N Bonnefoy

C Remacle
email: c.remacle@uliege.be

Biolistic transformation of Chlamydomonas reinhardtii and Saccharomyces cerevisiae mitochondria

Keywords: mitochondrial DNA, genetic transformation, biolistic technics, homologous recombination, mutagenesis and ectopic gene insertion, Chlamydomonas reinhardtii, Saccharomyces cerevisiae

Chlamydomonas reinhardtii and Saccharomyces cerevisiae are currently the two microorganisms in which genetic transformation of mitochondria is routinely made. The generation of a large variety of defined alterations as well as the insertion of ectopic genes in the mitochondrial genome (mtDNA) are possible, especially in yeast. Biolistic transformation of mitochondria is achieved through the bombardment of microprojectiles coated with DNA, which can be incorporated into mtDNA thanks to the highly efficient homologous recombination machinery present in S. cerevisiae and C. reinhardtii organelles. Despite a low frequency of transformation, the isolation of transformants in yeast is relatively quick and easy since several natural or artificial selectable markers are available, while the selection in C. reinhardtii remains long and awaits for new markers. Here we describe the materials and technics used to perform biolistic transformation, in order to mutagenize endogenous mitochondrial genes or insert novel markers into mtDNA. Although alternative strategies to edit mtDNA are being set up, so far insertion of ectopic genes relies on the biolistic transformation technics.

Introduction

Transformation of the mitochondrial genome can be achieved in two unicellular microorganisms: originally the yeast Saccharomyces cerevisiae and later on the green microalga Chlamydomonas reinhardtii [START_REF] Bonnefoy | Genetic transformation of Saccharomyces cerevisiae and Chlamydomonas reinhardtii mitochondria[END_REF].

In both systems, donor DNA is delivered into mitochondria by microprojectile bombardment (biolistic transformation) and the DNA is inserted into the recipient mitochondrial genome by homologous recombination. The use of a biolistic particle delivery system where thousands of microprojectiles are coated with multiples copies of the donor DNA is an efficient way to target the multicopy mitochondrial genomes since it increases the chances to deliver enough copies of the donor DNA to invade the endogenous polyploid genome. In addition, the use of the adequate size of the microparticles and rupture disk determining the power of the helium shockwave allows the particles to cross both the cell and mitochondrial membranes, as discussed in [START_REF] Boynton | Chloroplast transformation in Chlamydomonas[END_REF].

In other organisms, mitochondrial transformation has been unsuccessful, but DNA editing of the mitochondrial genome is becoming feasible in both animals [START_REF] Lee | Mitochondrial DNA editing in mice with DddA-TALE fusion deaminases[END_REF] and plants [START_REF] Li | Advancing organelle genome transformation and editing for crop improvement[END_REF], using sequence-specific nucleases targeted to mitochondria. However so far insertion of novel genes into the mitochondrial genome is still only possible through biolistic transformation, i.e. in S. cerevisiae and C. reinhardtii, and should potentially be achievable in Candida glabrata where biolistic transformation has also been reported [START_REF] Zhou | Mitochondrial DNA heteroplasmy in Candida glabrata after mitochondrial transformation[END_REF].

Chlamydomonas mitochondrial transformation

The first demonstration of mitochondrial transformation in Chlamydomonas has been published in 1993 [START_REF] Randolph-Anderson | Further characterization of the respiratory deficient dum-1 mutation of Chlamydomonas reinhardtii and its use as a recipient for mitochondrial transformation[END_REF]. The authors used a biolistic delivery system to transform a mutant called dum1, that harbors a 1.5 kb terminal deletion of the linear 15.8 kb mitochondrial genome comprising the telomere (500 bp) and most of the cob gene (1 kb) encoding the apocytochrome b (Figure 1). In addition to this deleted genome, the mutant also contains dimers which result from head-to-head fusions between deleted monomers [START_REF] Randolph-Anderson | Further characterization of the respiratory deficient dum-1 mutation of Chlamydomonas reinhardtii and its use as a recipient for mitochondrial transformation[END_REF][START_REF] Matagne | Induction and characterization of mitochondrial DNA mutants in Chlamydomonas reinhardtii[END_REF]. The dum1 mutant is unable to grow in the dark in the presence of acetate because of the loss of the activity of complex III and is a non-reverting mutant, which after selection in the dark allows the recovery of rare respiratory competent cells with intact mtDNA.

The next step in the set-up of mitochondrial transformation was reached in 2005 when Yamasaki and coauthors [START_REF] Yamasaki | Shared molecular characteristics of successfully transformed mitochondrial genomes in Chlamydomonas reinhardtii[END_REF] reported that the same deletion mutant could be transformed using plasmids containing fragments of mtDNA as well as PCR products, again using a biolistic equipment. However, the transformation rates were still low (Table 1). In 2006, we published an optimized protocol for mitochondrial transformation using both PCR fragments and plasmids containing mitochondrial genome fragments of various sizes, using the same strategy of selection in the dark and another deletion mutant as recipient strain (dum11) which is essentially the same as dum1 except that the deletion in the cob gene is shorter (0.7 kb) [START_REF] Remacle | High-efficiency biolistic transformation of Chlamydomonas mitochondria can be used to insert mutations in complex I genes[END_REF] (Table 1, Figure 1). 1 Transformation rates based on data found in the respective papers 2 Transformation rates expressed in number of transformants per 10 7 cells 3 Several combinations of gold particles of different sizes have been tested and this combination gave the best results 4 Best transformation rate for undigested plasmid containing a 5.0 kb mtDNA fragment (left telomere-cob-Ctermnd4/Ntermnd5-cox1/right telomere) using dum1 as recipient strain 5 PCR product from the left telomere to nd4 6 Best transformation rate for linearized plasmid containing a 6.6 kb mtDNA fragment (left telomere to cox1, see Figure 1) using dum11 as recipient strain, expressed in number of transformants/µg DNA 7 Best transformation rate for a 6.6 kb mtDNA PCR fragment (left telomere to cox1, see Figure 1) using dum11, expressed in number of transformants/µg DNA Mutations in genes encoding subunits of respiratory complex I and a mutation conferring resistance to myxothiazol in cob could be introduced. At last, a mutant with intact telomeres at both sides of the genome could be transformed. Afterwards, another recipient strain with a longer deletion extending until the nd4 gene (Figure 1) was used to force the introduction of mutations in this gene [START_REF] Larosa | Reconstruction of a human mitochondrial complex I mutation in the unicellular green alga Chlamydomonas[END_REF][START_REF] Salinas | Co-evolution of mitochondrial tRNA import and codon usage determines translational efficiency in the green alga Chlamydomonas[END_REF], although the transformation rates were much lower (Table 1). In general, most of the transformants were homoplasmic for the mitochondrial genome although heteroplasmic ones could also be recovered, that ultimately could reach the homoplasmic state after subcloning [START_REF] Salinas | Co-evolution of mitochondrial tRNA import and codon usage determines translational efficiency in the green alga Chlamydomonas[END_REF].

Table 1: Experimental conditions for mitochondrial transformation in C. reinhardtii Strain

Beads

Finally, two reporter genes, gfp and ble conferring resistance to zeomycin, have since then been successfully inserted between the left telomere and cob using the dum1 strain as recipient [START_REF] Hu | Stable expression of antibiotic-resistant gene ble from Streptoalloteichus hindustanus in the mitochondria of Chlamydomonas reinhardtii[END_REF][START_REF] Hu | Successful expression of heterologous egfp gene in the mitochondria of a photosynthetic eukaryote Chlamydomonas reinhardtii[END_REF]. Although these results open the way to ectopic gene insertion, there was no follow up.

Yeast mitochondrial transformation

S. cerevisiae mtDNA, called the rho factor, has proved to be particularly amenable to manipulation, for several reasons. First S. cerevisiae can be fully devoid of mtDNA (rho 0 ) but can also tolerate and maintain rhogenomes, that contain large deletions. Second, there does not seem to be any clear requirement for a specific replication origin in mitochondrial rhosequences, thus allowing even foreign DNA to be maintained. In rho-, the remaining sequences are reiterated so that the rhocells contain the same amount of mtDNA as wildtype cells. Third, rhogenomes can be expressed in trans and are able to freely undergo homologous recombination with mtDNA sequences in crosses. These properties have proved to be invaluable to set up mitochondrial transformation in S. cerevisiae in order to manipulate the mitochondrial genome.

The first proof of concept of mitochondrial transformation has been obtained using a particle gun to bombard sequences of the wild-type COX1 gene into a strain with a full mtDNA (so-called rho+), harboring a non-reverting deletion in the mitochondrial COX1 (oxi3) gene. A fully wild-type strain with normal respiratory function was restored [START_REF] Johnston | Mitochondrial transformation in yeast by bombardment with microprojectiles[END_REF].

The same year, a similar biolistic transformation of a rho 0 strain (i.e. devoid of mtDNA) with a plasmid containing the mitochondrial COX2 (oxi1) was achieved, and shown to rapidly form what was called a synthetic rhogenome. This artificial rho-mtDNA was carrying the COX2 gene and could restore respiratory function upon cross with a rho+ strain harboring a point mutation or deletion in COX2.

