Jonathan Siau

Olivier Chadebec

Ronan Perrussel

Jean-René Poirier

Hybrid Cross Approximation for a Magnetostatic Formulation

Keywords: H-matrix, Hybrid Cross Approximation (HCA), Integral equation, Magnetostatic

A particular integral equation is considered for magnetostatic problems. A comparison of the adaptive cross approximation and the hybrid cross approximation, and the fast multipole method is done. The relevancy and the versatility of the HCA is assessed on magnetostatic examples.

I. INTRODUCTION

I NACTIVE regions, like the air, have to be taken into account in modeling electromagnetic devices. The integral equations enable us to avoid the discretization of these regions, but keeping the possibility to compute the magnetic field at any point in these inactive regions. However an integral equation leads to a dense matrix G ∈ R n×n of the form

G ij = Ω Ω ϕ i (x)g(x, y)ψ j (y)dxdy (1)
with n the number of Degrees Of Freedom (DOF), ϕ i and ψ j respectively test and trial functions and g a kernel function. Matrix G a priori requires a storage and a matrix-vector product complexity proportional to n 2 .

To avoid this n 2 complexity, solutions have been proposed for integral equations. Most of these solutions approximate the kernel g by a degenerate kernel g. The multipole method is such a solution but requiring an explicit expansion of g. This expansion is known for the kernel considered here, but we prefer an algebraic approach that adaptively controls the error.

We have retained the H-matrix format that enables the compression of the matrix G block-wise: the blocks of G that correspond to distant interactions are approximated by lowrank matrices. This method constrains the storage and complexity to scale as n log n, and permits in particular to compute efficient preconditioners. The Adaptive Cross Approximation (ACA) [START_REF] Bebendorf | Approximation of boundary element matrices[END_REF] is the most classical technique to build the low-rank approximations. Many articles show great compression results for classical kernels, as the Single Layer Potential (SLP) or the Double Layer Potential (DLP). However some limitations of the ACA are pointed out with simple examples (see [START_REF] Börm | Hierarchical Matrices[END_REF] for instance): it is shown that we do not have a reliable error estimation for geometries with edges and kernels that contain a differential operator. The solution we promote instead is the Hybrid Cross Approximation (HCA) [START_REF] Börm | Hybrid cross approximation of integral operators[END_REF].

The purpose of this paper is to compare the well-known ACA to the HCA on a simple example, and to show how HCA can be straightforwardly implemented on a more complicated kernel. Section II introduces the clustering method and the two cross approximation techniques. Then, in Section III a comparison between the HCA and the Multi-Level Fast Multipole Method (MLFMM) is performed for the solution of an electrostatic problem on a thick sphere. A magnetostatic formulation is presented in Section IV and the ACA and HCA are considered for tackling this formulation: first on a spherical shell and finally on a more complicated geometry.

II. H-MATRIX

The H-matrix representation is a data-sparse matrix format that enables the memory storage and the complexity of the arithmetic operations, as the matrix-vector product, to scale as n log(n). Thus, it allows solving large-scale problems efficiently using integral equation methods.

A. Clustering and low-rank approximation

The H-matrices, like the FMM, use a clustering technique to define which DOF can be grouped in a cluster. This clustering is mainly obtained by a recursive bisection of a bounding box containing the computational domain; it then leads to define a binary tree, see Fig. 1a. The FMM are based on an oct-tree, thus the clustering is different.

Once the hierarchical clustering is performed, we compress the interactions between distant clusters of DOF; see green blocks in Fig. 1b. Let us denote by t and ŝ the set of DOF indices in two such distant clusters t and s. The compression can be done by replacing the kernel g by a degenerate g or by approximating directly G| t×ŝ , the submatrix of G for the indices in t × ŝ, by a low-rank matrix with a prescribed error . Both approaches lead to a low-rank approximation

G| t×ŝ ≈ AB T (2)
with G| t×ŝ ∈ R m×n , A ∈ R m×k , B ∈ R n×k and k the rank of the approximation. This format requires a storage of k(m + n) instead of mn and the complexity of the matrix-vector product is proportional to k(m + n) instead of mn. The rank k weakly depends on usually as k ≈ -log , thus k m, n, and storage and complexity of arithmetic operations are reduced significantly.

B. Adaptive cross approximation

The ACA only uses the knowledge of some matrix entries to build a low-rank approximation of a matrix M . Briefly, the ACA is an iterative method where at each iteration an algorithm chooses a "well-suited" couple of pivot indices (t * , s * ) and computes the row t * and the column s * of M in order to generate a rank-one update which is added to the previous approximation. At the end of each iteration, the error is estimated and the algorithm is stopped when the required accuracy is reached. Details can be found in [START_REF] Bebendorf | Hierarchical Matrices: A Means to Efficiently Solve Elliptic Boundary Value Problems[END_REF].

A problem is that the heuristics of the ACA may fail for domain with edges and kernels with differential operators [START_REF] Börm | Hierarchical Matrices[END_REF]. This problem can be avoided using the HCA.

C. Hybrid cross approximation

Let us assume that

g(x, y) = D x D y γ(x, y), ∀(x, y) ∈ Ω t × Ω s (3) with D x , D y two differential operators, Ω t = ∪ i∈ tsupp(ϕ i )
and Ω s = ∪ j∈ŝ supp(ψ j ), the supports of the distant clusters t and s and γ a smooth generator function on Ω t ×Ω s . The SLP and DLP can easily be expressed using [START_REF] Börm | Hybrid cross approximation of integral operators[END_REF]. We also define two bounding boxes B t and B s such that Ω t ⊂ B t and Ω s ⊂ B s .

The HCA is an algorithm based on the cross approximation of the function γ contrary to the ACA where the cross approximation is applied at the matrix level. For this purpose, we define a grid of points in B t (resp. B s ) where γ can be evaluated. Here, we consider a tensorization of order p Chebyshev's interpolation point (x i ) K i=1 (resp. (y j ) K j=1 ) with K = (p + 1) d but other choices could be considered. Then we perform the ACA on the matrix S whose entries are

S i,j = γ(x i , y j ), ∀i, j ∈ {1, . . . , K} (4)
to find k pivots. The computation cost of this ACA is low, because the rows and columns of S are easily assembled.

Once the pivots xi l and ỹj l are chosen (x i l ⊂ x i ), the functional cross approximation we consider is

γ(x, y) =    γ(x, ỹi1 ) . . . γ(x, ỹi k )    T Γ -1 k    γ(x j1 , y) . . . γ(x j k , y)    , (5)
where Γ k a k×k-matrix whose coefficient (q, l) is γ(x iq , ỹj l ). Γ -1 k is not practically used, but the LU-factorization of Γ k can be computed in the ACA subroutine [START_REF] Bebendorf | Recompression techniques for adaptive cross approximation[END_REF]. Thus we replace Γ -1 k by the solutions of systems with an upper U k and a lower L k triangular matrices.

Finally, the degenerate kernel is

g = D x D y γ, (6)
thus the approximation G| t×ŝ of G| t×ŝ is given by

G| t×ŝ = AB T = ( ÃU -1 k )(L -1 k BT ), (7)
with à ∈ R m×k and B ∈ R n×k define as follow

Ãiq = Ωt ϕ i (x)D x γ(x, ỹjq )dx, (8)
Bil = Ωs ψ j (y)D y γ(x i l , y)dy. (9)
Note that we have to compute two simple integral instead of a two-fold integral, leading to a lower complexity. It is also easier to implement compared to the ACA: the ACA needs to assemble a row/column at each iteration but it is not directly provided by the usual mesh-element-loop approach.

III. ELECTROSTATIC EXAMPLE

Let us consider an electrostatic equation on a sphere with a standard Galerkin discretization of the SLP 1 4π

j S S ϕ i (x)ϕ j (y) x -y dydx q j = S ϕ i (x)V 0 (x)dx, (10)
where ϕ j are piece-wise constant functions, q j are the charge density unknowns and V 0 a real constant. To compare the FMM and the H-matrices, in the assembly step we set ACA , the relative error in Froebenius norm for each block, to 10 -4 so that both methods gives an equivalent accuracy. The relative norm of the residual in the iterative solver GMRes has to be lower than 10 -6 . The histogram in Fig. 2 shows the assembly and solution times for both methods. We can see that the H-matrix is faster than the FMM, even if it is slower to be assembled. In return, the FMM's iterations are about 10 times slower than the H-matrix's, because the chosen FMM variant consists in storing only the near-interactions. This variant allows us to use little memory as shown in Fig. 3. Note also that the H-matrix storage (here in kb/DOF) is close to the theoretical logarithmic complexity.

