Sholpan Sumbekova
email: sholpan.sumbekova@legi.grenoble-inp.fr

Alberto Aliseda
email: alberto.aliseda@u.washington.edu

Alain Cartellier
email: alain.cartellier@legi.grenoble-inp.fr

Mickael Bourgoin
email: mickael.bourgoin@ens-lyon.fr

⋅ A Cartellier

Clustering and Settling of Inertial Particles in Turbulence

Particle laden flows are of a great interest in many industrial and natural systems. When the carrier flow is turbulent, a striking feature named preferential concentration takes place: particles denser than the carrier fluid tend to inhomogeneously distribute in space, forming clusters and depleted regions. The present study aims at giving new insight into the possible connection between preferential concentration and particles settling in homogeneous isotropic turbulence. Preferential concentration is quantified using Voronoï diagrams, while 2D Lagrangian Particle Tracking is used to access particles dynamics and in particular statistics of settling velocity. Independent and joint statistics of local concentration and settling velocity are discussed. More paticularly, the possible dependence of particle settling rates on clustering properties are investigated.

Introduction

A striking feature of turbulent flows laden with inertial particles is the so-called preferential concentration (or clustering) that leads to strong inhomogeneities in the particles concentration field at any scale. This has now been widely observed in many experimental and numerical realizations including homogeneous and isotropic turbulence [START_REF] Fessler | Preferential concentration of heavy-particles in a turbulent channel flow[END_REF][START_REF] Salazar | Experimental and numerical investigation of inertial particle clustering in isotropic turbulence[END_REF][START_REF] Squires | Preferential concentration of particles by turbulence[END_REF]. Several properties of preferential concentration seems now to be robustly established with consensual results in the litterateture [START_REF] Monchaux | Analyzing preferential concentration and clustering of inertial particles in turbulence[END_REF][START_REF] Obligado | Preferential concentration of heavy particles in turbulence[END_REF]. We can mention among others: (i) the fractal geometry of clusters, (ii) the fact that the typical clusters scale is of the order of ten times the Kolmogorov scales of the carrier turbulence and (iii) the fact the maximal clustering effects are observed for particles whose Stokes number (defined as the ratio of the particle relaxation time to the Kolmogorov viscous time of the carrier turbulence) is close to unity. However, a clear theoretical description of preferential concentration remains a challenge, in particular regarding the specific mechanism(s) leading to clustering [START_REF] Goto | Sweep-stick mechanism of heavy particle clustering in fluid turbulence[END_REF][START_REF] Maxey | The gravitational settling of aerosol-particles in homogeneous turbulence and random flow-fields[END_REF] and the further impact clustering may have on the collective dynamics (and particularly on the settling rate) of particles [START_REF] Aliseda | Effect of preferential concentration on the settling velocity of heavy particles in homogeneous isotropic turbulence[END_REF]. An important question addressed in the present study concerns the possible enhancement of settling velocity within clusters, as reported by Aliseda et al. [START_REF] Aliseda | Effect of preferential concentration on the settling velocity of heavy particles in homogeneous isotropic turbulence[END_REF], and more generally the connection between local particle concentration and settling properties.

Recently, our group has introduced Voronoï tesselations analysis, obtained from snapshots of the particles in a given area in a flow field, to quantify clustering [START_REF] Monchaux | Preferential concentration of heavy particles: a Voronoi analysis[END_REF][START_REF] Monchaux | Analyzing preferential concentration and clustering of inertial particles in turbulence[END_REF].

Here we propose to take benefit of the capacity of such analysis to be coupled to Lagrangian tracking of particles, in order to investigate the interplay between clustering and settling. Our experimental study gives access to unprecedented statistics of settling velocity conditioned to particles local concentration field, whose first results are presented in this article.

Experimental Setup

Experiments are conducted in a large wind tunnel with a square cross-section of 0.75 m × 0.75 m (see Fig. 1a). Turbulence is generated with an active grid [START_REF] Obligado | Preferential concentration of heavy particles in turbulence[END_REF], capable to produce moderate to high turbulence with still relatively good homogeneity and isotropy levels. The mean stream velocity U of the wind was varied from 2.5 to 10 m/s. The active generates velocity fluctuations with a turbulence intensity (defined as the ratio between the standard deviation of velocity 𝜎 U to the mean stream velocity U) of the order of 20 %. The corresponding of Reynolds number based on Taylor microscale, Re 𝜆 = 𝜎 U 𝜆∕𝜈 ranges between 200 and 500 (𝜆 = √ 15∕𝜈 𝜖 𝜎 U is the Taylor microscale, with 𝜈 =1.5e -5 m 2 s -1 the kinematic viscosity of air at working temperature and 𝜖 the turbulent energy dissipation rate per unit mass). Figure 1b represents a typical velocity spectrum of the turbulence produced by the active grid. Inertial particles are water droplets generated by 18 injectors located in a transverse plane 15 cm downstream the grid. The average diameter of the particles is constant for all velocities, with a typical mean sauter diameter D p ≃60µm (although size distribution has a significant polydispersity of the order of 50 %). We define the typical Stokes number of the particles as St = 1 36

