M Salvador Castell

Dr Phil M Oger
email: philippe.oger@insa-lyon.fr

Judith Peters
email: peters@ill.eu

Pr Judith Peters

High-pressure Adaptation of Extremophiles and Biotechnological Applications

Keywords: high pressure, deep biosphere, stress, piezophile, adaptation

published or not. The documents may come   L'archive ouverte pluridisciplinaire

Introduction

High pressure (HP) characterizes many habitats on Earth, such as deep sea, sub sea floor and continental subsurface. Ocean covers approximately 70% of world's surface and its average pressure is 38 MPa (1 MPa = 10 bar). This hydrostatic pressure originates from the weight of the water column and corresponds to 10 MPa/km [START_REF] Bartlett | Microbial Adaptations to the Psychrosphere/Piezosphere[END_REF]. Part of the oceans are defined as "deep sea", which encompasses the entire biosphere below 1000 m from the water surface and consequently with pressure higher than 10 MPa. The highest hydrostatic pressure detected in the ocean is approximately 110 MPa at 11,000 m depth at the Challenger Deep of Mariana Trench in the Pacific Ocean. Though, pressures above 110 MPa have been found at the sub sea floor. In this case, pressure is caused by hydrostatic pressure but also by the weight of material (lithostatic pressure) and it increases roughly by [START_REF] Oliveira | Arc repressor will not denature under pressure in the absence of water[END_REF] MPa/km in sediments and by about 28 MPa/km in oceanic rocks. Sub sea floor contains a substantial part of the Earth biomass and can potentially influence global biochemistry [START_REF] Schrenk | Microbial Provinces in the Subseafloor[END_REF][START_REF] Hazael | Pressure as a Limiting Factor for Life[END_REF][START_REF] Daniel | Origins of life and biochemistry under high-pressure conditions[END_REF].

All high-pressure habitats are occupied by microorganisms and other complex organisms and they highly contribute to the Earth's biomass [START_REF] Orcutt | Microbial Ecology of the Dark Ocean above, at, and below the Seafloor[END_REF][START_REF] Kallmeyer | Global distribution of microbial abundance and biomass in subseafloor sediment[END_REF]. Pressure impact on organisms' growth allow to divide them in different categories. Organisms that cannot tolerate ambient pressure are designed as strict or obligate piezophiles, inversely, facultative piezophiles or just piezophiles are organism that tolerates ambient pressure, but their optimal growth pressures are higher than 10 MPa (figure 1). On the other side, organism that withstand optimal pressures lower than 10 MPa are called piezotolerants and, finally, organisms which their growth is inhibited by pressure are designed as piezosensitive [START_REF] Oger | La vie sous pression des microorganismes piézophiles[END_REF].

Figure 1. Schema/c growth curves of microorganisms according to pressure (MPa) [START_REF] Oger | La vie sous pression des microorganismes piézophiles[END_REF].

The interest in piezophiles has begun more than a century ago [START_REF] Simonato | Piezophilic adaptation: a genomic point of view[END_REF][START_REF] Demazeau | High hydrostatic pressure and biology: A brief history[END_REF], but the technological difficulties and the need of specialized equipment have done that highpressure studies are not, currently, developed in most laboratories. However, the interest and the applications of the pressure biotechnological applications have been growing during last decades.

Bacteria and Archaea domains contain facultative and obligate piezophiles. Examples of obligate piezophiles are the bacteria Shewanella benthica and Colwellia marinimaniae, with their optimal pressure at 70 MPa and 120 MPa respectively [START_REF] Nogi | Taxonomic studies of extremely barophilic bacteria isolated from the Mariana Trench and description of Moritella yayanosii sp. nov., a new barophilic bacterial isolate[END_REF][START_REF] Kusube | Colwellia marinimaniae sp. nov., a hyperpiezophilic species isolated from an amphipod within the challenger deep, Mariana Trench[END_REF], and the archaeon Pyrococcus yayanossi, withstanding an optimal pressure of 50 MPa [START_REF] Birrien | Pyrococcus yayanosii sp. nov., an obligate piezophilic hyperthermophilic archaeon isolated from a deep-sea hydrothermal vent[END_REF]. The technical constraints to isolate obligate piezophiles may provoke their underestimation.