For these results, we did not use the H-LU preconditioning [START_REF] Bebendorf | Hierarchical Matrices: A Means to Efficiently Solve Elliptic Boundary Value Problems[END_REF] of the H-matrices because it is a well-preconditioned problem. However for a more general case, when we used the general, efficient and light H-LU preconditioning we conclude that the H-matrices are expected to be faster than the FMM in the solution step. It is especially relevant for problems with multiple right-hand sides.

IV. MAGNETOSTATIC FORMULATION

A. Formulation

Let us consider the ferromagnetic behavior law

M(x) = χ(x, H)H(x), ∀x ∈ Ω, (11)
with M the magnetization, H the magnetic field, and Ω an isotropic ferromagnetic material with χ its magnetic susceptibility. We define χ as a constant for the next experiments.

The magnetic field can be written as H = H red + H 0 with H red the reduced magnetic field created by the ferromagnetic material and H 0 the magnetic source field created by current flows.

We consider the following integral equation [START_REF] Kalimov | Application of a hybrid integrodifferential method for analysis of thin magnetic shields[END_REF] using the total scalar potential Φ Φ

(x) + 1 4π Ω χ(y, H) ∇Φ(y) • (x -y) x -y 3 dy = Φ 0 (x) (12)
with Φ 0 the scalar potential deriving from the magnetic field H 0 . Using a P 1 finite element approximation for Φ and the Galerkin discretization leads to the linear system

([M ] + [G]) Φ = D (13)
with the mass matrix and the right-hand-side vector

M ij = Ω ϕ i (x)ϕ j (x)dx, D i = Ω ϕ i (x)Φ 0 (x)dx, (14)
and the volume integral

G ij = 1 4π Ω ϕ i (x) Ω χ∇ϕ j (y) • ∇ y 1 x -y dydx. ( 15
)
Solving (13) permits to compute the magnetic field H in the air [START_REF] Carpentier | Resolution of Nonlinear Magnetostatic Problems With a Volume Integral Method Using the Magnetic Scalar Potential[END_REF]. The mass matrix is a finite element matrix with a sparse storage, so the limitation here is the dense matrix G.

B. Degenerate expansion

We use the HCA to compress the matrix G. In (15) the

kernel function g(x, y) = ∇ y 1 x -y enables to identify γ(x, y) = 1 x -y , D x = Id, D y = ∇ y . ( 16
)
The two single integrals to compute are then

Ω ϕ i (x)
xỹj l dx and

Ω χ∇ϕ j (y) • ∇ y 1 xiq -y dy. ( 17
)
The computational cost of these integrals is proportional to the number of quadrature points per element instead of the quadratic cost of the twofold integral, allowing us to have a lower complexity than the ACA for the same rank.

C. Numerical results

Two cases are considered in this section, the first consists in solving (13) on a spherical shell meshed with tetrahedra; the solution is shown in Fig. 4. The second consists of a numerical study of the ACA and the HCA for the assembly of (15) on the TEAM Problem 13 geometry [START_REF] Nakata | Summary of results for team workshop problem 13 (3-d nonlinear magnetostatic model)[END_REF].

For the first results, we keep the same parameters as for the electrostatic example and we can compare the results with those obtained with the full-storage. We can see in Table I that the HCA is faster than the ACA to assemble an Hmatrix, but unfortunately the HCA requires more memory compared to the ACA. These results match to these found in the literature, which also treat spherical geometries for other integral equations.

These memory requirements can be reduced using two algebraic re-compression techniques, the SVD of the low-rank matrices and the Agglomeration. Both methods are explicitly described in [START_REF] Bebendorf | Hierarchical Matrices: A Means to Efficiently Solve Elliptic Boundary Value Problems[END_REF]. These methods do not change the asymptotic complexity of the assembly step, and they allow us to reduce the complexity of the arithmetic operations with the matrices. We do not detail these methods but we can see in Fig. 5 the simplification of the structure and the reduced storage.

Table II is obtained using the agglomeration after the assembly of the H-matrix. The assembly times have only slightly increased compared to Table I, but the major improvement is that both methods have an equivalent and reduced storage.

Finally, we compare the accuracy of both methods (without using the agglomeration) applied to the TEAM Problem 13 geometry (see Fig. 6), meshed with 29, 362 tetrahedra. For a matter of ease, the "Error" column is computed comparing the matrix-vector products Gx -Gx / Gx , where x is a random vector. It is not the conventional way to do, but since we are using an iterative solver this information is relevant for the solution quality.

The results of the assembly step are presented in Table III, where we see that the ACA needs an unusually higher storage than the HCA. However the HCA is less accurate than the ACA, but still of the same order as ACA . On this realistic problem, the HCA is more efficient comparing the storage requirements and the assembly times. V. CONCLUSION This paper first demonstrates the efficiency of the Hmatrix to solve an electrostatic problem compared to the FMM. Thanks to the general, efficient and light H-LU preconditioning, we note that the H-matrices are generally more efficient for other considered problems. Then we focus on the construction of the low-rank approximations, and we compare ACA and HCA considering two cases. The ACA does not fail for this formulation and the considered examples. Comparing ACA and HCA, we show that the HCA is faster but requires a larger storage. Adding an agglomeration step, storage is equivalent for both methods. At the end the HCA, seems more efficient and reliable than the ACA for this formulation.

Fig. 1 .

 1 Fig. 1. (a) Domain bisection and corresponding binary tree. (b) Two levels of the matrix partition induced. Green and red blocks are "far" and "near" interaction. Clusters in interaction are identified in each block.

Fig. 2 .Fig. 3 .

 23 Fig. 2. Comparison of the FMM and H-matrix times on a DELL PRECISION M4800 with an Intel(R) Core(TM) i7-4800MQ CPU @2.70GHz

Fig. 4 .

 4 Fig. 4. Isovalues of the solution on a Spherical shell.

Fig. 5 .

 5 Fig. 5. Example of a H-matrix agglomeration. (a) Initial H-matrix. (b) Agglomerated H-matrix. The number in the green blocks indicates the rank.

Fig. 6 .

 6 Fig. 6. TEAM Problem 13 geometry.

TABLE I COMPUTING

 I TIMES, STORAGE AND ERROR WITHOUT AGGLOMERATION.

		NbDOF	Assembly (s) Solver (s) Storage (kb/DOF)	Error
	A	3,846	718	0.5	15.4	1.53%
	C	12,938	7321	10	24.3	0.85%
	A 48,154	56,112	89	37.6	0.37%
	H	3,846	278	0.6	18.3	5.23%
	C	12,938	2,333	12	29.4	1.33%
	A 48,154	18,154	109	44.9	1.42%

TABLE II COMPUTING

 II TIMES, STORAGE AND ERROR WITH AGGLOMERATION.

		NbDOF	Assembly (s) Solver (s) Storage (kB/DOF)	Error
	A	3,846	952	0.5	7.0	1.66%
	C	12,938	9,774	10	10.3	0.27%
	A 48,154	74,604	95	14.9	0.45%
	H	3,846	287	0.5	7.0	1.63%
	C	12,938	2,586	12	10.4	0.54%
	A 48,154	18,876	98	14.9	0.72%

TABLE III ASSEMBLY

 III TIME, STORAGE AND ERROR. TEAM PROBLEM 13.

		ACA	Assembly (s) Storage (kb/DOF)	Error
	A 1E-02	8,574	12.4	1.950E-3
	C	1E-03	11,961	15.5	1.643E-4
	A 1E-04	15,690	19.2	1.346E-5
	H 1E-02	2,607	10.4	4.070E-2
	C	1E-03	2,773	13.9	3.300E-3
	A 1E-04	2,857	18.6	8.890E-4

Manuscript received June 8, 2014; revised XX, XX. Corresponding author: J. Siau (email: jonathan.siau@g2elab.grenoble-inp.fr).

ACKNOWLEDGMENTS

We thank the GDR SEEDS for its financial support.