These two founding experiments opened the way to the manipulation of the yeast mitochondrial genome virtually at will, and today, 33 years later, these two strategies are still used to mutagenize or insert new genes into mtDNA. Direct transformation into a rho+ strain requires that a selection can be applied to identify transformation and recombination events. To contrary, transformation in two steps does not depend on a selection. First a synthetic rhois obtained after transformation of a rho 0 strain, then the modified or ectopic sequences are inserted into an otherwise rho+ genome by homologous recombination, through crosses (generating first a diploid) or cytoduction (generating directly a haploid strain without mixing the nuclear backgrounds).

Although a recent publication reported the use of the Crispr Cas9 system to mutagenize yeast mtDNA [START_REF] Yoo | Cas9/gRNA-mediated genome editing of yeast mitochondria and Chlamydomonas chloroplasts[END_REF], the only way so far to modify the genome arrangement or introduce ectopic genes in mtDNA depends on biolistic transformation. A maybe not exhaustive overview of the ectopic genes inserted into S. cerevisiae mtDNA is provided in Table 2. Amongst these various reports, the two main fundamental breakthroughs were the engineering of a site upstream of COX2 for insertions of ectopic genes while preserving respiratory function [START_REF] Thorsness | Nuclear mutations in Saccharomyces cerevisiae that affect the escape of DNA from mitochondria to the nucleus[END_REF], and the recoding of a mitochondrially-expressed ARG8 auxotrophy gene [START_REF] Steele | Expression of a recoded nuclear gene inserted into yeast mitochondrial DNA is limited by mRNA-specific translational activation[END_REF], that has been the most widely used mitochondrial reporter, inserted at many sites in the mtDNA and used for a wide variety of studies (Table 2). However, since synthetic genes can nowadays be ordered easily, many novel genes and locations of the genome are open to the creativity of researchers working on mitochondrial functions. Below we will describe the materials needed, protocols to follow and useful tips to achieve efficient mitochondrial transformation using the biolistic method. Many steps are common between C. reinhardtii and S. cerevisiae, however there are some specificities; in this case, when numbering some steps which are organism-specific, we will precise "Cr" for Chlamydomonas or "Sc" for budding yeast.

Materials

Strains

1. Cr. Deletion mutant are typically used, such as dum1, dum11, dum14, dum16 and dum22 (see Note 1). 2. Sc. The rho 0 or rho+ recipient strain used are preferably kar1-1 (to allow subsequent cytoduction) and of DBY947 genetic background (For example DFS160 [START_REF] Steele | Expression of a recoded nuclear gene inserted into yeast mitochondrial DNA is limited by mRNA-specific translational activation[END_REF], see Note 2).

Media and cell preparation

1. Cr. Liquid and solid TAP (Tris-Acetate-Phosphate) medium: 2.42‰ (w/v) Tris buffered at pH 6.8-7.0 with 1‰ (v/v) glacial acetic acid, 0,108‰ (w/v) K2HPO4 and 0.056‰ (w/v) KH2PO4, supplemented with a complex set of oligo-elements [START_REF] Harris | The Chlamydomonas sourcebook: a comprehensive guide to biology and laboratory use[END_REF]. Acetate is used as a reduced organic carbon source to sustain growth in the dark (see Note 3). 2. Cr: Solid TAP plates: as liquid TAP but with 1.5% (w/v) agar, 1 cm thick and freshly made (1-2 days before the bombardment). 3. Sc. YPR: 1% (w/v) yeast extract, 2% (w/v) Bacto™ peptone, 2% (w/v) raffinose, 0.1% (w/v) glucose, 40 μg/mL adenine (see Note 4). 4. Sc. YPD: 1% (w/v) yeast extract, 2% (w/v) Bacto™ peptone, 2% (w/v) glucose, 40 μg/mL adenine. 5. Sc. SD-sorbitol: 0.67% (w/v) yeast nitrogen base, 5% (w/v) glucose, 1 M sorbitol, 100 μg/mL adenine, 3% (w/v) agar, other supplements as required 40 μg/μL except for the selection marker. Pour a few days before the bombardment and dry moderately. 6. Sc. YPEG-Sorbitol: 1% (w/v) yeast extract, 2% (w/v) Bacto™ peptone, 3% (w/v) glycerol, 3% (w/v) ethanol, 0.1% (w/v) glucose, 1M sorbitol; 3% (w/v) agar (see Note 5). 7. Sterile glass tubes and flasks and shaker at 28 to 30°C. 8. Cr. White light neon/LED tubes at 75 µmol photons. m-2 .s -1 in a cultivation chamber at 23-25°C. 9. Room temperature centrifuge with sterilized bottles or sterile plastic 50 ml tubes.

Preparation of microprojectiles and DNA precipitation

1. Room temperature and refrigerated centrifuge to spin 1.5 ml tubes at 16000 g. 2. Tungsten powder <1 μm 99.95% (metals basis; Alfa Aesar/Johnson Matthey, item 44210, CAS 7440-33-7; see Note 6). 3. 70% (v/v with sterile water) or 100% Ethanol, room temperature. 4. Sterile water. 5. Glycerol: 50% (w/v), sterilized by autoclaving, -20°C. 6. DNA for mitochondrial transformation of Chlamydomonas or yeast (Note 7). 7. Sc. Required for yeast rho 0 strains and optional for rho+ strains: Qiagen-based midi-or maxipreps (>2 μg/μL) of a plasmidic DNA for nuclear transformation, carrying a nuclear selection marker and a nuclear replication origin; we often use the vector YEp351 [START_REF] Hill | Yeast/E. coli shuttle vectors with multiple unique restriction sites[END_REF] or pRS315 [START_REF] Sikorski | A system of shuttle vectors and yeast host strains designed for efficient manipulation of DNA in Saccharomyces cerevisiae[END_REF] for the LEU2 marker. 8. Spermidine free base (Sigma): 1 M in sterile water. 9. Calcium chloride: 2.5 M in water, kept at -20°C (Note 8), filter-sterilized. 10. Ethanol 100%, freezer stock (Note 9).

Bombardment

Identification of mitochondrial transformants

1. Cr. Proteinase K (20 mg/mL in water), stored at -20 °C. 2. Cr. Thermocycler and PCR consumables 3. Sc. rho+ mittester strain carrying a mitochondrial mutation in the region covered by the DNA for mitochondrial transformation and of mating type opposite to the recipient rho 0 strain, for example of strain currently used see [START_REF] Bonnefoy | In vivo analysis of mutated initiation codons in the mitochondrial COX2 gene of Saccharomyces cerevisiae fused to the reporter gene ARG8 m reveals lack of downstream reinitiation[END_REF] (Note 13). 4. SD plates: 0.67% (w/v) yeast nitrogen base, 2% (w/v) glucose, 2% (w/v) agar; supplements as required at 40 μg/μL. 5. YPD plates: like liquid YPD but with 2% (w/v) agar. 6. YPEG plates: 1% (w/v) yeast extract, 2% (w/v) Bacto peptone, 3% (w/v) glycerol, 3% ethanol, 2% agar (Note 14).

Mating and isolation of yeast recombinant cytoductants

For mating and isolation of recombinant cytoductants, use a recipient wild-type rho+ strain or a mit-rho+ strain. To recover haploid recombinants, this strain must carry the kar1-1 mutation unless the rho0 strain used for the transformation (Subheading 2.1., item 2) is already kar1-1.

Methods

Overview of the transformation procedure

The device used for the microprojectile bombardment is the PDS-1000/He System, available from Bio-Rad, Inc. A detailed protocol is available on line (https://www.biorad.com/webroot/web/pdf/lsr/literature/M1652249.pdf). The system uses a high helium burst in an evacuated chamber to break a rupture disk which propels at optimal velocity a macrocarrier disk, loaded with DNA-coated microscopic metal particles, towards target cells spread on a Petri plate. Upon bombardment, the DNA introduced into the cells is taken up by the nucleus, and to a lesser extent by the mitochondria.

Cell culture

1. Cr. Cultivate the recipient strain for 2-3 days in liquid TAP medium under continuous light (50-70 µmol photons. m-2 .s -1 ) and mild agitation to reach mid-exponential phase (cell density of 2-5 10 6 cells/mL). Typically, 300 mL of culture give between 6 and 15 plates at 10 8 cells/plate. 2. Cr. Harvest the cells (2700 g, 10 min, 25ºC) in sterilized bottles, resuspend the pellet in sterile TAP at 5 x 10 8 cells/mL. 3. Cr. Spread 200 µL on freshly prepared plates. Let the plates dry moderately in the laminar hood. 4. Sc. Inoculate a colony of the cell to bombard in a few ml of liquid YPR and grow for 1 night to 2 days at 30°C with agitation. 5. Sc. Inoculate a flask containing a larger YPR volume at 1/100 and grow for 2 to 3 days at 30°C under agitation (Note 15). A 30 ml to 50 mL culture will be enough for 6 bombardments. 6. Sc. Harvest cells (3000 rpm, 5 minutes, room temperature) in sterile 50 mL tubes and resuspend in sterile YPD to reach a cell density of 1-5.10 9 cells/mL before spreading on plates. Typically, 30 mL of YPR culture resuspended with 600 µl of YPD are spread onto 6 plates (Note 16) of appropriate Sorbitol medium (SD-Sorbitol with supplements except for the nuclear transformation marker, or YPEG-Sorbitol), between 1 and 3 hours before the bombardment.