( D p 𝜂 ) 2
(1 + 2𝜌 p ∕𝜌 f ),with𝜂 the Kolmogorov dissipative scale of the carrier turbulence and 𝜌 p ∕𝜌 f the particle to fluid density ratio (which in our case, water particles in air, is of the order 815). In our experiments, the size of the particles is kept constant, and variations of Stokes numbers are explored by varying the dissipation scale 𝜂 of the flow, which decreases as the wind velocity increases. Stokes and Reynolds number are therefore linked parameters, which do not vary independently. Further experiments will be carried in near future to disentangle more precisely Stokes and Reynolds number effects. Note also that in our experiments the flow rate of water through the atomizing injectors producing the droplets is kept constant. This means that the particle volume fraction given by

Q L Q L +Q G
, where Q L is an injected flow rate and Q G is a main air flow rate within the flow decreases linearly as the wind speed increases. Finally, a longitudinal laser sheet (with millimetric width) is used to illuminate the particles (as shown in Fig. 1). Images were recorded with one high speed camera Phantom V12 from Vision Research Inc., with a field of view of the order of the integral scale of the carrier flow of 12 cm.

Results

Voronoï Tesselations

Voronoï analysis is used for quantifying the phenomenon of preferential concentration [START_REF] Monchaux | Preferential concentration of heavy particles: a Voronoi analysis[END_REF][START_REF] Monchaux | Analyzing preferential concentration and clustering of inertial particles in turbulence[END_REF]. Voronoï diagrams give a tesselation of 2D space (though the analysis can be extended to any spatial dimension) where each cell of the tesselation is assigned to a particle, with all points of one cell closer to its associated particle than to any other particle. Figure 2 shows the tesselation of one instantaneous recorded image. Each visible cell (we shall refer to cells as Voronoï cells) contains one unique particle. Consequently, the inverse area of each Voronoï cell represents the local concentration particles. The higher the local concentration of the particle, the smaller its corresponding Voronoï area A . Figure 3 shows the probability density function (PDF) of normalized Voronoï cell areas V = A ∕A (where ⋅ represents the ensemble average over the full set of images for each experiment) for the several experiments we carried at different wind velocities (hence different Stokes and Reynolds numbers). The deviation of the experimental PDF compared to the PDF expected for a pure random Poisson process (RPP), represented by the solid line in the figure, is representative of the existence of preferential concentration, as small areas (dense regions) and large areas (depleted regions) are over-represented in the experiments with respect to the RPP case. A detailed description of trends of clustering with Stokes and Reynolds number in our experiment goes beyond the scope of the present article, but can be found in [START_REF] Obligado | Preferential concentration of heavy particles in turbulence[END_REF]. We can just emphasize here that, in the range of parameters explored here, the level of clustering is significant (this can be quantified by the standard deviation of the PDFs shown in Fig. 3, which is of the order 𝜎 V ≃1.2, while the corresponding value for a RPP is 𝜎 RPP V ≃0.53) with a weak dependency on the explored parameters, although small areas (over-concentrated regions) are slightly over-represented for the largest Stokes numbers explored and large areas (under-concentrated regions) are for the lowest Stokes numbers explored (note however that this effect appears amplified by the log-log representation of the figure, but contributes marginally to the change of 𝜎 V , which remains in the range 𝜎 V ∈[1.1, 1.25] for the full set of experiments).

We use now the comparison between the measured PDFs and the RPP PDF to determine clusters and voids, following the definition proposed by Monchaux et al. [START_REF] Monchaux | Preferential concentration of heavy particles: a Voronoi analysis[END_REF]: clusters and voids are composed, respectively, by Voronoï cells feeding the small and large area tails of the PDF, corresponding to over-represented events with respect to the RPP case. Intersections of the measured PDFs and the RPP PDF near the maximum therefore define two thresholds V c and V v (with V c < V v ) such that cells with normalized area below V c belong to clusters and cells with normalized area above V v belong to voids. In our experiments, we found that these thresholds do not depend significantly on the experimental parameters (Stokes number or Reynolds number) and we have V c ≃0.6 and V v ≃2.1 (see Fig. 3).

Joint Analysis of Concentration and Settling Velocity

Particle Tracking Velocimetry was used to reconstruct particle trajectories and measure the instantaneous velocity of each individual particle. In the present case, the use of one single camera and a laser sheet illumination allows to access 2 components of particles velocity in the illumination plane. The key point of the present study is that besides this information on particles velocity, the above mentioned Voronoï analysis allows to attribute to each individual particle an instantaneous Voronoï area (representing the instantaneous local particle concentration in the vicinity of each particle). We can therefore consider accessing joint statistics of particles local concentration and velocity.