Eduardo José Peña
email: eduardop@biol.unlp.edu.ar

Gabriel Robles

Manfred Heinlein
email: manfred.heinlein@ibmp-cnrs.unistra.fr

In vivo imaging of tagged mRNA in plant tissues using the bacterial transcriptional antiterminator BglG Authors

Keywords: RNA localization, RNA transport, BglG, MS2, λN, TMV, movement protein, cell-to-cell movement, RNA labeling, RNA imaging

RNA transport and localization represent important post-transcriptional mechanisms to determine the subcellular localization of protein synthesis. Plants have the capacity to transport mRNA molecules beyond the cell boundaries through plasmodesmata (PD) and over long distances in the phloem. RNA viruses exploit these transport pathways to disseminate their infections and represent important model systems to investigate RNA transport in plants. Here, we present an in vivo plant RNA-labeling system based on the E.coli RNA-binding protein BglG. Using the detection of RNA in mobile RNA particles formed by viral movement protein (MP) as a model, we demonstrate the efficiency and specificity of mRNA detection by the BglG system as compared to MS2 and λN systems. Our observations show that MP mRNA is specifically associated with MP in mobile MP particles but hardly with MP localized at PD. MP mRNA is clearly absent from MP accumulating along microtubules. We show that the in vivo BglG labeling of the MP particles depends on the presence of the BglG-binding stem-loop aptamers within the MP mRNA and that the aptamers also enhance the coprecipitation of BglG by MP, thus demonstrating the presence of a MP:MP mRNA complex. The BglG system also allowed us to monitor the cell-to-cell transport of the MP mRNA, thus linking the observation of mobile MP mRNA granules with intercellular MP mRNA transport. Given its specificity demonstrated here, the BglG system may be widely applicable for studying mRNA transport and localization in plants.

Introduction

mRNA localization is an evolutionary conserved mechanism observed in a wide range of organisms and cell types, and by which mRNA species carrying RNA localization signals ("zipcodes") are localized to discrete locations in the cytoplasm (Ryder and Lerit, 2018). By dictating the specific localization at which encoded proteins are translated and operate, RNA localization processes play central roles in the determination of cell polarity, embryonic patterning, asymmetric cell-fate decisions, neuronal morphogenesis, cell migration, or epithelial polarity (Ryder and Lerit, 2018). Initial evidence for RNA compartmentalization was based on laborious in situ hybridization methods for the detection of individual mRNAs (Jeffery et al., 1983). In more recent years the advent of transcriptome-wide approaches showed that RNA localization is much more widespread than previously assumed, thus indicating that RNA localization has a prominent role in the spatial regulation of gene activity [START_REF] Lécuyer | Global analysis of mRNA localization reveals a prominent role in organizing cellular architecture and function[END_REF]Holt and Bullock, 2009). RNA localization is a multistep process involving mRNA export from the nucleus, protection from degradation, cytoplasmic transport, and localized anchorage. Upon transcription and mRNA maturation, the zipcodes are recognized by RNA-binding proteins and play important roles during nuclear export and cytoplasmic sorting [START_REF] Eliscovich | RNP transport in cell biology: the long and winding road[END_REF]. It is accepted that localized RNAs are transported as ribonucleoprotein complexes (RNPs) or "RNA granules". These granules are transported by molecular motors along the cytoskeleton to their final destination, where the mRNAs are anchored and eventually translated (Holt and Bullock, 2009;[START_REF] Martin | mRNA localization: gene expression in the spatial dimension[END_REF][START_REF] Buxbaum | In the right place at the right time: Visualizing and understanding mRNA localization[END_REF][START_REF] Mofatteh | SnapShot: Subcellular mRNA localization[END_REF].

Cell-to-cell communication is a fundamental process essential for the distribution of signaling molecules during the coordination of developmental programs and during adaptation to environmental or intrinsic stimuli. Plants allow intercellular transport and exchange of nutrients and signaling molecules through plasmodesmata (PD). These gatable 4 membranous pores extend through the cell wall and provide cytoplasmic and membrane continuity between cells [START_REF] Zambryski | Plasmodesmata: Gatekeepers for cell-to-cell transport of developmental signals in plants[END_REF][START_REF] Boyko | Intramolecular complementing mutations in tobacco mosaic virus movement protein confirm a role for microtubule association in viral RNA transport[END_REF]. The system of PD is connected to the phloem sieve elements, thus forming a cell-to-cell and long-distance communication network [START_REF] Lucas | Plasmodesmata as a supracellular control network in plants[END_REF]. Similar to other systems, also plants have the capacity to transport and localize RNA molecules to specific cellular compartments [START_REF] Okita | mRNA localization in plants: Targeting to the cell's cortical region and beyond[END_REF]Tian et al., 2020). Moreover, given the communication network established by PD and connected phloem, RNA molecules can also move between cells and tissues [START_REF] Kehr | Long distance transport and movement of RNA through the phloem[END_REF][START_REF] Nazim Uddin | Intercellular and systemic spread of RNA and RNAi in plants[END_REF][START_REF] Kehr | Long distance RNA movement[END_REF].

Although PD are essential for intercellular communication and plant life, the underlying mechanisms by which RNA molecules are targeted to PD and transported between cells remain poorly understood. One way to address these mechanisms is by studying the cellto-cell transport of single-stranded RNA viruses. Viruses encode specialized proteins known as movement proteins (MPs) that localize to PD, modify PD regulation and allow the passage of the viral RNA (Heinlein, 2015;[START_REF] Peña | RNA transport during TMV cell-to-cell movement[END_REF]. Tobacco mosaic virus (TMV), undoubtedly the most studied plant virus (Scholthof et al., 2011), is a singlestranded, positive-sense RNA virus. The viral genome encodes two subunits of the replicase as well as the MP and the coat protein (CP) [START_REF] Goelet | Nucleotide sequence of tobacco mosaic virus RNA[END_REF]. TMV cell-to-cell movement is independent of CP (Holt and Beachy, 1991) and likely occurs as a non-encapsidated ribonucleoprotein complex (RNP) that contains viral RNA in association with MP and replicase (Heinlein, 2015). Therefore, studies on TMV movement promise to reveal the mechanisms underlying the intra-and intercellular transport of RNA complexes in plants [START_REF] Peña | RNA transport during TMV cell-to-cell movement[END_REF].

The use of TMV derivatives that express MP as a fusion to fluorescent reporters led to the discovery that the protein interacts with the endoplasmic reticulum (ER) and the microtubule (MT) cytoskeleton (Heinlein et al., 1995;Heinlein et al., 1998). Several subsequent studies showed that the interaction with MTs is tightly associated with MP function in virus movement (Boyko et al., 2000a;[START_REF] Boyko | Validation of microtubule-associated tobacco mosaic virus RNA movement and involvement of microtubule-aligned particle trafficking[END_REF][START_REF] Ouko | Tobacco mutants with reduced microtubule dynamics are less susceptible to TMV[END_REF][START_REF] Niehl | Microtubules in viral replication and transport[END_REF].

Conditional mutations in MP were used to correlate the ability of MP to support the cell-tocell movement of the virus with the formation of MP particles showing stop-and-go movements in the cortical cytoplasm (Boyko et al., 2000b;[START_REF] Boyko | Validation of microtubule-associated tobacco mosaic virus RNA movement and involvement of microtubule-aligned particle trafficking[END_REF][START_REF] Boyko | Intramolecular complementing mutations in tobacco mosaic virus movement protein confirm a role for microtubule association in viral RNA transport[END_REF]. Mobile MP particles are also formed during ectopic expression of MP and their further analysis revealed that the particles localize to the cortical ER-actin network, at positions associated with intersecting MTs ("cortical MT-associated ER sites", cMERs [START_REF] Peña | Cortical microtubule-associated ER sites: Organization centers of cell polarity and communication[END_REF]). The movements of the particles occur along the ER-actin network, apparently guided by MTs and pausing their trajectories at ER-MT intersection points [START_REF] Brandner | Tobacco mosaic virus movement protein interacts with green fluorescent protein-tagged microtubule end-binding protein 1[END_REF]. Further studies demonstrated that TMV movement and the targeting of MP to PD depends on specific class XI and class XIII myosins (Amari et al., 2014). The association of MP particles with cMERs is consistent with the association of MP with membranes [START_REF] Peiro | The tobacco mosaic virus movement protein associates with but does not integrate into biological membranes[END_REF] and its affinity to bind MTs [START_REF] Ashby | Tobacco mosaic virus movement protein functions as a structural microtubule-associated protein[END_REF] and MTassociated proteins [START_REF] Brandner | Tobacco mosaic virus movement protein interacts with green fluorescent protein-tagged microtubule end-binding protein 1[END_REF][START_REF] Ferralli | Disruption of microtubule organization and centrosome function by expression of tobacco mosaic virus movement protein[END_REF]. Further studies suggested that the cMERs and PD may share features of ER:PM contact sites and that these features may be targeted for localization of MP and the MP-associated viral RNA [START_REF] Pitzalis | The roles of membranes and associated cytoskeleton in plant virus replication and cell-to-cell movement[END_REF]. A likely candidate for such a common feature is synaptotagmin 1 (SYT1).