Particle sterilization

1. Weigh 10 to 50 mg of tungsten particles (1mg = 1 shot) in a 1.5 mL microcentrifuge tube. 2. Add 1.5 mL of 70% ethanol (100% ethanol for gold particles), vortex and incubate 10 min at room temperature. 3. Spin down the particles for 15 min at 13000 rpm at room temperature and carefully remove the supernatant. 4. Add 1.5 mL of sterile water, vortex and repeat step 3 immediately. 5. Resuspend at 60 mg/mL in frozen 50% glycerol and keep on ice until needed (Note 17).

DNA coating

The protocol given below is intended for 6 bombardments (Note 18). Precipitation is conducted on ice, with ice-cold or freezer-stocks of reagents, and the rotor of the centrifuge should be pre-cooled at 2-4°C and used at full speed (e.g. 16000 g). Vortex each time a reagent is added. 1. Cr. Prepare a tube with 20 µg of linearized plasmid (1-2µg/µL) or 10 µg of PCR product (1-2 µg/µL) (Note 18) 2. Sc. Prepare a tube with 5 µg of plasmid with the nuclear selection marker (e.g. YEp351) and 15 to 30 µg of plasmid or PCR DNA for mitochondrial transformation (Note 19). Omit the first plasmid if bombarding cells plated on YPEG-Sorbitol. 3. Add 100 µL of tungsten particles (Note 20) to the DNA and mix, precipitate the DNA onto the particles by adding of 4 µL of 1 M spermidine and 100 µL of freezer-stored 2.5 M CaCl2, vortexing immediately after each addition. Incubate on ice 10 to 15 min and vortex occasionally (every 2-3 min). 4. Spin down 30s in the pre-cooled centrifuge; discard the supernatant and add 200 µL of freezer-stored 100 % ethanol. The aggregates of particles on the side of the tube should be scraped and broken using the pipette tip (Note 21). Repeat this step at least once and possibly more until the particles resuspend easily. 5. Spin down the tube briefly in the pre-cooled centrifuge; discard the supernatant, taking extreme care not to pipette tungsten particles, and add 60 µL of 100% ethanol. The suspension is distributed evenly on 6 macrocarriers (Note 22) placed in their holders and the ethanol is evaporated under the hood.

Bombardment (supplementary video)

1. Wash the chamber and all removable parts with 70% ethanol before use (Note 23). Carefully wipe dry all parts since the remaining ethanol will prevent building the strongest possible vacuum. 2. Open the helium gas connection and switch on the vacuum pump. 3. Insert the rupture disk (1100 psi = 7584 kPa) in the retaining cap using sterile forceps and tighten the assembly using the torque wrench (Note 24). 4. Cr. For Chlamydomonas transformation only, insert a stopping screen on the macrocarrier assembly system (Note 25). 5. Take the macrocarrier loaded in its holder and load it onto the assembly system so that the face with DNA looks down. 6. Remove the lid or the 100 mm Petri dish with the lawn of cells and place at 7 cm (Chlamydomonas, level 3) or 5 cm (yeast, level 2) from the macrocarrier assembly system (Note 26). 7. Close the door with the latch 8. Press the "vac" button to raise the vacuum and reach a partial vacuum corresponding to 29-29.5 inches Hg (736.6 mm Hg) on the PDS1000-He gage (the higher the better, Note 27). 9. Block the chamber evacuation by quickly pressing on the "hold" button. 10. Press the "fire" button. The PDS1000-He gage pressure gage will raise until the rupture disk bursts open noisily; release the "fire" button immediately. 11. Ventilate the chamber with the "vent" button. If applicable eliminate large macrocarrier fragments laying on the Petri plate with a sterile forceps. 12. Cr. Incubate Chlamydomonas plates for one day under continuous light (50-70 µmol photons. m-2 .s -1 ) at 23-25 °C, transfer in the dark and incubate 4-8 weeks for selection of respiratory competency. 13. Sc. Incubate yeast plates at 28 to 30°C until transformants appear (usually 3 to 5 days).

Identification of Chlamydomonas transformants

1. Design primers for amplification of a PCR product allowing the identification of the recombinant clones. 2. Check the plates after 4-6 weeks under a stereoscopic microscope to detect microcolonies. 3. After 8 weeks, transfer the plates to dim light (5 µmol photons. m-2 .s -1 ) for one week. Transformants are usually found at the periphery of the plates while the control plates are empty. 4. Transfer the colonies to fresh TAP plates. 5. Prepare whole cell extracts (adapted from Randolph et al 1993): transfer 0.5 cm 2 of patches of Chlamydomonas into a microcentrifuge tube and add 35 mL of H2O, 5 mL of PCR buffer 5X, and 10 mL of proteinase K (20 mg/mL in H2O). 6. Incubate 1 h at 58°C and 1 h at 95 °C. 7. Spin down briefly with a benchtop microfuge and store the crude extracts at 4°C. 8. Use 1/10 (v/v) of the crude extract for PCR experiments. 9. Sequence the PCR fragment if needed.

Identification of yeast transformants

After bombardment of a rho 0 recipient strain (Figure 2) 1. Three days after the bombardment, inoculate an appropriate rho+ mutant (mit-) tester strain in liquid YPD to set up a fresh culture. 2. When transformants are grown, prepare plates of SD medium with appropriate supplements and spread a lawn of tester strain on YPD plates. 3. Replica plate the transformants first onto the minimal SD medium, to keep the transformants, and then onto the tester strain lawn to cross the transformants. Mark each plate very precisely to facilitate step 6. 4. Incubate 2 days at 30°C to allow mating and recombination. 5. Replica-plate the YPD cross plate to YPEG medium (or another appropriate medium if scoring another phenotype), in order to detect respiring diploids (see Note 28). Incubate for 2 to 3 days at 30°C (see Note 29). 6. Compare the position of the respiring diploids and of the original transformants to pick, off the bombarded plate (or its direct replica), colonies that correspond to the position of respiring recombinants. Favor positions where transformants are not too dense. 7. Streak the transformants on YPD (on which they will lose the plasmid carrying the nuclear marker) and incubate at 30°C to get single colonies. Repeat the cross and marker rescue analysis with the tester strain as in steps 3 to 6. Usually these cycles of subcloning and retesting must be done three times before obtaining pure stable synthetic rho-clones.

After bombardment of a rho+ recipient strain 8. After a bombardment of cells plated directly on YPEG-Sorbitol medium, only the few colonies that underwent mitochondrial transformation restoring respiratory function will grow. 9. After a bombardment of cells plated on SD-Sorbitol medium to first select nuclear transformants, replica-plate the nuclear transformation directly onto glycerol medium (YPEG or N3) (Figure 3). 10. Homologous recombination will be achieved upon growth on glycerol. After streaking cells on YPD and replica-plating on YPEG, a maximum of 1 to 5% of cells should be unable to grow on glycerol.

Mating and isolation of recombinant cytoductants starting from synthetic rho-clones

1. Pick a single colony of the synthetic rhoand recipient rho+ strains (see Note 30) and inoculate 5 ml liquid YPD culture overnight. At least one of these two strains (usually the synthetic rho-) must carry the kar1-1 mutation that delays karyogamy, thus allowing mtDNA exchange to occur before nuclear fusion. 2. Mix 0.5 mL of each parent (alternatively 1 mL of synthetic rhoand 0.2 mL wildtype rho+; see Note 31) in a microfuge tube, spin, remove the supernatant, resuspend in residual liquid, and spread the mixture as a lawn onto a YPD plate. 3. Incubate at 30°C for 4-5 h (typically 4h and 30 min). Check zygote formation under the microscope, you should observe the formation of schmoo projections. Scrape the mating cells from the plate (we use a toothpick) and inoculate a fresh 5 mL culture of liquid YPD medium. Incubate at 30°C with agitation for a few hours to overnight. 4. Make a few serial dilutions of the culture and plate them to obtain single colonies on the appropriate minimal medium, e.g. to select for the recipient nuclear genotype and against the donor nuclear genotype, if possible. Alternatively, plate on selective medium if recombination restores e.g. growth on glycerol or without arginine, or on YPD medium if no selection is available. Densities of 50-200 colonies per plate should be obtained. 5. Replica-plate these colonies to medium that reveals the altered phenotype of the recipient strain due to integration of the mutant donor sequences into its mtDNA. For example, print to YPEG to identify clones that have become respiratory deficiency upon integration of a new mutation. 6. To eliminate cytoductants that acquired only the synthetic rho-mtDNA by cytoduction, candidate clones can be crossed to other tester strains, like a known rhoclone that contains the region you are seeking to mutate. Expected rho+ recombinant cytoductants will produce respiring diploids upon mating to this known rhostrain.