Figure 4 shows the dependence of the average settling velocity V z on Stokes number (or equivalently on Reynolds number) for three types of particles in each experiment: (i) V z,all refers to the average settling velocity of the full set of particles, regardless of the value of their associated Voronoï area (ii) V z,cluster refers to the average settling velocity of particles belonging to clusters only (i.e. we condition Fig. 4 Mean settling velocity calculated (i) for the full set of particles for each experiment (V z,all ), (ii) for particles belonging to clusters only (V z,clusters )and (iii) for particles belonging to voids only (V z,voids ) the average of settling velocity by keeping only particles with Voronoï area V < V c ) and (iii) V z,voids refers to the settling velocity of particles belonging to voids only (i.e. we condition the average of settling velocity by keeping only particles with Voronoï area V < V c ). Several points are worth being highlighted and discussed from this plot:

1. All curves follow a similar trend, with globally a reduction of settling velocity with increasing Stokes number (or equivalently in our case with increasing Reynolds number). For the largest Stokes numbers explored we may observe a saturation, or even a decrease (except for particles in voids). Since the particle diameter is kept constant, this effect is purely due to the interaction between the particles and the carrier turbulence. Further studies are required to draw robust conclusions on these trends. In particular, when turbulence and inertia contribute to enhance or reduce settling is a subtle matter, as shown by a recent study focusing on the case of individual particles (in very dilute conditions) in turbulence [START_REF] Good | Settling regimes of inertial particles in isotropic turbulence[END_REF].

A systematic study, where seeding concentration, Stokes number and Reynolds number are varied independently is a requisite to identify robust generic trends. This goes beyond the scope of the present article, but will be addressed in coming studies. 2. The most striking feature on the curves in Fig. 4 is the systematic difference between the settling rate of particles in clusters compared to particles in voids. Clustered particles settle systematically faster than particles in voids. Besides, this difference increases with increasing Stokes number (or increasing Reynolds number). This observation is consistent with measurements reported by Aliseda et al. [START_REF] Aliseda | Effect of preferential concentration on the settling velocity of heavy particles in homogeneous isotropic turbulence[END_REF]. A natural question now concerns the possible origin of this effect. Two main mechanisms can be considered: (i) a signature of preferential sampling of the turbulent flow by the particles, in the spirit of the preferential sweeping mechanism, where clusterized particles would for instance explore preferentially high strain downward regions [START_REF] Maxey | The gravitational settling of aerosol-particles in homogeneous turbulence and random flow-fields[END_REF] or (ii) a role of collective effects, where the collective settling of clusters would enhance the individual settling of particles [START_REF] Aliseda | Effect of preferential concentration on the settling velocity of heavy particles in homogeneous isotropic turbulence[END_REF]. In our experiment, we observe that the level of clustering (which can be quantified for instance by the standard deviation of the normalized Voronoï areas) does not vary by more than 10 % over the range of Stokes number (or Reynolds number) explored here. We therefore do not expect a significant difference in the way particles preferentially sample the turbulence and believe that the observed trends are mostly attributable to collective effects. A possible dominant role of collective effects is further supported by the fact that the particle volume fraction is reduced by almost a factor 4 between the experiments at lowest wind-speed (i.e. lowest Stokes number) and those at higher wind-speed (i.e. higher Stokes number)), that impacts the overall concentration, but also the concentration within clusters and voids, and hence is expected to modify the collective settling behavior [START_REF] Aliseda | Effect of preferential concentration on the settling velocity of heavy particles in homogeneous isotropic turbulence[END_REF]. 3. Even if this may seem obvious, it is still worth pointing that the fact that the global mean settling velocity follows more closely the settling velocity of clustered particles (rather than particles in voids) is expected as, by essence, particles in voids are less numerous and therefore contribute less to the global mean.

Conclusion

By combining Voronoï analysis of the concentration field of inertial particles in turbulence and Lagrangian particle tracking, we have shown that in presence of preferential concentration, particles within clusters settle systematically faster than particles in voids, what we tend to attribute to the additional influence of collective settling of clusters. Further studies are now required to support this interpretation and to understand other observed trends (as the diminution of settling rate with increasing Stokes number). In particular, carrying experiments where Stokes number, Reynolds number and seeding concentration are varied independently is of primary importance. Such experiments are in progress at the moment.

Fig. 1 a

 1 Fig. 1 a Sketch of the experimental setup. b Typical velocity spectrum of turbulence produced by the active grid (red solid line represents a k -5∕3 trend)

Fig. 2

 2 Fig. 2 Example of Voronoï tesselation. Each Voronoï cell contains a unique particle

Acknowledgments This work has been supported by the LabEx Tec 21 (Investissements dAvenir -grant agreement ANR-11-LABX-0030), the French Agence Nationale pour la Recherche (project ANR-12-BS09-011-03) and the COST action on "Flowing Matter" MP1305.