This protein occurs at ER:PM contact sites as well as at PD (Ishikawa et al., 2020;[START_REF] Levy | Synaptotagmin SYTA forms ER-plasma membrane junctions that are recruited to plasmodesmata for plant virus movement[END_REF] and is a target of MP required for virus movement [START_REF] Levy | Synaptotagmin SYTA forms ER-plasma membrane junctions that are recruited to plasmodesmata for plant virus movement[END_REF][START_REF] Yuan | The plasmodesmal localization signal of TMV MP is recognized by plant synaptotagmin SYTA[END_REF][START_REF] Lewis | Arabidopsis synaptotagmin SYTA regulates endocytosis and virus movement protein cell-to-cell transport[END_REF] While proteins can be localized by translational fusion to fluorescent reporters, the development of comparable techniques to localize RNA molecules by transcriptional fusion to a fluorescent RNA molecule remain to be developed. Instead, RNA molecules are observed upon in vitro labeling with fluorochromes and delivery to cells by transfection or microinjection [START_REF] Wilkie | Drosophila wingless and pair-rule transcripts localize apically by dynein-mediated transport of RNA particles[END_REF], or upon in vivo labeling by indirect methods. These indirect methods are based either on hybridization of native RNA molecules with fluorescent probes that are delivered into the cell, or on the use of RNA-binding proteins (RBPs) or chemical ligands that tag the RNA upon binding to RNA motifs transcriptionally fused to the RNA under study. Whereas the RBPs are used as fusions to fluorescent proteins to image the localization of the tagged RNA, the chemical ligands become by themselves fluorescent upon binding to the RNA (Tilsner, 2015;[START_REF] George | Intracellular RNAtracking methods[END_REF][START_REF] Christensen | Advances in imaging RNA in plants[END_REF]. One of the most frequently applied RNA visualization techniques in animal cells is the MS2 method [START_REF] Bertrand | Localization of ASH1 mRNA particles in living yeast[END_REF]. This method is based on the high affinity binding of the MS2 coat protein (MS2-CP) to a specific RNA stem-loop (MS2-SL) structure that initiates encapsidation of the bacteriophage RNA genome. The MS2 method utilizes a tandem array of this SL by transcriptionally fusing it to the mRNA under investigation and by expressing the SL-tagged mRNA together with the MS2-CP translationally fused to a fluorescent reporter, such as GFP. In order to reduce cytoplasmic background, the MS2-CP:GFP protein is fused to a nuclear localizing sequence (NLS). Thus, once transcribed, the mRNA carrying the SL array will be recognized and bound by the RBP (NLS:MS2-CP:GFP) forming a complex exported to the cytoplasm as a trackable fluorescent mRNP [START_REF] Bertrand | Localization of ASH1 mRNA particles in living yeast[END_REF].

Using the MS2 system in plants, an SL array-tagged mRNA encoding the MP of TMV in fusion to red fluorescent protein (RFP) was shown to localize to the mobile MP particles in the cortical cytoplasm [START_REF] Brandner | Tobacco mosaic virus movement protein interacts with green fluorescent protein-tagged microtubule end-binding protein 1[END_REF]. The mRNA signal in the MP particles depended on the presence of the SL array in the MP mRNA, thus showing that the signal reported by NLS:MS2-CP:GFP was specific [START_REF] Brandner | Tobacco mosaic virus movement protein interacts with green fluorescent protein-tagged microtubule end-binding protein 1[END_REF]. Moreover, the tagged MP mRNA was also observed to colocalize with MP at the PD. These findings indicated that the MP particles contain MP mRNA and that the MP facilitates the targeting of its own mRNA to the PD [START_REF] Brandner | Tobacco mosaic virus movement protein interacts with green fluorescent protein-tagged microtubule end-binding protein 1[END_REF]. However, although these results were robust, the sensitivity of the method was low. Thus, only a small number of RNA-containing MP particles could be detected.

To improve the sensitivity of RNA detection, we tested two other RNA labeling systems.

Whereas the 'λN system' uses the antitermination protein N of bacteriophage lambda as RBP [START_REF] Daigle | λN-GFP: An RNA reporter system for live-cell imaging[END_REF], another system that was not previously used in plants uses the antiterminator protein G of the E. coli bgl operon as RBP and is referred to as the 'BglG system' [START_REF] Chen | High efficiency of HIV-1 genomic RNA packaging and heterozygote formation revealed by single virion analysis[END_REF]. Both systems allowed us to confirm the presence of MP mRNA in MP particles and to detect MP mRNA together with MP at PD. With the aim to further increase the strength of the specific mRNA signal, we decided to increase the cytoplasmic accumulation of the RBPs by removing the NLS. Whereas the MS2 and λN RBPs formed large cytoplasmic aggregates under these conditions, the RBP of the BglG system showed a homogenous cytoplasmic distribution. The BglG protein was still homogenously distributed in the cytoplasm when co-expressed with MP. However, this localization changed dramatically when the MP encoding mRNA carried the cognate BglGbinding SLs. In this case, BglG was recruited to multiple mobile MP particles but not to highly MP-decorated MTs, thus specifically identifying the MP particles as RNPs.

Coimmunoprecipitation experiments confirmed these observations by showing that BglG coprecipitates efficiently with MP if MP is expressed from a SL-tagged mRNA but not if MP is expressed from mRNA without the SLs. Furthermore, we show that the BglG-labeled mRNA moves cell-to-cell thus linking the detection of cytoplasmic, mRNA-containing, mobile MP particles with mRNA movement.

Results

Imaging of MP mRNA with MS2, λN and BglG systems

Using the MS2 method to study TMV RNA transport and localization, it was shown that MP colocalizes with MP mRNA (tagged with 12 MS2-SLs) in mobile particles and at PD, thus suggesting that the mobile particles are RNPs destined to PD [START_REF] Brandner | Tobacco mosaic virus movement protein interacts with green fluorescent protein-tagged microtubule end-binding protein 1[END_REF].

However, further studies were hampered by the low sensitivity of mRNA detection. When we repeated these experiments, the localization of the MP mRNA at PD was observed regularly, whereas the association of the mRNA with the mobile MP particles in the cytoplasm was much harder to detect (Figure 1a and Movie S1). Increasing MP expression to levels observed during infection compromised the background of our system. Therefore, to achieve RNA visualization with higher sensitivity and resolution, we decided to test other RNA localization methods.

We started by testing the λN system, which is based on the specific binding of a 22 amino acids long peptide derived from the phage lambda protein N (λN, 3.1 kDa) to an RNA hairpin formed by 15 nucleotides (boxB) derived from the nut N binding site in the N mRNA. Similar to the MS2 system, the λN system allows mRNA detection by fusing the mRNA of interest with boxB SLs and by co-expression of the RBP (λN) targeted to the nucleus and fused to a fluorescent reporter [START_REF] Daigle | λN-GFP: An RNA reporter system for live-cell imaging[END_REF]Schönberger et al., 2012). Coexpression of the RBP (λN:eGFP:NLS) with MP:mRFP carrying a transcriptional fusion to 16 copies of the boxB-SLs (MP:mRFP-boxB) led to detection of the MP:mRFP mRNA in MP particles (Figure 1b), thus confirming that mobile MP particles indeed contain MP mRNA (Movie S2). The MP mRNA was also detected at PD together with MP, albeit with low frequency (Figure 1b).

As a second alternative, we tested the BglG system. This system has been developed for the detection of HIV RNA within HIV virions [START_REF] Chen | High efficiency of HIV-1 genomic RNA packaging and heterozygote formation revealed by single virion analysis[END_REF] and has not been applied to plants. We adapted this system to plants by translationally fusing 57 aminoacids of the Nterminal portion of the BglG protein (BglG, 6.3 kDa) carrying the RNA-binding domain to the red fluorescent protein mCherry and to an SV40 NLS. The resulting RBP construct, BglG:mCherry:NLS, was then expressed together with MP:GFP encoded by mRNA tagged with 18 copies of the 29 nt long BglG-SL. This again led to the localization of the MP mRNA in MP particles (Figure 1c and Movie S3). However, this method barely detected any MP mRNA-specific signal at PD (Figure 1c).

The three tested RNA labeling methods allowed the imaging of MP mRNA colocalizing with MP particles in the cell cortex. Some of these MP RNA-containing MP particles remained stationary while other particles where mobile and displayed a stop-and-go movement, as previously noted [START_REF] Brandner | Tobacco mosaic virus movement protein interacts with green fluorescent protein-tagged microtubule end-binding protein 1[END_REF]. In between short stationary intervals, the particles showed movements that alternated between episodes of random motion over short distances at intermediate velocities and directed movements for long trajectories with higher velocities of up to 3 -3,5 µm/s (Figure S1). These intervals of MP particle movement at higher velocities are compatible with the reported velocities of myosin-driven transport [START_REF] Nebenführ | Stop-and-go movements of plant Golgi stacks are mediated by the acto-myosin system[END_REF] and are also consistent with the role of class XI myosins in the targeting of MP to PD as well as in TMV movement (Amari et al., 2014).