Obviously, piezophiles excel in sustaining pressure conditions beyond the usual limits for humans; however, the reasons for that adaptation are still debated.

Effects of Pressure on Macromolecules and Cells

Pressure alters the biomolecules by changing their volume. Thermodynamically, the variation in Gibbs free energy (G) is defined by the equation 1

d ( ∆ G ) =-∆ SdT +∆VdP, (1)
where ΔS is the difference in entropy, ΔV the change in volume and T and P represent temperature and pressure, respectively. At constant temperature, dT = 0 and thus

( ∂ G ∂ P ) T =∆ V . (2)
This equation, according to the Le Châtelier principle [START_REF] Chatelier | Sur un enoncé général des lois des équilibres chimiques[END_REF], states that an increase in pressure will cause a shift to the state that occupies the smallest volume, meaning, for example, to the unfolded state for most globular proteins, where ΔG < 0. As a result, pressure modifies the volume of the system but not its internal energy (as temperature does). Water with its low compressibility is a crucial partner for pressure action.

Notably hydration water (water bound at the surface of macromolecules) is very sensitive to pressure and it can reorganize its network under pressure implying an effect on the macromolecule [START_REF] Mentré | Effects of high hydrostatic pressures on living cells: A consequence of the properties of macromolecules and macromolecule-associated water[END_REF]. Moreover, macromolecules present an extraordinary stability against pressure under low hydration conditions [START_REF] Oliveira | Arc repressor will not denature under pressure in the absence of water[END_REF][START_REF] Rayan | Pressure-induced helix-coil transition of DNA copolymers is linked to water activity[END_REF].

Relatively low pressures affect the interatomic molecular bindings (such as Van der Waals, hydrogen bonding and hydrophobic interactions) altering conformations and structures of biomolecules and therefore, their physical properties (solubility, melting point, density), equilibrium states and processes' rate [START_REF] Silva | High-Pressure Chemical Biology and Biotechnology[END_REF]. Electrostatic and hydrophobic interactions are specifically affected by pressure. In contrast, pressures above 2 GPa are needed to impact non-covalent interactions [START_REF] Balny | High pressure effects on biological macromolecules: from structural changes to alteration of cellular processes[END_REF][START_REF] Balasubramaniam | Principles and Application of High Pressure-Based Technologies in the Food Industry[END_REF].

Nucleic acids

Although the unfolding volume of DNA duplexes is small, pressures up to 1 GPa have, in general, a stabilizing effect on canonical DNA (e.g. with common pair bases). This stabilizing effect may be explained by the decrease of hydrogen bonds' distance.

Consequently, pressure increases the duplex-single-strand transition temperature [START_REF] Mentré | Effects of high hydrostatic pressures on living cells: A consequence of the properties of macromolecules and macromolecule-associated water[END_REF].

Only in specific cases of synthetic polymers ( e.g adenine -thiamine copolymer) and salt concentrations, pressure can lead to double-stranded melting [START_REF] Rayan | Comparison of the heat-and pressure-induced helixcoil transition of two DNA copolymers[END_REF]. Regardless, the mechanism is not the same as the heat-induced DNA melting, as under pressure, water molecules penetrate DNA base pairs destabilizing their interactions [START_REF] Takahashi | Effect of pressure on thermal stability of G-Quadruplex DNA and double-stranded DNA structures[END_REF][START_REF] Rayan | Pressure-induced helix-coil transition of DNA copolymers is linked to water activity[END_REF].

There is a lack of information about the effect of pressure on RNA but, generally, it has been observed that RNA is more pressure sensitive than DNA. For example, pressure induces a structure reorganization of tRNA [START_REF] Giel-Pietraszuk | A nature of conformational changes of yeast tRNAPhe: High hydrostatic pressure effects[END_REF][START_REF] Schuabb | Exploring the free energy and conformational landscape of tRNA at high temperature and pressure[END_REF] and it destabilizes small RNA oligomers [START_REF] Garcia | Simulation of the pressure and temperature folding/unfolding equilibrium of a small RNA hairpin[END_REF].

Non-canonical pair structures (different from the usual Watson-Crick pair bases), such as G-quadruplexes or stem-loops, are less stable under pressure than canonical structures by a factor of 10 [START_REF] Takahashi | Effect of pressure on thermal stability of G-Quadruplex DNA and double-stranded DNA structures[END_REF][START_REF] Takahashi | Pressure-dependent formation of i-motif and Gquadruplex DNA structures[END_REF].