Notes

1. The dum1, dum11, dum14 and dum16 mutants present a deletion including the left telomere and part or the entire cob gene [START_REF] Matagne | Induction and characterization of mitochondrial DNA mutants in Chlamydomonas reinhardtii[END_REF][START_REF] Dorthu | Biochemical, genetic and molecular characterization of new respiratory-deficient mutants in Chlamydomonas reinhardtii[END_REF], the dum22 mutant presents a deletion extending until the nd4 gene [START_REF] Salinas | Co-evolution of mitochondrial tRNA import and codon usage determines translational efficiency in the green alga Chlamydomonas[END_REF] (Figure 1). The dum19 dum25 mutant is not a deletion mutant and possesses a frameshift deletion in cox1 (dum19 mutation) and an in-frame 3 bp deletion in nd1 (dum25 mutation) [START_REF] Remacle | Mutants of Chlamydomonas reinhardtii deficient in mitochondrial complex I: characterization of two mutations affecting the nd1 coding sequence[END_REF]. dum1 (CC-2255), dum11 (CC-4098) and dum22 (CC-4738) are available at 'chlamycollection.org'. dum14, dum16 and dum19 dum25 are available at the Remacle's lab.

2. The genetic background of the strain is a very important factor for the biolistic transformation efficiency [START_REF] Bonnefoy | Genetic transformation of Saccharomyces cerevisiae mitochondria[END_REF]. Mostly we use cells from the S288c background, especially strains derived from DBY947 [START_REF] Neff | Isolation of the β-tubulin gene from yeast and demonstration of its essential function in vivo[END_REF], such as DFS160, a strain harboring an arg8 deletion which is at once very convenient for bombardment of constructs with the ARG8 m reporter [START_REF] Steele | Expression of a recoded nuclear gene inserted into yeast mitochondrial DNA is limited by mRNA-specific translational activation[END_REF]. Strains derived from W303 [START_REF] Thomas | Elevated recombination rates in transcriptionally active DNA[END_REF], (ATCC 200060) give lower but satisfactory efficiencies whereas strains in the D273-10B (ATCC 24657) background are very difficult to transform. In addition bombarding a rho+ strain is 10 to 20 less efficient than bombarding a rho 0 (compare Figures 2 and3).

3. TAP medium (1 ‰ (v/v) glacial acetic acid, corresponding to 17.3 mM sodium acetate) is the most common used medium but initial experiments have been made with HSHA high-salt high-acetate medium which contains 29.4 mM sodium acetate [START_REF] Harris | The Chlamydomonas sourcebook: a comprehensive guide to biology and laboratory use[END_REF]. Some authors add ampillicin 50 mg/mL in the dark to avoid contaminations. We have not tried and found no excessive contaminations provided that the chamber where the bombardment takes place is sterilized just before use.

4. The 0.1% glucose accelerates growth, it is optional. Since raffinose is expensive, galactose can also be used without significant decrease in transformation efficiency. Adenine is added even for Ade+ prototrophs, since it seems to increase the transformation efficiency.

5. YPEG supplemented with 1M sorbitol can be used for direct selection of respiring transformants into rho+ cells; the 0.1% glucose supplement accelerates growth. Alternatively, rho+ cells can also be plated on the usual SD-Sorbitol medium for bombardment and directly replicated onto YPEG medium to select respiring transformants (see Figure 3). In addition, when selecting for transformants that are expected to produce the functional Arg8 protein by expression of the Arg8 m reporter gene, cells can be directly plated on SD-Sorbitol medium lacking arginine.

6. Tungsten powder M10 ~0.7 μm and gold powder 0.6 μm are also available from Bio-Rad (catalog #1652266 or #165-2262, respectively). However we always use Alfa Aesar tungsten powder <1 μm which is inexpensive and works as well as gold particles, and 3 times better than the tungsten powder from Bio-Rad which is the process to be discontinued. [START_REF] Matagne | Induction and characterization of mitochondrial DNA mutants in Chlamydomonas reinhardtii[END_REF]. In Chlamydomonas, the transforming DNA is preferably a linearized plasmid DNA or a PCR fragment, although super-coiled DNA can also be used, albeit with lower efficiency. In yeast, the DNA used for mitochondrial transformation is preferably a concentrated plasmid DNA, although successful transformation has also been achieved with PCR fragments (minimal size for recombination=28 bp). When plasmids are used, Qiagen plasmid midi or maxi kits (tip-100 or tip-500) give the best results in our hands. The transforming DNA concentration should be at least 2 μg/μL for plasmids.

8. Frozen 2.5M CaCl2 remains liquid, so no need to aliquote. 9. If available use a brand new bottle of absolute ethanol bottle and freeze 15 ml aliquotes.

10. Usually we soak macrocarriers holders for one night in absolute ethanol after each bombardment, then drain and sterilize them in a Pasteur oven. 11. It is possible but very unconvenient to make during the same experiment more shoots than the number of available macrocarrier holders. Thus for routine use, investing in more holders will rapidly be worth the price to optimize the number of strains constructed per biolistic transformation experiment.

12. Rupture disks of 1350 psi (pound per square inch) are also efficient for biolistic transformation of yeast, but in our hands 1100-psi disks rather give better results. If rupture disks are very old (several years), they can still be used, however they tend to be less resistant and will break at lower helium pressure, then 1350 psi might be more efficient than 1100 psi.

13. If possible it is best to use a tester in a background like DBY947 rather than D273-10B, since for tester crosses (subheading 3.7 step 1) the lawn will be less thick and will not mask the print of the transformants, making it easier to identify positive colonies 14. Alternatively we use N3 medium: 1% Yeast Extract, 1% Bacto-Peptone, 2% glycerol, 0.05 M Potassium Phosphate pH= 6.2.

15. In our hands a 3-day culture is optimal for rho0 or rho+ cells from the DBY947 background. [START_REF] Yoo | Cas9/gRNA-mediated genome editing of yeast mitochondria and Chlamydomonas chloroplasts[END_REF]. We prefer spreading the cells with a glass rod rather than beads because plating will be more heterogenous, thus providing zones of optimum cell density for the transformation as well as an irregular pattern of transformants: this provides useful identification marks when comparing the plates to pick the mitochondrial transformants at Subheading 3.7., step 6. 17. Sterilized tungsten particles can be kept frozen in 50% glycerol for several months to several years without loss of transformation efficiency. Discard if the supernatant turns yellowish. We have not tried gold.

18. Precipitations can be upscaled by a factor of 1.5 or even 2, still in the same tube; however keep using a 200 µl tip to resuspend and break the precipitate with ethanol at step 4, even if the ethanol volume is upscaled, because using 1 mL tips will not yield a finely dispersed DNAparticle precipitate. Using siliconized tubes for the precipitation did not make a significant difference for us. [START_REF] Thorsness | Escape of DNA from mitochondria to the nucleus in Saccharomyces cerevisiae[END_REF]. Keeping the DNA volume to a minimum optimizes the precipitation. 20. Particles settle extremely rapidly, vigorously vortex the tungsten suspension each time before pipetting. 21. Conduct this step thoroughly to obtain a finely dispersed precipitate that will be crucial to efficiently transform the cells.

22. We usually work under a laminar flow hood, using sterile forceps or the specific Bio-Rad tool to insert the macrocarrier membranes into the holders placed into sterile sterile Petri dishes. Do not prewash the macrocarriers nor desiccate them after coating; the ethanol will quickly evaporate.

23. For a very effective sterilization, the chamber and parts can be soaked with 70% ethanol before starting the cell preparation and DNA precipitation, allowed to sit, and wiped dry just before the bombardment. Using a 70% ethanol spray is convenient. The chamber has to be washed in a similar way after the bombardment and even more thoroughly, to remove any residues of agar medium that would lead to contaminations later on.

24. Contrary to Bio-Rad recommendations, we never soak rupture disks in isopropanol just before use. This reduces the transformation efficiency and is unnecessary to prevent contamination. Be careful not to use two rupture disks stacked together, they won't break at the desired helium pressure. It is easy to notice because in this case the hole for tightening the retaining cap with the torque wrench will not be exactly at the usual place. [START_REF] Funes | Partial suppression of Oxa1 mutants by mitochondria-targeted signal recognition particle provides insights into the evolution of the cotranslational insertion systems[END_REF]. In yeast we have found that more transformants were obtained when omitting to assemble the stopping screen before the microcarrier. However, in this case it is important to use 3% agar in the sorbitol plates, as advised in Subheading 2, otherwise the flying carrier disk will severely damage the agar surface.

26. Shorter distances result in very high colony densities in the center of the plate and few colonies at the periphery while longer distances decrease the transformation efficiency.

27. Failure to draw the greatest possible vacuum dramatically reduces the transformation efficiency. Teflon sealing tape can be used to reduce air leakage in the connections between the chamber and the pump. Cell viability is not significantly affected by an extended stay under these vacuum conditions. 28. For plates with a large number of transformants, it is useful to replicate the mated cells also on glucose medium selecting for the diploids; these diploid plates may by comparison facilitate the identification of the mitochondrial transformants on the original bombarded plates.