Improving sensitivity of mRNA detection

We wondered whether we could achieve stronger RNA labeling by removal of the NLS sequences and thereby increasing the cytoplasmic level of the RBP. However, expression of either MS2-CP:GFP or λN:GFP without NLS sequences and in the absence of the specific RNA binding target led to aberrant and unspecific accumulations of the proteins in the cytoplasm. Both proteins accumulated in the nucleus and in irregular cytoplasmic bodies visible already at low magnification (Figure 2a, left and center). In contrast, BglG:mRFP did not accumulate in any particularly dense cytoplasmic location and showed a clear homogenous nucleocytoplasmic distribution (Figure 2a, right). Imaging cortical planes at high magnification demonstrated that MS2-CP:GFP and λN:GFP were seen in the form of small granules with high motility, thus structures that undoubtedly could mask the specific labeling and detection of any SL-tagged RNA (Figure 2b, left and center panels; Movies S4 and S5). In contrast, BglG:mRFP showed a homogenous distribution in the cortical cytosol (Figure 2b, right panel), with movements according to cytoplasmic streaming (Movie S6).

Thus, unlike MS2-CP:GFP and λN:GFP, BglG:mRFP turned out to be suitable for expression without NLS in further RNA labeling experiments.

When co-expressed with MP:GFP translated from untagged MP:GFP mRNA (without BglG SLs), BglG:mRFP retained its homogenous distribution in the cytoplasm (Figure 3a). The BglG:mRFP fluorescence again followed cytoplasmic streaming (Movie S6 and Movie S7) independent of MP:GFP that typically decorated MTs or localized to particles showing intermittent movement along or between MTs, as previously described [START_REF] Brandner | Tobacco mosaic virus movement protein interacts with green fluorescent protein-tagged microtubule end-binding protein 1[END_REF][START_REF] Boyko | Validation of microtubule-associated tobacco mosaic virus RNA movement and involvement of microtubule-aligned particle trafficking[END_REF]. In contrast, expression of MP:GFP from BglG-SL-tagged MP:GFP mRNA (MP:GFP-SL) led to recruitment of BglG:mRFP to the MP:GFP particles (Figure 3ab; Movie S7 and Movie S8) resulting in the labeling of the RNP granules with high contrast (Figure 3b). Fluorescence signal intensity analysis confirmed this specific recruitment of BglG:mRFP to the SL-tagged RNA (Figure S2). Interestingly, the RNA signal colocalized exclusively with MP within MP particles and was absent from MP localized along MTs. This observation confirms the role of MP particles as RNPs and also shows that the MP by itself either is insufficient for in vivo RNA binding or exists in different isoforms, an RNA bindingcompetent form present in the mobile MP particles and an RNA binding-incompetent form localized to MTs. The homogeneous, non-localized distribution of BglG:mRFP in the absence of its cognate RNA-SL is also observed in median planes of the cell, in close proximity to MP:GFP localized at PD (Figure 3c, arrowheads). In contrast, in the presence of MP:GFP tagged with of BglG-SLs, BglG:mRFP co-localized with MP:GFP at the PD, thus indicating an association of MP:GFP mRNA with PD-localized MP:GFP (Figure 3d, arrows).

However, several of the MP:GFP spots occurred without a corresponding RNA labeling (Figure 3d, arrowheads), indicating that MP:GFP mRNA does not always colocalize with MP:GFP at the PD. We did not observe any significant difference in the motions of free-and BglG-tagged MP:GFP particles, suggesting that neither the RNA tag nor the binding of the RBP to this tag affects particle behavior (Figure S3).

While agroinfiltration ensures strong protein expression in almost all the cells of the infiltrated N. benthamiana leaf patch [START_REF] Kapila | An Agrobacteriummediated transient gene expression system for intact leaves[END_REF], conditions need to be optimized for protein co-expression experiments. To facilitate the visualization of tagged RNA molecules, we developed a BglG:mRFP-transgenic N. benthamiana line. Transient expression of MP:GFP or MP:GFP-SL in this line revealed the presence of MP:GFP mRNA in MP:GFP particles, and, as expected, the mRNA signal in the particles was dependent on the SLs and, thus, absent when the MP:GFP construct without SLs was used (Figure S4).

The requirement of the SLs for efficient binding of BglG:mRFP to the MP:GFP mRNA was further confirmed by in vitro experiments showing that the presence of the SLs within the MP:GFP mRNA enhances the co-precipitation of BglG:mRFP by MP:GFP from plant extracts above experimental background (Figure 3e andf).

MP mRNA moves cell-to-cell

Having shown that the BglG system can be used for mRNA detection and the analysis of mRNA movements inside cells, we next wanted to know if the same approach can also be used for the detection of mRNA movements between cells. A hallmark feature of viral MPs is their ability to modify the size exclusion limit of PD and to translocate through PD into the neighboring cells [START_REF] Lucas | Plant viral movement proteins: Agents for cell-to-cell trafficking of viral genomes[END_REF]. The cell-to-cell movement of MPs can be monitored in N. benthamiana leaves upon expression in isolated cells using diluted agrobacterium cultures for infiltration (Robles Luna et al., 2013;[START_REF] Brunkard | The cytosol must flow: Intercellular transport through plasmodesmata[END_REF], however, to our knowledge, such a simple in vivo method has not been reported for assessing the cell-to-cell trafficking of mRNA molecules. Instead, previous approaches used fluorescent in vitro-labeled RNA molecules that were delivered into the cell by microinjection. Such assays demonstrated that E. coli produced cucumber mosaic virus MP facilitated the cell-to-cell movement of co-injected, fluorescently-labelled KNOTTED mRNA [START_REF] Lucas | Selective trafficking of KNOTTED1 homeodomain protein and its mRNA through plasmodesmata[END_REF]. To determine whether the observed MP mRNA-containing MP particles are associated with intercellular movement of MP mRNA, we performed co-agroinfitration experiments in which we combined high concentrations of agrobacterium (OD 600 = 0,4) for the expression of MP:GFP or MP:GFP-SL in all cells of the infiltrated tissue with highly diluted cultures (OD 600 = 1,0x10 -4 ) for coexpression of BglG:mRFP only in isolated cells. We expected that when MP mRNA is transported from a cell that expresses BglG:mRFP to neighboring cells that do not express this protein, BglG:mRFP should highlight the transported mRNA by co-transport of its red fluorescent signal due to its binding to the BglG SLs in the mRNA. Figure 4a shows that BglG:mRFP is indeed transported between cells and that this event depends on the presence of BglG-SLs in the MP:GFP mRNA. We considered a cell-to-cell transport event when two or more neighboring cells showed accumulation of BglG:mRFP fluorescence.

From all the red fluorescent, BglG:mRFP-containing spots, only 11 % showed red fluorescence in more than one cell when the RBP was expressed together with MP:GFP from untagged MP:GFP mRNA. In contrast, this number increased to 32 % when BglG:mRFP was expressed together with MP:GFP-SL, thus in the presence of BglG-SL tagged mRNA (Figure 4b and Table S1). Considering that the distribution of independent transformation events upon infiltration of a low concentration of agrobacteria follows the Poisson distribution, the occurrence of adjacent cells that express BglG:mRFP to a frequency of 32 % of all transformation events by independent transformation of the adjacent cells is unlikely (P < 2,3 x 10 -13 ; see Materials and Methods). The increased frequency of adjacent cells that contain BglG:mRFP seen in the presence of SL-tagged MP:GFP mRNA therefore suggests the visualization of intercellular, MP-facilitated RNA transport. The occurrence of foci with more than one cell showing mRFP fluorescence (11 %) in the absence of the SL tags may indicate a background of intercellular BglG:mRFP (33 kDa) diffusion as similar values were previously observed for the cell-to-cell movement of free GFP (27 kDa) [START_REF] Borniego | Bioinformatic and mutational analysis of ophiovirus movement proteins, belonging to the 30K superfamily[END_REF]Robles Luna et al., 2013). In essence, the observations indicate that (a) transiently expressed MP mRNA is transported between cells, (b) this event can be imaged with the BglG system presented here, and (c) mRNA movement occurs in the form of an RNP complex into which BglG:mRFP is incorporated and co-transported through PD.

