion probability in a long-lived host species, putting important efforts on recapture (to insure a high δ) as part of a longitudinal setting, and integrating additional cross-sectional data could greatly improve the precision of the seroconversion estimators (Figure S6 b, top panel). In contrast, if the main interest is on the survival probability, putting more effort on resighting (independently of recapture) could improve the precision of the estimates [START_REF] Lahoz-Monfort | Exploring the consequences of reducing survey effort for detecting individual and temporal variability in survival[END_REF][START_REF] Lieury | Designing cost-effective capture-recapture surveys for improving the monitoring of survival in bird populations[END_REF]), but integrating cross-sectional data will provide no added benefit (Figure S6 a, middle panel). In any case, as already advocated in other papers [START_REF] Albert | Sampling in ecology and evolution -bridging the gap between theory and practice[END_REF][START_REF] Garnett | Mathematical models in the evaluation of health programmes[END_REF][START_REF] Restif | Model-guided fieldwork: practical guidelines for multidisciplinary research on wildlife ecological and epidemiological dynamics[END_REF]), but still seldom done [START_REF] Herzog | Mathematical models used to inform study design or surveillance systems in infectious diseases: a systematic review[END_REF], we recommend a priori modelling based on available knowledge when designing eco-epidemiological studies, notably to account for host demography, immune response characteristics and sampling costs (Fig. S9).

In addition to the assumption of independence of the datasets, the approach we used relies on the same assumptions as the ones classically made by the chosen capture-recapture and compartmented epidemiological models, and thus the same limitations apply. Notably, it is important to note that, because we used a simulated dataset, the performances of the three presented approaches could have been overestimated. For instance, we did not consider the effect of potential heterogeneities between individuals included in the cross-sectional and longitudinal datasets (e.g., mean age differences or differences in age variances between the marked and unmarked individuals). If they cannot be avoided, these sources of heterogeneities could be accounted for in the modelling process. Finally, considering the recent advances made in quantitative ecology, this approach could be applied to more complex scenarios than the one we considered here, by being combined with methods accounting for state misclassification by repeating sampling [START_REF] Mcclintock | Seeking a second opinion: uncertainty in disease ecology[END_REF][START_REF] Lahoz-Monfort | Statistical approaches to account for false-positive errors in environmental DNA samples[END_REF], using the information contained in quantitative measurements [START_REF] Choquet | Estimating transitions between states using measurements with imperfect detection: application to serological data[END_REF], combining assays such as serology and direct detection [START_REF] Viana | Integrating serological and genetic data to quantify cross-species transmission: brucellosis as a case study[END_REF][START_REF] Buzdugan | Inference of the infection status of individuals using longitudinal testing data from cryptic populations: towards a probabilistic approach to diagnosis[END_REF] or by integrating individual traits more explicitly [START_REF] Plard | IPM2: Towards better understanding and forecasting of population dynamics[END_REF].

  using a specifically developed individual based model (see Appendix S2 for codes). To compare the performances of both designs under various scenarios, one cross-sectional sample and one longitudinal sample of 50 individuals (nCS = nLG) per year were then taken per simulated population following the designs described above. In the case of integrated modelling, several combinations of cross-sectional (nCS = 20, 40 or 60) and longitudinal (nLG = 20 or 40 or 60) sample sizes were tested. Unless otherwise stated, the resighting (p) and recapture (δ) probabilities were set to 0.80 and sampling was conducted over five years after having reached the endemic equilibrium (Fig.S1). Within a time step, samples were collected after exposure. The performances of the estimators were then compared based first on their bias, and second on their Mean Square Error (MSE = bias 2 + variance) in order to account for the bias and the precision of the estimators; the lower the bias or MSE, the more accurate the estimator. In the three cases (cross-sectional, longitudinal and integrated), ecoepidemiological parameters were estimated from the data by maximization of likelihood using a frequentist approach. This method was preferred due to reduced computation time compared to Bayesian inference. Sensitivity analyses were conducted to explore the validity of the results for ranges of biological and observation parameters. All simulations and analyses were run within R 3.3.3. Simulation codes are provided in Appendix S2, including examples of frequentist and Bayesian estimations of the parameters.

  cross-sectional data to longitudinal data increased the precision of the estimator for any fixed longitudinal sample size. For instance, when antibody level persistence was long, adding 20 unmarked individuals to 20 marked individuals at each sampled occasion allowed the standard error of the estimated values to be divided by 1.7. The results are not trivial though: for instance, for intermediate antibody level persistence, sampling longitudinally 20 marked individuals and a novel batch of 20 unmarked individuals at each yearly sampling occasion

FIGURESFIGURE 1 .

 1 FIGURES

FIGURE 2 .

 2 FIGURE 2. The longitudinal design overall leads to no bias but low precision in the estimation

FIGURE 3 .

 3 FIGURE 3. The integrated estimator leads to higher precision in the estimation of the

ACKNOWLEDGEMENTS

We are thankful to Rémi Choquet and Roger Pradel for discussions, to Emmanuelle Robardet, Evelyne Picard-Meyer and Florence Cliquet for the serotine bat data, and to three anonymous reviewer's for their suggestions. This work used computational and storage services associated with the shared clusters provided by CEFE-CNRS and UCLA Institute for Digital Research and Education's Research Technology Group (Hoffman2). This paper is a contribution to the French Polar Institute IPEV programs ECOPATH 1151 and PARASITO-ARCTIQUE 333 and to the ECOPOP observation service of the OREME scientific observatory. AG was supported by a PhD fellowship from French Ministry of Research and the DARPA, project PREEMPT # D18AC00031. The content of the article does not necessarily reflect the position or the policy of the U.S. government, and no official

TABLES TABLE 1. Eco-epidemiological parameters estimated from a bat colony exposed to a rabies virus using the longitudinal or integrated design. The estimates are presented with their 95% confidence interval between brackets. Note that the confidence interval of seroconversion probability (in bold) is smaller when using the integrated design.

Parameter