Lionel Alletto

Aline Vandewalle

Patrice Mahieu

Olivier Micos

Léo Thorand

Co-design and multicriteria assessment of low-input cropping systems in the South-West of France: a 8-years experimentation with farmers

Introduction

Food production in the last decades has been greatly increased thanks to intensification of agriculture but this intensification also led to numerous undesirable environmental impacts such as biodiversity loss, soil degradation, and soil, water, air, and food contamination by pesticides [START_REF] Foley | Global consequences of land use[END_REF][START_REF] Mea | Millennium Ecosystem Assessment Synthesis Report[END_REF]. Moreover, in many cases, economical performances of intensive cropping systems (CS) decreased that weakens the sustainability of farms. In the South-West of France, two main traditional CS could be identified: a rainfed durum wheat-sunflower CS and a maize monoculture CS (mainly irrigated). The main objective of this study was to evaluate the sustainability of CS designed as alternatives of these two traditional CS that aimed at reducing the negative externalities of agriculture while answering specific objectives of the involved farmers.

SF  SW  SF DW  SG  SB  SW  SF  FB  SW  SF

Materials and methods

Workshops, mixing farmers, advisers and scientists, were organized in 2009 and 2010 in order to (i) identify strengths and weaknesses of initial CS; (ii) design alternative CS (based on formalized decision rules) in order to answer to common and specific objectives; (iii) define/choose the indicators to monitor the CS performances and, after an ex-ante assessment (iv) experiment and perform an ex-post evaluation of the designed CS for each involved farm [START_REF] Papy | Interdépendance des systèmes de culture dans l'exploitation[END_REF][START_REF] Debaeke | Iterative design and evaluation of rule-based cropping systems: methodology and case studies. A review[END_REF]. A 8-farms network was involved in this 8-years study (2010-2017) (Table 1). The main indicators chosen to monitor performances are mentioned in Table 2. Furthermore, in order to test the robustness of cropping systems, 8 crop price scenarii, chosen over the period 2007-2014, were applied and a comparison between the economic performances of the initial CS and the alternative CS was made. Crop diversification was found to enhance CS economical performances if associated with rigorous control of the mechanization loads (Fig. 1a). In most cases, diversification led to a better distribution of workload during the year allowing the development of high added value production (such a vegetables with direct selling).

Results and discussion

Selected indicators illustrated that diversification could lead to lower environmental impacts (Table 3), except if seed production (with high pesticide use) was integrated in the CS. For the project DiverIMPACTS, further investigations are performed with economical stakeholders to increase economical performances of diversified CS. In parallel, a multicriteria assessment (using the indicator list of DiverIMPACTS) of the initial (mainly maize monoculture) and diversified CS of the 20 farms involved in the Case Study 5 has started and should strengthen conclusions on the effects of crop diversification on the sustainability of South-West of France farms.

Fig. 1 .

 1 Fig. 1. Semi net margin (a) and robustness toward crop price fluctuations (b) of initial and alternative CS.

Table 1 . Main characteristics of the 8 farms involved in the study.

 1

	B: barley; CS: carrot seed; DW: durum wheat; FB: Fababean;

 SW  CS  SW  RS  SW Farm 6 (31) Calcareous clay on hillsides DW  SF DW  SF  SW  SG Farm 7 (46) Calcareous clay on hillsides RS  SW  B RS  SW  SF  B Farm 8 (81) Calcareous clay on hillsides DW 

Table 2 . List of selected indicators for cropping system performance assessment.

 2

	Dimension	Criteria	Indicator
	Economic	Profitability	Semi net margin (SNM) at rotation level (€/ha)
	Economic	Weed management	Weed abundance in the field
	Economic	Dependency on external inputs	Input use efficiency
	Economic	Productivity	Energetic yield (MJ)
	Environmental	Water quality (pesticide)	Treatment Frequency Index (TFI)
	Environmental	Water quality (pesticide)	I-phy
	Environmental	Water quality (pesticide)	Number of toxic pesticides for aquatic systems
	Environmental	Water quality (nitrate)	Nitrogen indicator (NI)
	Environmental	Water quantity	Irrigation amount (m 3 /ha)
	Environmental	Soil quality	Organic matter indicator (IMO)
	Environmental	Soil quality	Length of bare soil period (%)
	Environmental	Fossil energy	Total fuel consumption at CS level (MJ/ha)
	Environmental	Climate change mitigation	GHG emission (equivalent t CO2/ha)
	Social	Farmers' quality of life	Labor time (h/ha)
	Social	Farmers' quality of life	Workload distribution (h/ha/month)
	Social	Famer and public health	Number of pesticide applications

Table 3 . Multicriteria assessment of alternative CS

 3 (selected indicators from the list of Table2)