29. Respiratory-proficient colonies will appear earlier if the transformed DNA allows complementation of the mitmutation already in trans. [START_REF] Bonnefoy | In vivo analysis of mutated initiation codons in the mitochondrial COX2 gene of Saccharomyces cerevisiae fused to the reporter gene ARG8 m reveals lack of downstream reinitiation[END_REF]. If available, it is convenient to use as recipient a rho+ strain carrying a deletion mutation in the region of interest. This deletion should be large enough to include the change created and small enough to still allow recombination with the synthetic rhosequences.

31. If the synthetic rhodonor and the rho+ recipient strains have similar auxotrophies and thus cannot be easily distinguished, selected or counter-selected on minimal glucose medium, mating mixtures should contain equal numbers of cells of both strains. If some markers allow selection e.g. against the synthetic donor, then a fivefold excess of donor cells should be used compared to the recipient strain. In the cytoduction experiment, it might be convenient to recover at once recombined rho+ cells in both parent backgrounds. We are indebted to Claudia Serot for generating the supplementary video showing the bombardment procedure, including filming, choosing the sequences and editing the video clip. N. B. is supported by the Centre National pour la Recherche Scientifique, and C.R. is supported by Fonds National de la Recherche scientifique (FNRS, CDR J.0175.20) and Fonds Wetenschappelijk Onderzoek -Vlaanderen (FWO) and FNRS under EOS Project No. 30829584. rho 0 strain DFS160 [START_REF] Thorsness | Nuclear mutations in Saccharomyces cerevisiae that affect the escape of DNA from mitochondria to the nucleus[END_REF] was bombarded with a mixture of a LEU2 plasmid YEp351 and a plasmid containing the COX2 gene. The tester strain NB97 carried a ~130nt deletion in COX2 (cox2-60, [START_REF] Bonnefoy | In vivo analysis of mutated initiation codons in the mitochondrial COX2 gene of Saccharomyces cerevisiae fused to the reporter gene ARG8 m reveals lack of downstream reinitiation[END_REF]. Steps are numbered as indicated in the text in the method section. A lawn of the rho+ cox2-62 strain NB41 [START_REF] Bonnefoy | In vivo analysis of mutated initiation codons in the mitochondrial COX2 gene of Saccharomyces cerevisiae fused to the reporter gene ARG8 m reveals lack of downstream reinitiation[END_REF] that carries a 660 bp deletion in the upstream region of COX2 was spread onto medium lacking leucine and bombarded with a plasmid containing roughly 3 kb of ectopic sequences inserted upstream of COX2 in a region contained within the cox2-62 deletion, as well as a plasmid with the nuclear LEU2 marker gene (YEp351). [Leu+] transformants obtained after a 5 dayincubation at 30°C were directly replica-plated onto glycerol medium to select mitochondrial transformants, for which respiratory growth is restored.

3

 3 Partially purified C. reinhardtii mitochondrial genome (0.8 µg per bombardment) Plasmids containing from 1.8 to 5.0 kb mtDNA fragments (5 µg per bombardment) Plasmids containing 1.8 to 6.6 kb mtDNA fragments (3 µg per bombardment) ~ 100 -220 6 (10 -22 2 ) [9] PCR product (1.5 µg per bombardment) ~ 133 -266 7 (13 -26 2 )

  COX2RNA component of E. coli SRP Expression of RNA part of the SRP particle

Figure 1 :

 1 Figure 1 : Physical map of the 15.8 kb linear mitochondrial genome (GenBank accession EU306622). Horizontal blue arrows indicate the bidirectional transcription origin. The blue boxes represent the eight protein-coding genes: cob (apocytochrome b of complex III), cox1(subunit 1 of cytochrome c oxidase or complex IV), nd1, 2, 4, 5, and 6 (subunits of NADH:ubiquinone oxidoreductase or complex I) and rtl (reverse transcriptase-like protein). The boxes in two shades of green represent LSU (L1-L8) and SSU (S1-S4) genes encoding the structural ribosomal RNAs fragments. The vertical bars correspond to the 3 tRNA genes with the one-letter code (W, Q, and M). The light blue arrows at each end of the genome represent the inverted terminal repeats.

Figure 2 :

 2 Figure 2 : Production of a synthetic rho-yeast strain by biolistic transformation. The [Leu-]rho 0 strain DFS160[START_REF] Thorsness | Nuclear mutations in Saccharomyces cerevisiae that affect the escape of DNA from mitochondria to the nucleus[END_REF] was bombarded with a mixture of a LEU2 plasmid YEp351 and a plasmid containing the COX2 gene. The tester strain NB97 carried a ~130nt deletion in COX2 (cox2-60,[START_REF] Bonnefoy | In vivo analysis of mutated initiation codons in the mitochondrial COX2 gene of Saccharomyces cerevisiae fused to the reporter gene ARG8 m reveals lack of downstream reinitiation[END_REF]. Steps are numbered as indicated in the text in the method section.

Figure 3 :

 3 Figure 3: Example of rho+ yeast mitochondrial transformants obtained directly after bombardment of a rho+ mit-strain.A lawn of the rho+ cox2-62 strain NB41[START_REF] Bonnefoy | In vivo analysis of mutated initiation codons in the mitochondrial COX2 gene of Saccharomyces cerevisiae fused to the reporter gene ARG8 m reveals lack of downstream reinitiation[END_REF] that carries a 660 bp deletion in the upstream region of COX2 was spread onto medium lacking leucine and bombarded with a plasmid containing roughly 3 kb of ectopic sequences inserted upstream of COX2 in a region contained within the cox2-62 deletion, as well as a plasmid with the nuclear LEU2 marker gene (YEp351). [Leu+] transformants obtained after a 5 dayincubation at 30°C were directly replica-plated onto glycerol medium to select mitochondrial transformants, for which respiratory growth is restored.

Table 2 : Example of relocation of ectopic genes or tag within mtDNA using biolistic mitochondrial transformation Ectopic sequences Loci in mtDNA Specific features/remarks Purpose First publication Year Reference URA3

 2

						Arg8 m		TRP1
	other studies	ATP8, VAR1 in	COX3, ATP6,	COX1, COX2,	Then CYTb,	Initially COX3.		5' of COX2	COX2	Plasmid with
			available	strains; mutant version also	prototrophy in nuclear ∆arg8	Recoded gene restoring arginine		upon DNA escape into the nucleus	universal code; expressed only	Wild-type auxotrophy genes in
	-Positive selection for mtDNA	-Cargo for translocation across membrane	-Analysis of genetic instability	-Study of translation/assembly feed back loop	-Reporter gene for gene expression	-Gene replacement, introduction of mutations	escape	Selection of mutations modulating mtDNA		First proof of mtDNA escape
						1996 [18]		1993 [17]		1990 [19]

Bonnefoy & Remacle, Figure3

Acknowledgments

Supplementary material: Bombardment.mp4


A Amor

A Fernandes
email: amancio.fernandes@sorbonne-universite.fr

J Pouget

Numerical and experimental investigations of bistable beam snapping using distributed Laplace force

Keywords: Bistable mechanism, buckling, snapthrough, Laplace force, bifurcation, experimental tests

We report a study of an elastic buckled beam undergoing a contactless magnetic actuation of Laplace type. The beam model is based on the elastica beam theory including the beam extensibility. The Laplace force is produced by an electric current travelling along the beam placed in a magnetic induction. The magnitude of the electric current is the control parameter and by increasing the electric current the beam switches from one stable state to the other one for a given beam end-shortening. The main purpose of the study is to investigate the bistable response, more precisely, the diagram of the electric current as function of the midpoint vertical displacement of the beam according to the magnet location along the elastic beam. The model equations are established and they are numerically solved using an algorithm developed for nonlinear boundary value problem.

A second part of the study is devoted to experimental validation of the model and comparisons with the results extracted from the numerical solutions to the model equations. Especially, the comparisons ascertain with good accuracy the approach of the proposed bistable beam model. In addition, the influence of the model parameters on the bistable response is clearly identified.

Introduction

In previous work the switching mechanism of an elastic bistable beam due to the Laplace force actuation was reported in [START_REF] Amor | Snap-through of elastic bistable beam under contactless magnetic actuation[END_REF]. In this work, the authors have presented (i) a beam model based on the elastica theory for slender beams, (ii) the actuation mechanism produced by the Laplace force, (iii) the numerical solutions to the model equations and (iv) the experimental validations of the bistable response subject to the Laplace force. In the study the magnetic induction produced by the magnets are placed parallel to the whole length of the beam. It means that the bistable beam was subject to the lineic density of force upon the entire length of the beam.