Discussion

Traditionally, mRNAs have been viewed as simple intermediates between the genomic information stored in the DNA and the encoded effector proteins. It is now clear that mRNAs are actively localized to distinct subcellular domains, with fundamental implications in embryonic patterning, asymmetric cell division, and cell polarity and, thus, in the determination of cell fate [START_REF] Chin | RNA localization: Making its way to the center stage[END_REF][START_REF] Buxbaum | In the right place at the right time: Visualizing and understanding mRNA localization[END_REF]. Plant mRNAs are also able to move between cells and to distant tissues, and this trafficking plays important roles in the regulation of plant development, growth, and defense [START_REF] Ham | Phloem-mobile RNAs as systemic signaling agents[END_REF].

Grafting experiments followed by high throughput transcriptomic analysis demonstrated that more than 2000 genes produce phloem-mobile mRNAs in Arabidopsis (Thieme et al., 2015).

Similar numbers of transported mRNAs where also identified following analysis of grafted grapevines or cucumber [START_REF] Yang | Messenger RNA exchange between scions and rootstocks in grafted grapevines[END_REF][START_REF] Zhang | Vascular-mediated signalling involved in early phosphate stress in plants[END_REF]. Phosphate stress treatments to cucurbit-grafted plants identified a population of transcripts involved specifically in early phosphate stress signaling. In addition, some of these molecules were selectively transported from the rootstock to specific tissues on the scion, thus highlighting specific signaling roles of mRNA transport [START_REF] Zhang | Vascular-mediated signalling involved in early phosphate stress in plants[END_REF]. However, while an increasing number of phloem-mobile RNA molecules is being identified and general features and functions are being elucidated [START_REF] Liu | Intercellular and systemic trafficking of RNAs in plants[END_REF][START_REF] Kehr | Long distance RNA movement[END_REF][START_REF] Reagan | RNA on the move: The plasmodesmata perspective[END_REF][START_REF] Yang | m5C methylation guides systemic transport of messenger RNA over graft junctions in plants[END_REF], the mechanisms underlying the selective entry of mRNAs into the phloem long-distance stream remains difficult to address. The transporting phloem sieve elements are devoid of nuclei. Thus, in order to function in signaling, the mRNAs must enter and exit the phloem sieve elements through PD, requiring active intracellular transport to these channels. Understanding the mechanism by which mRNAs are transported between cells and systemically requires approaches to visualize mRNAs and their transport complexes with subcellular resolution.

The MS2 method is broadly used to visualize RNA molecules in organisms ranging from bacteria to higher eukaryotes and to study the processes of transcription, transport, translation and decay [START_REF] Bertrand | Localization of ASH1 mRNA particles in living yeast[END_REF][START_REF] Vera | Single-cell and single-molecule analysis of gene expression regulation[END_REF]Tutucci et al., 2018;[START_REF] Grünwald | In vivo imaging of labelled endogenous β-actin mRNA during nucleocytoplasmic transport[END_REF][START_REF] Larson | Real-time observation of transcription initiation and elongation on an endogenous yeast gene[END_REF][START_REF] Wu | Translation dynamics of single mRNAs in live cells and neurons[END_REF]. The method has also been applied in plants and allowed to localize specific mRNAs and virus-derived RNAs [START_REF] Brandner | Tobacco mosaic virus movement protein interacts with green fluorescent protein-tagged microtubule end-binding protein 1[END_REF][START_REF] Luo | Selective targeting of mobile mRNAs to plasmodesmata for cell-to-cell movement[END_REF][START_REF] Michaud | Differential targeting of VDAC3 mRNA isoforms influences mitochondria morphology[END_REF]Hamada et al., 2003). In this work, we compared the MS2 system with the λN and BglG systems for the ability to label MP mRNA within mobile MP particles and MP-labelled PD (Figure 1). While all these systems equally suffer from low sensitivity, the sensitivity of the BglG system could be increased by expressing the BglG protein without NLS, thus by augmenting its cytoplasmic concentration. This approach failed with the λN and MS2 systems because the N and CP proteins formed cytoplasmic aggregates and could not be used under these conditions. Thus, using the BglG system under conditions of high levels of the BglG protein in the cytoplasm, we could distinguish the presence of MP mRNA in mobile MP granules and its absence from MP localized to other sites, thus clearly identifying the MP granules as RNA particles. The specific role of the SLs for efficient MP mRNA detection was confirmed by coimmunoprecipitation experiments showing that SLs in the MP:GFP mRNA strongly enhance the coprecipitation of BglG:mRFP with MP. By using affinity beads for MP:GFP, the coprecipitation of BglG:mRFP suggests the binding of BglG to a MP:MP mRNA-SL complex (Figure 3e), which is consistent with the observed in vivo detection of SL-tagged MP RNA in MP granules.

Previous results obtained with the MS2 system indicated that MP accumulates with its own RNA at PD [START_REF] Brandner | Tobacco mosaic virus movement protein interacts with green fluorescent protein-tagged microtubule end-binding protein 1[END_REF]. While our adapted BglG system allowed the detection of MP mRNA within MP particles with unprecedented sensitivity, the BglG:mRFP rarely localized with SL-tagged mRNA to PD. However, given that BglG:mRFP showed cell-to-cell movement together with the bound MP mRNA, it is understandable that mRNA may not accumulate in PD but rather moves through the channel without being retained in the channel cavities. The reason why the MS2 system shows more frequent colocalization of MP mRNA with MP protein in PD than the BglG system is unclear. However, unlike the BglG protein, the MS2 CP showed a strong tendency to aggregate when its concentration in the cytoplasm was increased by deletion of its NLS. This tendency to aggregate may be increased by a local MS2 CP accumulation within the PD cavities, thereby provoking local entrapment of mRNA in an mRNA-MS2 CP aggregate. In addition, both RBPs must dimerize to recognize and bind to their respective RNA motifs [START_REF] Wu | Fluorescence fluctuation spectroscopy enables quantitative imaging of single mRNAs in living cells[END_REF][START_REF] Amster-Choder | Modulation of the dimerization of a transcriptional antiterminator protein by phosphorylation[END_REF]). The MS2 system thereby involves a 90 kDa complex placed at each SL position of the tagged mRNA. The size of this complex is larger than the BglG complex (60 kDa) and could impose steric hindrances for the passage through the pore. Cell-to-cell transport of mRNA without accumulation in the transport channel is consistent with mRNA transport through the nuclear pore complex in animal cells, in which the passage through the pore is not the rate limiting step despite the size of the packed mRNA-protein complex [START_REF] Grünwald | In vivo imaging of labelled endogenous β-actin mRNA during nucleocytoplasmic transport[END_REF][START_REF] Mor | Dynamics of single mRNP nucleocytoplasmic transport and export through the nuclear pore in living cells[END_REF].

While MP mRNA was always associated with MP, not all localizations of MP were associated with the presence of MP mRNA. Importantly, MP mRNA was localized to mobile MP particles but not to MP accumulating along MTs. Given that MP binds RNA in vitro in a non-sequencespecific manner [START_REF] Citovsky | The P30 movement protein of tobacco mosaic virus is a single-strand nucleic acid binding protein[END_REF], MP may bind any RNA in its vicinity, thus most likely its own coding RNA upon translation in association with the ER. Thus, the absence of MP mRNA from MP-associated MTs could be explained by a specific mechanism that traps the mRNA into MP particles during translation. However, the absence of MP mRNA from MPassociated MT may also indicate that the MP fractions associated with particles and MTs differ in RNA-binding activity. For example, the two MP fractions could be associated with different host factors or be differentially modified by posttranslational modifications. Support for this hypothesis is provided by the previous finding that MP becomes ubiquitinated during infection [START_REF] Reichel | Degradation of tobacco mosaic virus movement improved MS2 system for accurate reporting of the mRNA life cycle[END_REF], except for MP associated with MT [START_REF] Ashby | Tobacco mosaic virus movement protein functions as a structural microtubule-associated protein[END_REF].

MP is also a target for phosphorylation [START_REF] Waigmann | Regulation of plasmodesmal transport by phosphorylation of tobacco mosaic virus cell-to-cell movement protein[END_REF][START_REF] Citovsky | Phosphorylation of tobacco mosaic virus cell-to-cell movement protein by a developmentally regulated plant cell wall-associated protein kinase[END_REF][START_REF] Haley | Multiple serine phosphorylation sites on the 30 kDa TMV cell-to-cell movement protein synthesized in tobacco protoplasts[END_REF][START_REF] Kawakami | Defective tobamovirus movement protein lacking wild-type phosphorylation sites can be complemented by substitutions found in revertants[END_REF][START_REF] Watanabe | In vivo phosphorylation of the 30-kDa protein of tobacco mosaic virus[END_REF] and may therefore be differentially phosphorylated when associated with MT.