	(a)	1130				1105						(b)
		1000		918					Initial CS Alternative CS	2300	Farm 1	Farm 2
														Farm 3	Farm 4
														Farm 5	Farm 6
	SNM (€/ha)	500	740	628	570 632	632	641	578 714	365		400 381	800 1300 Alternative CS 1800 (€/ha)	Farm 7	Farm 8
						218			308			SNM
													300
													-200 -200	300	800	1300	1800	2300
			Farm 1	Farm 2	Farm 3	Farm 4	Farm 5	Farm 6	Farm 7	Farm 8		SNM Initial CS (€/ha)
						SNM	TFI	NI		GHG	Labor time	Farmer satisfaction
					Farm 1	-35%	-39 %	+43%	-21 %		-67 %	SNM
					Farm 2	+46%	-62%	0%		0%			-72%	TFI; SNM; Labor time + Conservation agriculture
					Farm 3	-10%	-16%	0%		-17%		-37%	Labor time + diversification in vegetable production
					Farm 4	+189% +25% +156%	-62%		-45%	Labor time + Conservation agriculture; better weed management
					Farm 5	+72%	+71%	+15%	-17%		0%	TFI (seed contract); SNM
					Farm 6	-20%	-48%	+32%	+24%		+25%	Better weed management
					Farm 7	-15%	+10%	+24%	-30%		0%	Better weed management; low SNM
					Farm 8	+5%	-43%	+47%	-43%		+10%	Labor time; Better weed management


Antoine Couëdel

Lionel Alletto

Eric Justes

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

Rape

N AND S CYCLES IN CRUCIFER-LEGUME COVER CROP MIXTURES

Antoine COUËDEL 1,* -Lionel ALLETTO 1,2 -Éric JUSTES

  financially supported by the French Ministry of Agriculture (CASDAR project n° C-2013-05) and by the Occitanie Region (CLE project n°13053068)

Conclusion

  Our study confirms for a wide range of cover crop species and on four sites x years that crucifer-legume mixtures can provide multiecosystem services by mutualising services of N and S cycles. Couëdel, A., Alletto, L., Justes, É., 2018a. Plant Soil. doi:10.1007/s11104-018-3615-87 Couëdel, A., Alletto, L., Tribouillois, H., Justes, E., 2018b. Agric. Ecosyst. Environ. 254, 50-59. doi:10.1016/j.agee.2017.11.017 *The service is provided in proportion of the best sole crop family (average of all species tested), which then is considered at 100%. **Higher than the average of sole cover crops

bispecific mixtures 4 field experiments Nitrate Catch crop Sulphate catch crop N green manure S green manure Crucifers sole cover crops

  1,3 1 AGIR, Université de Toulouse, INRA, INPT, INP-EI PURPAN, Castanet-Tolosan, France; 2 Chambre Régionale d'Agriculture Occitanie, 31321 Castanet-Tolosan, France; 3 CIRAD, UMR SYSTEM, Univ. Montpellier, CIHEAM-IAMM, INRA, Montpellier SupAgro, 34980 Montpellier, France. Is their a mutualisation of services related to N and S cycles in crucifer and legume cover crop mixtures ?

				Strong	Strong	Weak	Strong
	Legumes sole cover crops		Weak	Weak	Strong	Weak
	Mixtures		?	?	?	?
	Crucifers	100%*	100%	18%	100%
	Legumes	66%	30%	100%	23%
	Mixtures	98%**	99%**	63%**	85%**

*corresponding author: antoine.couedel@inra.fr Crop (n) Crop (n +1) Multiservices Cover crop 98 Question :

Catch crop services during the growing period Nutrient recycling services after cover crop termination Background Material and methods Results

	Egyptian clover
	Crimson clover
	Purple vetch
	Common vetch
	Hairy vetch
	Soya bean
	Faba bean
	Pea
	Lupin


Isaure Chauvot De Beauchene

Sjoerd De Vries

Martin Zacharias
email: zacharias@tum.de

Modelling ssRNA-protein complexes at atomic resolution

Keywords: RNA-protein docking, fragment-based docking, ab initio modeling, singlestranded RNA, RNA-protein recognition Amarrage ARN-protéine, amarrage par fragment, modélisation ab initio, ARN simple brin, reconaissance ARN-protéine

RNA-protein specific binding underlies a large variety of fundamental cellular processes. An atomistic description of such binding processes would aid the rational conception of pharmaceutical modulators of those functions. However, while the field of protein-protein docking has achieved considerable improvements in the last decade, protein-RNA docking encounters specific difficulties. This is mainly due to the flexibility and the large conformational space of RNAs compared to proteins, and especially single-stranded RNAs (ssRNAs). Here, we present a novel and highly effective fragment-based approach to tackle this problem, capable of accurate prediction of the structure of a ssRNA bound to a protein, starting from the structure of the protein and the sequence of the RNA. As a proof-of-principle, we focus on the common case of a uniform ssRNA sequence. Without any information on specific contacts or the RNA structure, our method permitted to define accurately the binding site on the protein with 10 Å precision, through the use of a comprehensive fragment library. Moreover, the bound conformation of the ssRNA could be sampled with ~1.5 Å RMSD on heavy atoms, a precision never reached so far. In future research, the method will be extended to dock arbitrary ssRNA sequences to protein structures.