In the proposed study, we want to know what would be the bistable response if the length of the pair of the magnets is less than the total length of the beam. In this situation, the lineic density of the Laplace force is partly distributed over a section of the beam. Consequently, new configurational parameters must be considered, namely, the magnet length and the position of the magnet center. More precisely, a question may be raised, is there an (or several) optimal location(s) of the magnets for which the electric current traveling in the beam is less than any other position? One of the advantages of the bistable mechanism is that it requires low energy during the switching process from one stable configuration to the other one. Another advantage in actuation application is that they can be displayed in large strokes compared to straight and monostable structures. Both buckled beams and elastic arches possess this attractive property that allows them to be used in a large variety of applications in the area of micro and nano-electromechanical systems (MEMS/NEMS) [START_REF] Buchaillot | Postbuckling dynamic behavior of self-assembled 3D microstructures[END_REF][START_REF] Casals-Terre | Snapaction bistable micro-mechanisms actuated by nonlinear resonance[END_REF][START_REF] Qiu | A curved-beam bistable mechanism[END_REF][START_REF] Krylov | Bistability of curved microbeams actuated by fringing electrostatic fields[END_REF][START_REF] Seunghoon | Pre-shaped buckled-beam actuators : Theory and experiments[END_REF][START_REF] Saif | On a tunable bistable MEMS -Theory and experiment[END_REF]. Many applications in advanced technological devices continue to attract the attention of engineering researches such as micro-positioners, switches, valves, relays [START_REF] Chen | Synthesis of compliant multistable mechanisms through use of a single bistable mechanism[END_REF][START_REF] Fu | Design and fabrication of a magnetic bi-stable electromagnetic MEMS relay[END_REF], tactile display [START_REF] Hafez | n Tactile interfaces : Technologies, applications and Challenges[END_REF][START_REF] Niu | Bistable electroactive polymer for refreshable braille display with improved actuation stability[END_REF], and mechanical memory for data storage [START_REF] Roodenburg | Buckling beam micromechanical memory with on-chip readout[END_REF]. Most research works proposed in the literature have been focused on the use of static forces to produce the transition between the two stable states involving the inherent instability of the buckled beams. Those driving forces (extended to applied moments) can be mechanical [START_REF] Qiu | A curved-beam bistable mechanism[END_REF][START_REF] Camescasse | Bistable buckled beam: Elastica modeling and analysis of static actuation[END_REF][START_REF] Camescasse | Bistable buckled beam and force actuation : Experimental validations[END_REF][START_REF] Cazottes | Design of actuation for bistable structures using smart materials[END_REF][START_REF] Cazottes | Bistable buckled beam : modeling and actuating force and experimental validations[END_REF], magnetic [START_REF] Han | Parametric study and optimization of micro-optical switch with a laterally driven electromagnetic microactuator[END_REF][START_REF] Park | Pre-shaped buckled-beam actuators, Theory and experiment[END_REF], thermal [START_REF] Stanciulescu | A lower bound on snap-through instability of curved beams under thermomechanical loads[END_REF][START_REF] Alcheikh | A highly sensitive and wide-range resonant magnetic microsensor based on a buckled micro-beam[END_REF] including actuation using shape memory alloys [START_REF] Zaidi | Contactless and selective energy transfer to a bistable micro-actuation using laser heated shape memory alloy[END_REF] or electrostatic [START_REF] Zhang | Snap-through and pull-in instabilities of an arch-shaped beam under an electrostatic loading[END_REF][START_REF] Krylov | The pull-in behavior of electrostatically actuated bistable microstructures[END_REF][START_REF] Das | Pull-in and snap-through instabilities in transient deformations of microelectromechanical systems[END_REF][START_REF] Younis | Nonlinear dynamics of MEMS arches under harmonic electrostatic actuation[END_REF]. A bistable smart beam model has been proposed by [START_REF] Aimmanee | Piezoelectrically induced snap-through buckling in a buckled beam bonded with a segmented actuator[END_REF], the author examine the snap-trough action of a bistable beam equipped with piezoelectric actuators. An experimental validation is presented as well. Among the numerous applications of bistable beam, a special attention has been paid to applications involving bistable mechanisms to microrobotics [START_REF] Hafez | Optimized binary modular reconfigurable robotic devices[END_REF]. Some promising devices have been proposed, for instance, bistable gripper [START_REF] Palathingal | Analysis and design of fixed-fixed bistable arch-profiles using a bilateral relationship[END_REF]. An interesting and comprehensive review of buckled structures has been reported by [START_REF] Hu | Buckling-induced smart applications: recent advances and trends[END_REF], going from simple buckled beams or shallow elastic arches to morphing structures such as elastic shells, multistable cylinder lattices including the combination of multiple bistable beams for deployable structures. Some studies using electro-thermal actuation has been considered to control the axial force of bistable micro-beams [START_REF] Torteman | Electrothermal excitation of parametric resonances in doubleclamped micro beams[END_REF] for application to flow velocity sensors. Such an electro-thermal actuation has been used for the snap-through of a pre-shaped double beams with opposite curvatures [START_REF] Hussein | A monolithic tunable symmetric bistable mechanism[END_REF]. To close the present list of applications, we can quote the study of a buckled beam equipped with piezoelectric sensors for energy harvesting [START_REF] Xie | A piezoelectric energy harvester for broadband rotational excitation using buckled beam[END_REF]. In the present work, we want to analyze the complex and the rich response of the buckled elastic beam under an electromagnetic actuation. More precisely, the chosen actuation is the Laplace force [START_REF] Maugin | Continuum mechanics of electromagnetic solids[END_REF][START_REF] Jackson | Classical electrodynamics[END_REF] produced by an electric current traveling along the beam (in conductive material, metal) placed in magnetic induction.

The proposed model is based on the elastica theory of elastic beams [START_REF] Love | A Treatise on the Mathematical Theory of Elasticity[END_REF][START_REF] Frasier | Mathematical technique and physical conception in Euler's investigation of the elastica[END_REF][START_REF] Antman | Nonlinear Problems of Elasticity[END_REF][START_REF] Magnusson | Behaviour of the extensible elastica solution[END_REF]. Such an approach is particularly rich because the kinematic description accounts for large transformations of the beam (cross-section rotation of large amplitudes). Many studies analyzed in the literature have proposed alternative modeling for kinematic description and for the nonlinear response of the buckled beam in its post-buckling regime. Among the different approaches found in literature some are based on the modal projection, that is the solution to the equations of the buckled beam expanded on the base formed by the buckling modes. Vangbo [START_REF] Vangbo | An analytical analysis of a compressed bistable buckled beam[END_REF] has proposed interesting detailed analysis of the snap-through effect of the bistable buckled beam. His model accounts for the compressibility of the elastic beam for small deflections. Based on the modal projection model, a study of the snap-through and bistability of beams with arbitrarily pre-shaped deformation has been examined by [START_REF] Hussein | Analytical study of the snapthrough and bistability of beams with arbitrarily initial shape[END_REF]. Stability and bifurcation conditions are discussed in details. In other respects, a weakly nonlinear approach has been examined by [START_REF] Nayfeh | Exact solution and stability of postbuckling configurations of beams[END_REF]. The authors have carried out a model in which the nonlinearity is due to the coupling between deflection and beam stretching (geometrical nonlinear deformation). A study of the behaviour of post-buckled beams has been conducted by Jiao et al. [START_REF] Jiao | Small and large deformation models of post-buckled beams under lateral constraints[END_REF] based on small (modal projection) and large transformation (elastica) models. Extension to shallow elastic arches has been considered using this kind of approach [START_REF] Pinto | Active non-linear control of buckling and vibrations of a flexible buckled beam[END_REF][START_REF] Chandra | A numerical investigation of snapthrough in a shallow arch-like model[END_REF]. A more sophisticated model has been conducted by [START_REF] Tajaddodianfar | Size-dependent bistability of an electrostatically actuated arch NEMS based on strain gradient theory[END_REF], the authors consider the strain gradient formulation of the elasticity to model bistable beams or shallows arches under electrostatic actuation.

The chosen beam model is presented in the next section (Section 2). Especially, in this section, the beam kinematics is described by two strain measures which are the extensional strain along the beam axis and the beam curvature. Both descriptions in the fixed frame and local frame attached to the deformed beam are given in the same section. The model is based on previous work [START_REF] Amor | Snap-through of elastic bistable beam under contactless magnetic actuation[END_REF] and is briefly recalled in Section 3. Moreover, the boundary conditions -clamped-clamped beam -and continuity conditions at the magnet boundaries are specified in the same section. Section 4 deals with the numerical study of the nonlinear boundary value problem. The numerical problem thus stated relies on the model equations based on the shooting method associated with a numerical procedure to capture the unknown shooting parameters. One of the main results coming from the numerical investigations is the bistable response, in other words, the actuating driving force (or the electric current) as a function of the beam midpoint displacement. The results are discussed according to the chosen parameters (magnet position, the endshortening). A part of the work is devoted to the experimental validations of the proposed model reported in Section 5, the results thus obtained are discussed and compared with those deduced from the model.