We believe that the BglG RNA labeling method adapted to plants in this work will facilitate the dissection of pathways involved in RNA transport and localization in plants. With respect to TMV and other viruses, this tool may allow the identification of cellular factors associated with MP particles and viral RNA movement. Importantly, we show that the BglG system can be used to visualize the cell-to-cell transport of in vivo expressed mRNA and, therefore, represents a fundamental tool to characterize RNA elements required for the transport of mobile transcripts.

Methods

Plasmid constructs

Constructs for expression of MP:RFP-SL (pMDC32-MP:mRFP-SL) and NLS:MS2CP:GFP (pNLS:MS2CP:GFP) where previously described [START_REF] Brandner | Tobacco mosaic virus movement protein interacts with green fluorescent protein-tagged microtubule end-binding protein 1[END_REF]. The construct for expression of MP:RFP-BoxB was created by Gateway cloning using the entry vector pZeo-MP:mRFP [START_REF] Brandner | Tobacco mosaic virus movement protein interacts with green fluorescent protein-tagged microtubule end-binding protein 1[END_REF]pSCJ215 (Schönberger et al., 2012) as destination vector. The binary plasmid for expression of λN:eGFP:NLS was described elsewhere (Schönberger et al., 2012). BglG:mCherry:NLS was PCR-amplified from pBglG-mCherry [START_REF] Chen | High efficiency of HIV-1 genomic RNA packaging and heterozygote formation revealed by single virion analysis[END_REF] using primers pBglG:mCh:NLSfw

(5'ggggacaagtttgtacaaaaaagcaggcttaatgaacatgcaaatcacc-3') and pBglG:mCh:NLSrev (5'ggggaccactttgtacaagaaagctgggtatacctttctcttcttttttggc-3'). The PCR product was cloned into pDONR™/Zeo using a BP clonase reaction (Thermo Fisher Scientific). The resulting entry vector pZeo-BglG:mCherry:NLS was confirmed by Sanger DNA sequencing and then used in an LR recombination reaction with the destination vector pMDC32 [START_REF] Curtis | A gateway cloning vector set for high-throughput functional analysis of genes in planta[END_REF] to obtain the binary expression vector pBglG:mCherry:NLS. A tandem of 18 BglG-SLs was PCR amplified from pBglG-SL [START_REF] Chen | High efficiency of HIV-1 genomic RNA packaging and heterozygote formation revealed by single virion analysis[END_REF] using primers BglG-SLfw (5'-ggggacaagtttgtacaaaaaagcaggcttatttaattaatgcagaattcgcccttgg-3') and BglG-SLrev

Protein expression in plants.

Cultures were grown over night at 28 ºC and at 180-200 rpm agitation on LB medium under appropriate antibiotic selection. Rifampicin (50 µg/ml) and gentamicin (50 µg/ml) for all the constructs expressed from GV3101 A. tumefaciens strain, kanamycin (50 µg/ml) for bacteria containing pSCJ (for the expression of λN:eGFP:NLS and MP:RFP-BoxB) and pMDC based vectors or Spectinomicin (100 µg/ml) for pB7 or pH7 based vectors (for the expression of NLS:MS2CP:eGFP and all the non NLS tagged versions of the RBPs). Cultures were harvested by centrifugation during 5 min at 5.000 x g, resuspended in sterile distilled water and diluted to specific optical densities (OD 600 ) before agroinfiltration. For experiments shown in Fig. 1 and Fig. 2, cultures with an OD 600 of 0.4 were used. For experiments in Fig. 3, MP-and RBP-encoding bacteria were used with an OD 600 of 0.4 and 0.1, respectively.

For expression of BglG:mRFP in isolated cells (Fig. 4), the bacteria were diluted to an OD 600 of 1,0 x 10 -4 whereas the MP:GFP-encoding cultures were used with an OD 600 of 0.4 to ensure expression throughout the infiltrated tissue. Leaves were co-infiltrated with agrobacteria containing pBin61-P19 (kanamycin resistance) (Silhavy et al., 2002) for coexpression of the tombusvirus protein P19 (OD 600 = 0.2) to inhibit post-transcriptional RNA silencing and ensure efficient expression of the constructs. Cultures were infiltrated into the abaxial side of fully expanded leaves of 4-5 weeks old Nicotiana benthamiana plants using 2.0 ml syringes without needles. For all RNA labeling experiments, one-half of the leaf was used for the expression of the SL-tagged mRNA while the other half was used for the corresponding non-tagged control. Plants were grown and maintained in growth chambers, with 16h, 24 ºC / 8h, 22 ºC day/night cycles.

Transgenic N. benthamiana plants were generated by leaf disk transformation (Horsch et al., 1985) using A. tumefaciens GV3101 bacteria carrying the plasmid pH7-BglG:mRFP.

Fluorescence microscopy and image processing

For analysis of leaf tissues by fluorescence microscopy, leaf discs were excised and mounted on microscope slides and vacuum-infiltrated with water. The leaf discs were excised at 48-72 hours post infiltration (hpi) for experiments shown in Figure 1 and Figure 2, at 36-48 hpi for experiments shown in Figure 3, and at 96 hpi for experiments shown in Figure 4. Abaxial epidermal cells were imaged using a Zeiss LSM780 confocal microscope using excitation/emission wavelengths of 488/505-545 and 543/585-615 for the imaging of GFP and RFP/mCherry, respectively. Images were processed using ImageJ (version 1.51j8) software. Particles were tracked in movies using the ImageJ macro "OverlayCrosses" (https://git.io/fjnDF).

Immunoprecipitation and protein analysis

40 hs after infiltration, 0,5 g of tissue belonging to leaf material expressing either MP:GFP or MP:GFP-SL together with BglG:mRFP were ground to fine powder in liquid N2 and resuspended in 1.0 ml of ice-cold extraction buffer (20 mM Tris/HCL pH 7.5, 150 mM NaCl, 20 0.5 mM EDTA, 0.5 % NP40, 1 x Complete Protease Inhibitor (PI, Roche)). The lysate was cleared by 5 min centrifugation at 20.000 x g at 4 ºC, diluted with one volume of ice-cold dilution buffer (20 mM Tris/HCL pH 7.5, 150 mM NaCl, 0.5 mM EDTA 1 x PI), and an aliquot reserved for immunoblot analysis (input fraction, IN). Cleared and diluted protein extract was incubated with 20 µl of GFP-Trap agarose beads (Chromotek, Germany) for 1.5 h at 4 ºC with gentle end-over-end agitation. Beads were collected by centrifugation at 2500 x g for 2 min at 4 ºC. The supernatant was placed aside for immunoblot analysis (flow-through fraction, FT) and beads were washed two times with washing buffer (20 mM Tris/HCL pH 7.5, 150 mM NaCl, 0.5 mM EDTA). Beads were resuspended in 50 µl of 2X Laemmli buffer, boiled for 10 min and collected by centrifugation at 2500 x g for 2 min at 4 ºC, and proteins in the supernatant were collected and used for immunoblot analysis (immunoprecipitated fraction, IP). IN, FT and IP protein samples were separated by 12 % SDS-PAGE and blotted onto PVDF membrane for immunoblot detection. RFP-fused proteins were detected with anti-RFP 6G6 mouse monoclonal antibody (Chromotek, Germany), and the TMV MP was detected using MP antibodies raised in rabbit (Boyko et al., 2000c). Horseradish peroxidase conjugated anti-mouse (BioRad, USA) and anti-rabbit (Invitrogen) were used as secondary antibodies. Chemiluminescent reagent was used for detection of peroxidase activity according to the manufacturer's instructions (GE, ECL Plus Western Blotting Detection Reagents, UK). Chemiluminescent signal intensities of immunoreactive bands were determined within digital images using ImageJ (version 1.51j8) software. Average pixel intensities within specific regions of interest (ROI) applied to bands of samples containing BglG:SL-tagged mRNA were corrected by the average pixel intensities measured within the same ROI size above or below the measured bands (background). The same procedure was performed with ROIs of the same size applied to corresponding bands of samples containing non-tagged mRNA. Results are displayed as the ratio between the corrected average pixel intensities of both samples.

Statistical analysis

Intensity values of BglG:mRFP immunoreactive bands shown in Figure 3 were compared with an unpaired t-test using GraphPad prism (V 6.01).

The statistical probability for the independent transformation of adjacent cells by diluted agrobacteria was calculated by Poisson statistics expressed as P(x) = (e -µ µ x )/x!, where "x"

is the number of these occurrences [number of independently transformed cells (B) found adjacent to initially transformed cells (A)] and "µ" is the ratio between adjacent cells Nine of these spots consisted of more than one fluorescent epidermal cell (see Table S1).