Introduction

RNA-protein specific binding supports a large variety of fundamental cellular processes, from initiation/repression of gene transcription to inter-cellular communication [1 2 3]. However, the number of atomic structures of protein-RNA complexes remains quite low, therefore structural prediction methods are needed. Docking methods aim to assemble a complex from atomic structures of the free, unbound structures of the components (protein or RNA). The first task in docking is to sufficiently sample the space of possible conformations and relative orientations (i.e. poses) of the unbound components so as to include near-native structures. In this regard, RNA-protein docking encounters specific limits compared to protein-protein docking, due to the very high flexibility and conformational variety of RNA.

The conformational changes occurring upon RNA association with protein can involve global rearrangements, changes of secondary structure elements and/or flipping-out of bases from intra-to extrahelical position. Therefore, unbound RNA structures are a reasonable starting point only in a limited amount of cases. They are especially unreliable when some single-stranded loops participate in the binding. Due to the high flexibility of single-stranded RNA (ssRNA) regions, an experimental structure of the unbound form is very unlikely to exist, and even so would not provide enough data to infer the bound form. Even RNA molecules that are otherwise well-structured contain such single-stranded regions that carry the specificity in most specific RNA-protein recognition processes. Still, essentially all studies in the field have concentrated so far in docking unbound structures of RNAs with limited conformational sampling. They classically sample possible RNA conformations based on the unbound structure by use of coarse-grained models [START_REF] Setny | A coarse-grained force field for Protein-RNA docking[END_REF], normal modes analysis, elastic network models [5 6] or local conformational perturbations [P. Setny and M. Zacharias, in prep.] , then perform rigid [7 8] or nearly-rigid docking [START_REF] Mashiach | An integrated suite of fast docking algorithms[END_REF]. Such methods can perform well when only moderate conformational changes occur. The lack of methodology for modeling larger changes, in particular bound single-stranded loops, limits the accuracy of all current protein-RNA docking methods [START_REF] Fulle | Molecular recognition of RNA: challenges for modelling interactions and plasticity[END_REF].

Here we present a novel strategy for ab initio modeling of ssRNA bound to a protein that does not require any structural information on the ssRNA, assembling it from sequence alone. Our approach consists of cutting the RNA into small overlapping fragments, docking them separately, and assembling the compatible docking poses into a new realistic conformation (Fig. 1). Our strategy uses a large, home-made RNA fragment library to sample the conformational diversity of RNA, and relies on a coarse-grained force field to select effectively the most probable poses. With this strategy, a structure of poly-U ssRNA in complex with a protein was generated in close agreement with the published crystal structure (< 5 Å RMSD). To the best of our knowledge, this is the first time that a ssRNA has been successfully predicted at such high precision without use of any prior knowledge on the RNA structure.

Results

a. Bound docking on a test-case

In order to get a first idea of the validity of our fragment-based approach and an upper-limit of its performance, we performed docking tests on one case using not the fragment library but the bound form of the RNA. We chose the PDB structure 1B7F representing the sex-lethal protein bound to a 5'-UUUUUUUU single-stranded RNA, and which constitute a canonical case of a RRM-containing RNA-protein complex [10 11]. The RNA is bound to a deep cleft delimited by RNA-binding domains. Most nucleotides bind by their base and/or sugar, and establish 1 to 5 hydrogen-bonds with the protein (Fig. 2). We cut the RNA into trinucleotides that overlap by two nucleotides. Each trinucleotide fragment was docked onto the bound structure of the protein, both partners in coarse grained (CG) representation, using our docking program ATTRACT [9 10] , and the poses were assembled using distance restraints between overlapping nucleotides. . This test resulted in a quasi-native RNA chain (1.2 Å RMSD toward the bound form) (Table 1). This result was comparable to rigid bound-docking with the entire RNA in its bound form (0.7 Å RMSD , result not shown).

Bound docking Biased docking Unbiased docking

Fragment Best RMSD Nb poses < 5 Å n5-n7 1.7 Å 1.8 Å 3.3 Å 5 n6-n8 1.7 Å 1.8 Å 4.4 Å 7 n7-n9 0.5 Å 1.1 Å 4.4 Å 72 n8-n10 0.9 Å 1.4 Å 3.5 Å 255 n9-n11 0.7 Å 0.9 Å 3.5 Å 165 n10-n12 1.4 Å 2.7 Å 6.1 Å 0 Average / total 1.2 Å 1.6 Å 4.3 Å 496 ( 6% )
Table 1. Comparison to the bound form of the poses obtained by bound, biased or unbiased docking.

b. Biased docking using a fragment library

We built a fragment library sampling the conformational space for each of the 64 possible trinucleotide sequences, based on the available ssRNA-protein structures in the Protein Data Bank (to be published). The test-case 1B7F was not included in the library building process. In order to get a first evaluation of the capacity of our library to sample efficiently near-native solutions, with a reduced computational cost, we performed biased docking of a poly-U octo-nucleotide (noted n5-n12) on the sex-lethal protein, corresponding to complex 1B7F in our benchmark. After excluding from our library the fragments issued of this complex, we selected for each bound fragment the best fitting conformer in the UUU sub-library, ending up with six UUU conformers.