Although canonical DNA duplexes are stabilized by pressure, DNA-protein interactions may be perturbed due to changes in the electrostatic and hydrophobic interactions. Accordingly, pressure affects negatively all molecular reactions where DNA is involved, such as replication, transcription and recombination [START_REF] Heremans | Biology under extreme conditions[END_REF][START_REF] Abe | Exploration of the Effects of High Hydrostatic Pressure on Microbial Growth, Physiology and Survival: Perspectives from Piezophysiology[END_REF].

Proteins

Most of the knowledge about pressure effects on proteins is based on studies on globular proteins [START_REF] Panick | Structural characterization of the pressure-denatured state and unfolding/refolding kinetics of staphylococcal nuclease by synchrotron small-angle X-ray scattering and Fourier-transform infrared spectroscopy[END_REF][START_REF] Marion | Pressure-induced molten globule state of human acetylcholinesterase: Structural and dynamical changes monitored by neutron scattering[END_REF][START_REF] Sr | Influence of cosolvents, self-crowding, temperature and pressure on the sub-nanosecond dynamics and folding stability of lysozyme[END_REF]. Some of these studies reveal that structural transitions of globular protein due to pressure are based on a hydration mechanism that accompanies protein conformational changes. At higher pressure the hydration degree is increased by the penetration of water into the protein cavities causing the increase of the surface area in contact with the solvents thus contributing to the volume change [START_REF] Roche | Cavities determine the pressure unfolding of proteins[END_REF]. Pressure mainly alters tertiary and quaternary structures of proteins but secondary structures (α-helices, β-sheets, and turns) are much less sensitive to water penetration and to destabilization by pressure. For this reason, the state unfolded by pressure may be a hydrated globular structure with large amounts of folded structure [START_REF] Silva | High-Pressure Chemical Biology and Biotechnology[END_REF][START_REF] Meersman | High-Pressure Biochemistry and Biophysics[END_REF]. It is important to note again the role of water on pressure denaturation, since this volume change can only be observed for proteins in solution. Dry proteins are highly stable against pressure [START_REF] Balny | High pressure effects on biological macromolecules: from structural changes to alteration of cellular processes[END_REF][START_REF] Smeller | Pressure-temperature phase diagrams of biomolecules[END_REF][START_REF] Eisenmenger | High pressure enhancement of enzymes: A review[END_REF].

The unfolding of many monomeric proteins begins above 200 MPa [START_REF] Daniel | Origins of life and biochemistry under high-pressure conditions[END_REF], however enzymatic activities are usually modified at lower pressures. In fact, the application of pressures < 200 MPa confers higher thermostability to most proteins [START_REF] Smeller | Pressure-temperature phase diagrams of biomolecules[END_REF][START_REF] Winter | Exploring the temperature-pressure configurational landscape of biomolecules: from lipid membranes to proteins[END_REF].

Consequently, superposing pressure and temperature usually accelerates most of enzymatic reactions, such as hydrolases and transferases reactions [START_REF] Eisenmenger | High pressure enhancement of enzymes: A review[END_REF]. Moreover, few reactions can be enhanced by pressure even at low temperatures [START_REF] Eisenmenger | High pressure enhancement of enzymes: A review[END_REF]. For example, the efficiency of coconut husk hydrolysis by cellulases from Penicillium variable is increased at 300 MPa [START_REF] Dutra Albuquerque | Combined effects of high hydrostatic pressure and specific fungal cellulase improve coconut husk hydrolysis[END_REF].

Pressure generally changes the equilibrium between oligomers and their subunits or between two different proteins and this even at relative low pressures (about 50 MPa) [START_REF] Huang | Responses of microorganisms to high hydrostatic pressure processing[END_REF]. At this pressure, for example, ribosomes' subunits are dissociated [START_REF] Abe | Exploration of the Effects of High Hydrostatic Pressure on Microbial Growth, Physiology and Survival: Perspectives from Piezophysiology[END_REF] and larger protein assemblies as cytoskeletal proteins are disturbed resulting in reversible morphological changes [START_REF] Gao | On the Origin of Microtubules' High-Pressure Sensitivity[END_REF]. However, other oligomers are more resistant to pressures as the tetrameric urate oxidase, which dissociates at about 150-175 MPa [START_REF] Girard | Structurefunction perturbation and dissociation of tetrameric urate oxidase by high hydrostatic pressure[END_REF].