According to the Poisson distribution, the likelihood in a tissue of 1400 cells that nine out of (9 + 28 =) 37 transformation events occur in adjacent cells is extremely unlikely. Considering that each epidermal cell has four adjacent epidermal cells, the 28 initially transformed cells (A) have 112 adjacent cells (C). Given that and the number of all cells (D) is 1400, the number of cells that are initially transformed (A) is 28 and the number of neighboring cells concerned (C) is 112, the probability to obtain nine independent transformation events (B) adjacent to 28 initially transformed cells (A) is P( 9 ) = (e -(112/1260) x (112/1260) 9 )/9! = 8,73 x 10 -16 . Under the assumption that each cell has seven adjacent the probability is P( 9 ) = (e - (196/1176) x (196/1176) 9 )/9! = 2,3 x 10 -13 .

( 5 '

 5 -ggggaccactttgtacaagaaagctgggtagagctcaattcgcccttaggtcgc-3'). The PCR product was cloned into pDONR™/Zeo by a BP clonase reaction. The resulting vector was confirmed by Sanger DNA sequencing and digested with PacI/SacI restriction endonucleases (NewEngland Biolabs, MA USA). The released fragment was separated by electrophoresis, purified from the agarose gel and ligated by T4-DNA ligase into PacI/SacI digested pMDC32 destination vector[START_REF] Curtis | A gateway cloning vector set for high-throughput functional analysis of genes in planta[END_REF] to obtain pMDC32-BglG-SLs. Similarly, the coding region of MP:GFP was PCR-amplified from pTMV-MP:GFP5(Boyko et al., 2000c) using primers MPGFPfw (5'-ggggacaagtttgtacaaaaaagcaggcttaatggctctagttgttaaa-3') and MPGFPrev (5'-ggggaccactttgtacaagaaagctgggtatttagcggccgcgactagttca-3'). The product was cloned into pDONR™/Zeo by a BP clonase reaction (Thermo Fisher Scientific).The resulting entry clone was confirmed by Sanger DNA sequencing and used in an LR recombination together with either pMDC32 or pMDC32-BglG-SL to obtain the binary vectors pMP:GFP or pMP:GFP-SL. To express the different RBPs without NLS, the RBP-for BglG. The PCR products were cloned into pDONR™/Zeo, confirmed by Sanger DNA sequencing, and used for recombination with either pB7FWG2 or pH7RWG2[START_REF] Karimi | GATEWAY TM vectors for Agrobacteriummediated plant transformation[END_REF] using LR clonase to obtain the C-terminal fusion to eGFP (pB7-MS2CP:eGFP and pB7-λN:eGFP) or mRFP (pH7-BglG:mRFP). All plasmids were selected and propagated in E. coli (Top 10 strain, Invitrogen) and transferred by electroporation into Agrobacterium tumefaciens (strain GV3101).

  concerned (C) divided by the number of all cells (D) subtracted by the sum of initially transformed cells (A) and adjacent cells (C). Leaf disks analyzed (9,0 mm diameter) in the experiments described in Figure 4 contained approximately 1400 epidermal cells (abaxial side) and each cell had four to seven adjacent cells. MP:GFP-SL-expressing leaf discs transformed with diluted agrobacteria to express BglG:mRFP in isolated cells led to the occurrence of 28 red fluorescent, BglG:mRFP-expressing spots of cells on the average.

andFigure 1 .

 1 FIGURE LEGENDS

Figure 2 .

 2 Figure 2. Accumulation patterns of the different RBPs expressed without a NLS. a, Central views of N. benthamiana epidermal cells expressing MS2CP:eGFP (left), λN:eGFP (middle), or BglG:mRFP

Figure 3 .

 3 Figure 3. Detection of MP:GFP mRNA using the BglG system. a, cortical view of a N. benthamiana

Figure 4 .

 4 Figure 4. Cell-to-cell movement of MP mRNA. a, Agroinfiltrated N. benthamiana leaf sample

Figure 1 .

 1 Figure 1. Systems used for the in vivo imaging of TMV MP mRNA. a, MS2 system. Top, representation of transcripts (blue line) and encoded proteins (boxes). Middle, cortical view of a N. benthamiana leaf epidermal cell showing NLS:MS2CP:GFP-labeled MP:RFP mRNA (green channel, left), MP:RFP (red channel, center) and the colocalization of the RNA and protein (yellowish signal in the merged figure, right). The colocalization in MP particles is indicated by arrowheads. Bottom, central view of a cell from the same experiment showing NLS:MS2CP:GFP-labeled MP:RFP mRNA (left) and MP:RFP (center) colocalizing at PD (right; examples of PD localization are indicated by arrows). b, λN system. Top, representation of transcripts (blue line) and encoded protein (boxes). Middle, cortical view of a N. benthamiana epidermal cell showing λN:eGFP:NLS-labeled MP:RFP mRNA (green channel, left), MP:RFP (red channel, center), and colocalization of the RNA and protein (yellow signal in the merged channel, right). Examples of colocalization in MP particles are indicated by arrowheads. Bottom, central view of a cell from the same experiment showing the localization of λN:eGFP:NLS-labeled MP:RFP mRNA (green channel, left), MP:RFP (red channel, center), and colocalization in the merged image (yellowish signal, right; examples of PD localization are indicated by arrows). c, BglG system. Top, representation of transcripts (blue line) and encoded proteins (boxes). Middle,

Figure 3 .

 3 Figure 3. Detection of MP:GFP mRNA using the BglG system. a, cortical view of a N. benthamiana epidermal cell expressing MP:GFP (green channel, left) and BglG:mRFP (red channel, center). The merged image is shown on the right. b, cortical view of a N. benthamiana epidermal cell expressing MP:GFP from mRNA tagged with BglG-SLs (MP:GFP-SL, green channel, left) together with BglG:mRFP (red channel, middle). The merged image shows co-localization of MP:GFP mRNA and protein (yellow signal, right). c, central view of a N. benthamiana epidermal cell expressing MP:GFP (green channel, left) and BglG:mRFP (red channel, center). The merged image is shown on the right. d, central view of a N. benthamiana epidermal cell expressing MP:GFP-SL (green channel, left) and BglG:mRFP (red channel, center). The merged image is shown on the right. Arrowheads in c and d point to MP:GFP spots at the cell periphery consistent with localization to PD. Arrows in d point to colocalized signals of MP:GFP and BglG:mRFP indicating the colocalization of MP:GFP and MP:GFP-SL mRNA at PD. Scale bars, 5 µm. e, BglG:mRFP coimmunoprecipitates with MP:GFP and its SL-tagged mRNA. BglG:mRFP expressed together with either MP:GFP or MP:GFP-SL subjected to coimmunoprecipitation with GFP-specific antibodies. Input (IN), flowthrough (FT), and immunoprecipitated (IP) fractions were analyzed with anti-RFP (top) antibodies for

Figure 4 .

 4 Figure 4. Cell-to-cell movement of MP mRNA. a, Agroinfiltrated N. benthamiana leaf sample expressing BglG:mRFP in isolated cells and either MP:GFP (left) or MP:GFP-SL (right) in all cells (GFP channel not shown). b, Number of red fluorescent foci consisting of two or more epidermal cells. Average values and SD are shown. *, significantly different (unpaired t-test, p < 0,0001, n > 400). Data are derived from three experimental replicates. Scale bar, 100 µm.

The Plant Journal

Acknowledgements

We thank Thomas Dresselhaus (Cell Biology and Plant Biochemistry, University of Regensburg, Germany) for providing plasmids pSCJ215 and pSCJ206 and Wei-Shau Hu (HIV Drug Resistance Program, NCI Frederick, USA) for providing plasmids pBglG-SL and pBglG:mCherry. We thank Jérôme Mutterer (IBMP-CNRS, Strasbourg, France) for support in preparing movies and Claudia Ruscitti (Departamento de Matemática, Facultad de Ciencias Exactas-UNLP, Argentina) for help in the statistical analysis of mRNA cell-to-cell transport. This work was performed with financial support from the ANR (grant ANR-08-BLAN-0244-02) to MH, CONICET (grant PIP 2015-0847) to EJP, and CNRS-CONICET (PICS 2014) to MH and EJP.

Data availability statement

All relevant data can be found within the manuscript and its supporting materials. A documentation of the three replicate immunoprecipitation experiments used for quantification shown in Fig. 3f is available at https://doi.org/10.5281/zenodo.4047215.

Author contributions

EJP and MH conceived the idea and designed the study, EJP and GRL performed the experiments, EJP, GRL and MH analyzed the results, EJP and MH wrote the manuscript.

All authors approved the manuscript prior to publication.

Conflicts of interest

The authors declare no conflicts of interest.

Supporting