Quasi-native solutions (RMSD < 2.1 Å) were sampled for each fragment, among ~15,000 non-redundant poses per conformer . By selecting the 20 % best-scored solutions, we retained near-native solutions (RMSD < 3 Å) for each fragment (Table 1). The worst-docked fragment corresponded to nucleotides n10-n12, with the best solution in top 20% at 2.7 Å RMSD. The structure of this fragment corresponds to a conformation not closely sampled in the library. The best conformer for this fragment displays 1.8 Å RMSD when fitted to the bound form, versus 0.4 Å to 1.1 Å for the other fragments (results not shown). These results indicate that ATTRACT was able to sample and rank solutions very close to the optimal position of each conformer in the top 20 %.

We further tried to assembly the overlapping fragments into chains, by evaluation of the inter-atomic distances of the overlapping nucleotides (see Methods ). We retained the top 20 % poses for each of the six conformers, for a total of 18,222 poses. Out of the 18222 x 18222 possible pairings, 23,458 were found to possibly represent consecutive overlapping fragments. These pairs were arranged into 5-fragment chains by identifying pairs A-B, B-C, C-D, and D-E. The 75,311 resulting chains n5-n11 were filtered by total overlap energy, leaving 3,174 chains. The best chain had an average RMSD of 1.4 Å (Fig. 3). More importantly, this RMSD was representative for the whole result : 41 % of the chains were under 2.0 Å RMSD, and 97 % under 5.0 Å. In conclusion, the correct structure was essentially the only possible way to build a poly-U hepta-nucleotide onto the protein, when docking the proper conformers.

d. Unbiased docking with the complete fragment library

A similar procedure was applied considering not only the best conformers but the whole UUU library (1305 conformers) for each fragment in n5-n12. The docking produced more than 22 millions of non-redundant poses, from which we retained the top 20%. Out of this large pool, the fragments with the highest propensity to form complete hepta-nucleotide chains were selected. This resulted in 8,441 chain-forming fragments, corresponding to 7,798 poses (one pose can correspond to different fragments, by shifting its position in the chain). Some correct poses were found toward all bound fragment except n10-12, similarly to what was found by the biased six-conformer docking. In total, 6 % of the poses were correct (RMSD < 5 Å from at least one bound fragment) (Table 1).

In addition, these poses permitted to define accurately the binding site (Fig. 4). The worse pose was at only 15.4 Å from the closest fragment, and more than 50 % of the poses were under 10 Å. Moreover, the poses outside the binding site scored very poorly : the 8 poses at more than 15 Å from any fragment were ranked in the last 10 %. Therefore, our procedure for fragment assembly proved an efficient filter of wrong solutions.

Discussion and Perspectives

Here we present a method to predict the structure of ssRNA-protein complexes, based on the structure of the protein and the sequence of the RNA, using a fragment-based approach. For the first time, \ the structure of such a complex could be predicted at high precision.

First, we validated our fragment-based approach by preliminary tests of bound-bound ssRNA-protein docking. This led to a precision of ~1A RMSD towards to the crystal structure, comparable to the control run where the bound ssRNA was docked in its entirety. Therefore, the process of cutting the RNA into fragments does not lead to any loss in accuracy or precision. Subsequently, we performed "ab initio" RNA-protein docking, using the structure of the protein and the sequence of the RNA. The structures used for docking were obtained by extracting an exhaustive library of trinucleotide fragments out of the 571 available structures of ssRNA-protein complexes (PDB, July 2014). By a biased fragment docking, selecting the library conformers closest to the RNA structure, we generated accurate near-native structures at ~ 1.5 Å RMSD for an hexa-nucleotide, providing a proof-of-principle for ab initio fragment-based docking of ssRNA on protein. Using unbiased docking with the full library, we were able to select a pool of fragments that delineate accurately the binding site.

Our results constitute a considerable improvement compared to the limited success that had been achieved so far in docking ssRNA-protein complexes. Almost all current methods are limited to structured RNA. Only the RNA-lim method [START_REF] Hall | RNA-LIM: a novel procedure for analyzing protein/single-stranded RNA propensity data with concomitant estimation of interface structure[END_REF] has attempted to predict ssRNA-protein structures based on fragments, focusing the sampling around a pre-defined binding site. They predicted the position (but not the orientation) of RNA fragments at around 5 Å from the binding site. In contrast, our method achieves this precision for both position and orientation for unbiased docking , without the use of any knowledge on RNA conformation or binding site. For biased docking, our method predicts the RNA structure at high precision. We emphasize that the only difference between biased and unbiased docking is that in biased docking, the fragment library was limited to relevant conformations. All structures generated in biased docking were also generated in unbiased docking, albeit among millions of others.

In this study, we have focused on sampling, i.e. the generation of correct ssRNA-protein conformations. Although the fragment approach proved an efficient sampling strategy, the ranking of the best structures among decoys (scoring) remains an issue. As our test sequence is only made of UUU fragments, the 8,441 chain-forming fragments selected by our unbiased docking could theoretically form 4 x 10 19 chains for n5-n10. To reduce the number of chains to select for further refinement and scoring, we plan to use insights of specific RNA-protein contacts from conserved RNA-binding motifs in proteins.