2 Description of the system and modeling

Deformation measures, resultants in force and moment

We consider a clamped-clamped initially straight elastic beam [AB] of length L 0 , width b, and thickness h. The beam is isotropic and homogeneous. A fixed reference frame R 0 {A; e 1 , e 2 , e 3 } is attached to the fixed point A of the beam and the basis vector e 1 coincides with the beam axis in its state of rest. The beam transformation takes place in the {e 1 , e 2 } plane. Fig. 1.a gives a sketch of the beam in its configuration at rest. The left clamp is fixed while the right clamp undergoes a small endshortening ∆L caused by an axial compressive force. The distance between both clamps is now L = L 0 -∆L (Fig. 1.b). A material point G of the beam located at the curvilinear abscissa s measured along the deformed beam is considered. A local frame is attached at the current point G (s) of the beam and it is defined by {G; τ , n, e 3 } where τ is the unit vector tangent to the deformed beam, n is the unit vector perpendicular to τ and e 3 is identical to that of the fixed referential. Accordingly, the position of the material point G is given by

AG = q (s) = x(s)e 1 + y(s)e 2 , s ∈ [0, L 0 ] , (1a)
= q T (s) τ + q N (s) n , (1b)
either in the fixed frame or the local frame. We are able to compute the relationship between the coordinate {x, y} and {q T , q N } using the rotation of the beam cross-section θ = (e 1 , τ ). The beam kinematics are reported in detail in [START_REF] Amor | Snap-through of elastic bistable beam under contactless magnetic actuation[END_REF]. The different frame vectors and coordinates are shown in Fig. 1.b. The rectangular shaded in clear grey displayed in Fig. 1.b is the magnet that produces the magnetic induction in the e 3 direction. Fig. 1.c is the top view of the beam exhibiting both magnets and the abscissa of their center and length.

The deformation measures describing the beam deformation from its state at rest to the actual configuration are the curvature κ = dθ ds and the extensional strain given by ε = Λ -1 where Λ = ds ds is the ratio of the length of differential line element of the beam in the deformed configuration to that of the undeformed one. The actuation mechanism of the bistable beam is modelled by a uniform density of lineic force which is nonetheless permanently perpendicular to the deformed beam. Such an actuation is produced by Laplace force. In the present study the magnetic induction is applied only on a part of the beam length (see Fig. We consider a density of electric current per volume unit j running through an elementary volume dv of the beam. The latter is placed in a magnetic induction field B. According to the classical electrodynamics [START_REF] Jackson | Classical electrodynamics[END_REF] the infinitesimal Laplace force acting onto the elementary volume of the beam takes on the form

dF = j × B dv , (2)
We recall that the beam cross-section is rectangular, its area is given by A = b h. The elementary volume is then given by dv = Ad (where d is an elementary length of the beam). On using now the electric current per unit of length denoted by i such that iτ = Aj, the elementary Laplace force reads as

dF = i d τ × B . (3)
The magnetic induction field B in the present situation is produced by two identical magnets placed on both side of the beam (see Fig. 1.b and 1.c). Moreover, the magnetic induction is supposed to be uniform in the volume defined by the magnets. The internal force within the beam cross-section is R = (N, T ) where N is the axial force, and T is the shear force, and the bending moment is M e 3 . The internal actions (forces and moment) are depicted in Fig. 1.d.

Model equations

The equations of the model are deduced from the principle of virtual work accounting for the boundary conditions at the ends of the beam. The set of the equilibrium equations in dimensionless notation takes on the following form (the reader is welcome to refer to [START_REF] Amor | Snap-through of elastic bistable beam under contactless magnetic actuation[END_REF])

N -θ T = 0 , (4a)
θ N + T -f = 0 , (4b)
M + (1 + ε) T = 0 . (4c)
Along with the above equations, we must consider the geometrical compatibility relationships deduced from the definition of the tangent vector to the deformed beam. These equations written in the local frame are

Q T -θ Q N -(1 + ε) = 0 , (5a)
Q N + θ Q T = 0 , (5b)
where Q T and Q N are the tangential and normal displacements, respectively. We recall that Q = q/L 0 . At last, we write down the constitutive equations for the internal actions (in dimensionless notation) where the prime is the derivative with respect to the dimensionless curvilinear abscissa S defined by S = s L0 and k characterizes the beam extensibility the latter is given, in present case, by k

θ -M = 0 , (6a)
ε -k N = 0 , (6b)
= h 2 12L 2 0
. The dimensionless variables have been used by introducing appropriate characteristic force and moment (see [START_REF] Amor | Snap-through of elastic bistable beam under contactless magnetic actuation[END_REF]). For sake of consistency, we briefly recall how the dimensionless variables have been defined in the previous study (f, N, T, M) = (f L 0 /F 0 , N/F 0 , T /F 0 , M/M 0 ). The reference force and moment are given by F 0 = EAk, and M 0 = EI/L 0 , respectively (E is the Young modulus and I is the moment of inertia of the cross-section with respect to e 3 direction). We notice, here, that the force f = f n is a lineic density of force with f = i.B deduced from Eqns. (3) (see Fig. 1.d).

Boundary and continuity conditions

For sake of convenient, we introduce the following vector

ξ (S) =   θ (S) Q N (S) Q T (S)   . (7)
On using the above notation, the boundary conditions associated to the beam equations take on the form (clamp condition)

ξ (0) = 0 , ξ (1) =   0 0 X B   , (8)
where X B is the abscissa of the right point B after endshortening. The clamping actions are unknown.

The magnets are placed only on a part of the beam length (see Fig. 1), the applied lineic force is defined on three domains such as

f (S) = 0 S ∈ [0, S C [ ∪ ]S D , 1] , (9a)
f (S) = f n S ∈ [S C , S D ] . (9b)
The abscissa S C and S D are defined as the intersection of the deformed beam with vertical limits on the left side and right side of the magnets referred as to their abscissas X C and X D , respectively, and they are

X (S C ) = X C and X (S D ) = X D . (10)
Moreover, we have

X C = X M -L M and X D = X M + L M , (11)
where X M is the dimensionless abscissa of the middle location of the pair of magnets and 2L M is their length.

In addition, some continuity conditions at the boundaries of the domain must be satisfied at S C and S D ξ (S C ) = 0 and ξ (S D ) = 0 .

In Eqns. [START_REF] Roodenburg | Buckling beam micromechanical memory with on-chip readout[END_REF], the double square braces hold for the discontinuity defined by ξ = ξ +ξ -at the point to be considered. On using the notations introduced in (7), equations [START_REF] Roodenburg | Buckling beam micromechanical memory with on-chip readout[END_REF] amount to saying that the variables θ, Q N and Q T are continuous at the material points S C and S D .

Computing bistable response under actuation and results

Now, we have all the ingredients of the mechanical system to investigate the response of the bistable beam subject to the Laplace force. The latter is applied to a localized zone of the beam length as depicted in Fig. 1.b and Fig. 1.c. The first task is to solve the boundary value problem described by the set of Eqns. (4,5 and 6), by using a shooting method. The numerical simulations for bistable beam equations are based on the algorithm described in the previous study [START_REF] Amor | Snap-through of elastic bistable beam under contactless magnetic actuation[END_REF]. In this study the accuracy of the model equation was validated by a finite element computation on the 1D model. Accordingly, the model equations given by Eqns. ( 4) will be considered in the forthcoming sections.

The main numerical problem to overcome is the existence of unstable branches in the bistable response. In these zones we are faced with the divergence of the numerical algorithm and the discrepancy of the results could be arisen. In order to get around such a problem an efficient way of solving correctly the model equation is to control the displacement of a material point of the beam. We control the crossing point of the beam and a fixed vertical line located at the mid-point between the beam ends A and B and such that Q X (S J ) = (1 -∆L) /2 (in dimensionless unit), where S J is the curvilinear abscissa of the crossing point J (referred to Fig. 1). Therefore, the idea is to increment the vertical coordinate of the mid-point Q Y (S J ) = Y J by a very small amount at each iteration. Accordingly, the numerical algorithm provides a unique deformed beam while the beam is switching.

The numerical results are performed for an elastic thin beam made of an amagnetic stainless steel A316L having a Young modulus E = 207 GPa. The beam dimensions are the length 200 mm, the width 10 mm, and the thickness 0.2 mm. The magnetic induction between both magnets is about 0.2 Tesla. The numerical method allows us to compute the density of Laplace force due to the magnetic induction and the electric current is then computed from Eqn. (3) leading to i = f B . The first result is the response of the bistable beam, more precisely, the electric current as a function of the mid-point displacement (referred as to J, see Fig. 1.b) y m = Y J L 0 according to the magnet location. Fig. 2 shows the numerical results for three magnet positions, (i) at 50% (magnets at the mid-point), (ii) at 38% from the fixed left end of the beam and (iii) at 24% (no far from the left point). The solid line corresponds to the stable zone while the dashed line refers to the unstable region of the switching process. We note that the behavior of the bistable beam response is very sensitive to the magnet position. For the magnets at the middle of the beam ends an N -shaped curve is clearly observed. At first sight, the beam response for this symmetrical actuation is mainly characterized by the combination of the first three buckling modes. The bistable response is somewhat different for off-center actuation. In Fig. 2, for two different off-center actuations (0.24 and 0.38) we observe round-off of the curve when passing the maximum or minimum of the electric current. The nonsymmetrical actuation promotes a bistable switching involving the first two buckling modes and the third mode playing substantially no role. For off-center actuation the maximum of electric current to produce the snap-through is less than that of mid-span actuation. Fig. 3 reports the variation of the electric current needed to produce switching as a function of the magnet location. It is clear that the curve is symmetric with respect to the center. Nevertheless, there exists an optimal position of the magnets (the center of the pair of magnets) for which the electric current is minimum, i.e. around the off-center position 36.3%. However, once the length of the magnets no longer allows their center to be positioned in the vicinity of 36.3% (2L M > 72.6%), the optimal configuration is such that the right or the left edge of the pair of the magnets is placed at either end of the beam. For 2L M = 80% for instance the optimal position is X opt M = 40%. In addition, Fig. 4 shows that two other parameters have an impact on the optimal position of the magnets. Indeed, looking at Fig. 4, the position corresponding to the minimum of the switching electric current X opt M increases as the end-shortening ∆L and decreases as the slenderness L/h increases. The last series of numerical results are obtained by keeping the off-center actuation fixed and examining the influence of the end-shortening on the bistable beam response. The results are presented in Fig. 5 for three different end-shortening values (0.02%, 0.04% and 0.06%) leading to different mid-point deflections of the beam and maximums of the electric current.

Experimental validations

The present section reports a series of results extracted from the experimental tests performed on the bistable beam partly actuated by Laplace force. The experimental study is quite similar to that of the magnetic actuation along the entire length of the beam [START_REF] Amor | Snap-through of elastic bistable beam under contactless magnetic actuation[END_REF]. The experimental tests concern the results coming from the numerical solution to the bistable beam model, especially the electric current versus mid-point displacement according to the location of the pair of magnets for different end-shortening values. Comparisons with the results derived from the modeling study are discussed in details.

Technical details about the experimental set-up

Material

The thin elastic beams used for the experimental tests are described in the previous section. Nevertheless, a particular attention has been paid to obtain a high quality of planar beams with a minimum of residual stress. The magnetic induction field is produced by a pair of identical magnets of neodyme type (N dF eB alloy) their dimensions are a length of 80 mm, a width of 17 mm, and a thickness of 5 mm.

Experimental set-up

Different views (global test bench, instruments of measurement) of the experimental set-up are shown in Fig. 6. The elastic thin beam is fixed at each of its end in a clamp. One end of the beam is fixed, the other one is placed on a travel translator stage equipped with a micro-metric screw. On moving the translator we are able to control the end-shorting of the beam with a 10 µm adjustment. In addition, the end-shortening is measured correctly thanks to a digital comparator. The beam is placed between the pair of magnets as described above. The pair of magnets is allowed to move parallel to the beam axis. A Gaussmeter equipped with Hall effect probe allows measuring the magnetic induction produced by the magnets, the magnetic induction found is about 0.2 Tesla. An electric current supply device is used to deliver a given stable current running through the beam. A power resistor is connected in series to secure the power supply. Nevertheless, we can ask ourselves what is the impact of the Joule effect produced by the electric current passing through the beam. The temperature rise is about 0.0916 • C for a snap-through time of 50 ms. This variation of temperature gives rive to a beam elongation of 0.3 µm, which is small enough so that the effect is not noticeable in comparison to the beam imperfections and the accuracy of the measuring equipment. The experimental measurement process follows different steps : (i) the electric current is increased step by step from zero to the value which produces a strong enough Laplace force to trigger the beam snapping. For this value, the beam switches and gets the other stable position. (ii) The electric current is switched off as soon as the snap-through has been detected. (iii) The variation of the mid-point of the beam is measured using a laser profilometer (a Keyence ® ultra-speed laser profilometer model LJ-V7060). (iv) The profilometer signal is sent to the computer in order to be analyzed and correlated to the input signal for the electric current. (v) The signal is converted into an analogical value for the displacement of the beam center. (vi) The test is then repeated for switching in the opposite direction.

Experimental results and analysis

Three main results are obtained and analyzed from the experimental data. Among the first results are the measurements of the bistable response for different positions of the pair of magnets for a given end-shortening. The experimental results are shown in Fig. 2. In order to make a comparison with the results coming from the model, the experimental points are superposed to the solid lines corresponding to the numerical results. The switching process has been performed in both directions from one stable position to the other one and vise versa. At first sight, we observe that the experimental points are very close to those obtained with the analytical study, the discrepancy is less than 3.4%. The experimental study obviously does not allow to obtain data in the unstable zone (dashed line for analytical results). It is worthwhile noting that for the off-center actuation the instability phenomena is shifted with respect to that of the mid-point actuation. The off-set increases as the location of the magnets is close to the beam ends.

The second series of results are obtained for a given offset actuation when varying the end-shortening. Fig. 5 shows the experimental results (symbols in the figures) superposed to the curves extracted from the numerical results. Once again, the experimental study yields a clear validation of the numerical simulation performed using the model equations. The discrepancy between the experimental data and the results obtained with the numerical simulations does not exceed 2%.

The last results of the experimental study are illustrated in Fig. 3 exhibiting the variation of the maximum of electric current as a function of the position of the pair of magnets. The experimental data are referred as to the red points which are very close to the solid line for the analytical results. The optimal position for the pair of magnets is located at about 36.3% (X M = 0.363). At this position, the maximum of the electric current is less than any other maxima. The value of the electric current is about 4 A. Nevertheless, the present result is certainly dependent on the bistable parameters such as the end-shortening as well as the magnet length. Moreover, it is noteworthy drawing a parallel between the present results and those issued from a similar work concerning the bistable beam actuation using a punctual force. An identical result, but in the case of a punctual force actuation, was already pointed out by Cazottes et al. [START_REF] Cazottes | Bistable buckled beam : modeling and actuating force and experimental validations[END_REF] but on using the modal expansion method, and by Camescasse et al. [START_REF] Camescasse | Bistable buckled beam: Elastica modeling and analysis of static actuation[END_REF] in the framework of the elastica approach of the beam theory. The minimum of the actuating force is not located at the beam mid-point. The exact position is very dependent on the beam slenderness and end-shortening (see Fig. 4).

Discussion and concluding remarks

The essential results of the present model of bistable beam actuation using a distributed Laplace force lie with the validation of the model by experimental tests. The model thus adopted is based on the elastica theory of elastic beams which accounts for displacements and rotations of large amplitudes. Moreover, the model includes the beam extensibility which plays a crucial role in the switching process. One of the originalities of the study is the distributed Laplace force actuation on a finite zone of the beam which is permanently perpendic- ular to the deformed beam. The bistable beam response under Laplace force is investigated according to the position of the actuating zone (the center of the magnets) with respect to the beam length. In particular, the offcenter actuation needs a maximum of electric current to have the beam switched smaller than that of the central position for a given end-shortening. The experimental results reported in the present study ascertain the numerical predictive results for the bistable beam response and the influence of the magnet location.

An optimal position of the pair of magnets has been placed in evidence. For an actuation localized at about 36.3% from the left or the right of the beam we have the minimum of the electric current producing the Laplace for the beam switching. These results can be of relevant interest in engineering applications such as microswitches, micro-robotics, tactile interfaces, or MEMS.

A natural extension of the present work and forthcoming work would be to study the vibration of the bistable beam in the post-buckling regime, for instance, vibrations of small amplitudes around the buckled state, in first time. However, one of the most interesting studies would be the nonlinear dynamical response of the bistable system under time-dependent excitation of the Laplace force type. Especially, the latter can be easily controlled by varying in time the electric current running through the beam [START_REF] Nayfeh | Exact solution and stability of postbuckling configurations of beams[END_REF][START_REF] Chandra | A numerical investigation of snapthrough in a shallow arch-like model[END_REF][START_REF] Nayfeh | On the nonlinear dynamics of a buckled beam subjected to a primary-resonance excitation[END_REF]. The experimental validations of the vibrating behaviors of the buckled beam will be considered as well.

  1.b and Fig.1.c).

LFig. 1

 1 Fig. 1 Bistable beam : (a) beam at rest in the reference configuration, (b) buckled beam and magnetic actuation and (c) top view (d) internal and external actions on the beam.

Fig. 2

 2 Fig.2Bistable response : current as a function of the beam mid-point displacement for three different values of the magnet position for a given end-shortening set at ∆L = 0.04% (in solid line : the elastica; points, triangles and squares : experimental results).

Fig. 3

 3 Fig.3Variation of the electric current maximum versus the position of the pair of magnets for ∆L = 0.04% (in solid line : the elastica; points : experimental results).

Fig. 4

 4 Fig.4The variations of the optimal actuating location X opt M for 2L M = 0.4 : (a) as a function of the end-shortening ∆L, and (b) as a function of the slenderness L/h.

Fig. 5

 5 Fig. 5 Bistable response : electric current as a function of the beam mid-point displacement for three different values of the end-shortening for a given position (38%) of the pair of magnets (in solid line : the elastica; points, triangles and squares : experimental results).

Fig. 6

 6 Fig. 6 Photos of the experimental set-up : (a) view showing the beam placed between the both magnets (b) the global view of the bistable buckled beam fixed with the clamps at each of its ends and different instruments and (c) the measurement of the mid-point beam displacement owing the laser profilometer.

Acknowledgements The research work reported in the paper has been supported by the research project BISCEL-TECH funded by Fonds Unique Interministériel (FUI-APP21).

Conflict of interest

The authors declare that they have no conflict of interest.