Our method currently focuses on the common case of a binding ssRNA that is uniform in sequence. For this case, the current study provides in important proof-of-principle for RNA-protein modeling. The method is in principle extendable to arbitrary sequence, and this will be the direction of further research. If this method proves generalizable to a larger number of cases and sequences, it will constitute a major methodological breakthrough.

Methods a. Docking of the RNA fragments.

Both the protein and the RNA fragment were in coarse grain representation. The pyrimidine/purine were represented by 6/7 beads and the amino-acids by 5 to 8 beads [ 4 15] . Throughout this paper, the results are given in coarse-grain RMSD if not otherwise mentioned. A comparison tests on 21,704 RNA fragments showed a 10 % decrease in RMSD numbers when converting from CG to all-atoms for RMSD < 5 Å , with a regression coefficient of 0.90.

For each docked conformer, the starting positions of the ligand and receptors were produced by the "randsearch" procedure of ATTRACT [START_REF] De Vries | A Web Interface for Easy Flexible Protein-Protein Docking with ATTRACT[END_REF], generating starting positions and respective orientation of the two partners (protein and fragment). The positions of the center of mass (COM) of the fragment at each starting configuration are equidistant on a unit sphere of 75 Å radius centered on the protein. A gravity was applied, consisting in a harmonic distance restraint toward the COM of the protein with harmonic constant of 0.0015 kcal/Mol/Å. For each fragment were performed 1000 minimization steps, the pairwise interactions between ligand and receptor being approximated on a pre-calculated receptor grid. A final re-scoring was performed without grid, pairwise interactions being considered until a squared distance of 50 Å. The final poses were sorted by score, and the redundant poses (within 0.05 Å from a better scored pose) were discarded. We first docked each of the 6 best-fitting UUU conformers starting from 30 000 initial positions, producing ~ 15,000 non-redundant poses. Then each conformers of the whole UUU sub-library was docked starting from 200,000 position to account for the increased difficulty.

b. Overlap evaluation.

The overlap of two fragments was evaluated using ATTRACT scoring function with harmonic distance restraints and no force-field. The restraints were defined between the 2 nd and 3 rd nucleotides of the 1 rst fragment and the 1 rst and 2 nd nucleotides of the 2 nd fragment. The restraints were defined such that the two backbone beads and the most remote base bead must occupy the same position, with some margin. The margin was defined with smaller values for the backbone than for the base (2.3 Å and 2.8 Å respectively), to account for the necessity to further link the backbone atoms in the refinement procedure.

c. Sequential clustering.

For the unbiased docking, the 20% best-scored poses for each conformer were selected and grouped, then clustered by 2 Å. The centers of the 2Å-clusters were clustered into 3Å-clusters, and the centers of the 3Åclusters into 4Å-clusters. The overlap between the center of mass of the central structure of the 4Å-clusters was evaluated, and all pairs of clusters with low overlap-energy were kept. The same procedure was applied inside each pair of overlapping 4Å-clusters at the 3Å-clusters level, with a lower overlap margin. Each 3Åcluster belonging to the first 4Å-cluster was paired with each 3Å-cluster within the second 4Å-cluster. The same procedure was followed with each pair of overlapping 3Å-clusters at the 2Å-clusters level, then at the level of individual fragments, with decreasing overlap margins. According to optimization tests, we used distance restraints of 2.23 Å for backbone and 2.83 Å for side chain, with decreasing margins for overlapenergy for the different clustering levels.

Figure 1 .

 1 Figure 1. Strategy for homology-driven ssRNA-protein docking with structural fragments.

Figure 2 .

 2 Figure 2. Crystallographic structure of the human sex-lethal protein (surface) bound to a 5'-U8 ssRNA (sticks) (PDB ID 1B7F). Left: The RNA backbone and bases are distinguished in black and white respectively. Right: The nucleotides and amino-acids establishing hydrogen bonds (dashes) are represented as black sticks and gray lines respectively, with the same orientation as on the left picture.

Figure 3 .

 3 Figure 3. Best prediction obtained by unbound fragments docking. The amino-acids interacting with the RNA in the crystal structure are represented in white sticks. The bound RNA is represented both in black lines and in spheres corresponding to beads of ATTRACT coarse-grain representation, with one color per bead type. Each bound nucleotide n is approximated by up to 3 docked fragments {n, n+1, n+2}, {n-1, n, n+1} and {n-2, n-1, n}. The beads of the docked fragment are represented as asterisks, with same color code as for the bound nucleotides.

Figure 4 .

 4 Figure 4. Delineation of the RNA-binding site of sex-lethal protein by unbiased docking. The 8441 chain-forming poses are represented as dark lines, the bound RNA as white sticks and the protein as surface. None of the poses are occluded by the protein.