Actually, a protein in its native state possesses distinct, nearly isoenergetic conformational substates, which may have similar or dissimilar functions or the same function with different rates (statistical substates). As pressure can decrease the folding rate and increase the unfolding one, it can shift the population of different protein substates on the basis of their volumetric differences [START_REF] Silva | Pressure induces folding intermediates that are crucial for protein-DNA recognition and virus assembly[END_REF][START_REF] Luong | Pressure -A Gateway to Fundamental Insights into Protein Solvation, Dynamics, and Function[END_REF][START_REF] Librizzi | The effects of pressure on the energy landscape of proteins[END_REF]. This capability allows the characterization of various intermediate substates by pressure, which may act in the folding process [START_REF] Collins | High-Pressure Protein Crystallography and NMR to Explore Protein Conformations[END_REF]. Moreover, pressure can change the reaction rates, providing new information about the dynamics and reactions of proteins [START_REF] Meersman | High-Pressure Biochemistry and Biophysics[END_REF]. This was confirmed, for example, by a dynamic study of myoglobin, where it has been shown that pressure reduces protein motions and facilitates access to different substates [START_REF] Librizzi | The effects of pressure on the energy landscape of proteins[END_REF].

Few studies have been done on pressure effects of non-globular proteins, as fibrous, disordered and membrane proteins. Examples are the studies on the collagen structure [START_REF] Potekhin | High pressure stabilization of collagen structure[END_REF], the intrinsically disordered protein alpha synuclein [START_REF] Roche | Impact of Hydrostatic Pressure on an Intrinsically Disordered Protein: A High-Pressure NMR Study of α-Synuclein[END_REF] and on the Lmr transmembrane protein [START_REF] Vogel | High pressure modulated transport and signaling functions of membrane proteins in models and in vivo[END_REF] or the ion channel MscS [START_REF] Macdonald | Effect of high hydrostatic pressure on the bacterial mechanosensitive channel MscS[END_REF]. An important point is that the behaviour of transmembrane proteins against environmental stresses is affected by the protein structure but also by its lipids surrounding [START_REF] Ulmer | Effects of Pressure-Induced Membrane Phase Transitions on Inactivation of HorA, an ATP-Dependent Multidrug Resistance Transporter, in Lactobacillus plantarum[END_REF].

Membrane proteins and membrane lipids form an ensemble; they influence each other as a result of biochemical or environmental changes which can compromise membrane process as energy production or ion flux. For example, the transporter efficiency of the tryptophan permease Tat2 from yeast cells is affected due to a membrane fluidity modification at HP [START_REF] Campanaro | Laterally transferred elements and high pressure adaptation in Photobacterium profundum strains[END_REF]. Therefore, membrane integrative studies are necessary to better describe the membrane protein behaviour. Its influence of the lipid matrix on the protein response to pressure have been studied for few proteins [START_REF] Powalksa | Fluorescence Spectroscopic Studies of Pressure and Temperature Effects on Na + , K + -ATPase Reconstituted into Phospholipid Bilayers and Model Raft Mixture Zur Erlangung des akademischen Grades[END_REF][START_REF] Periasamy | Effects of temperature and pressure on the lateral organization of model membranes with functionally reconstituted multidrug transporter LmrA[END_REF][START_REF] Kapoor | Revealing conformational substates of lipidated N-ras protein by pressure modulation[END_REF].

Phospholipids

Lipids, and specially their hydrocarbon chains, possess a highly compressible potential, which makes them very sensitive to pressure [START_REF] Brooks | Pressure effects on lipid membrane structure and dynamics[END_REF]. When pressure is applied on a phospholipid bilayer, the acyl chains from phospholipids get straighten resulting in a thicker and higher ordered bilayer. Due to this higher acyl chains' order, pressure promotes the lipid phase transition from liquid-crystallin (phase essential for the biological function of the membrane) to a more rigid phase called the gel phase.

Additionally, pressure can also promote the apparition of new phases, such as interdigitated phases or non-lamellar phases (i.e., cubic or hexagonal) [START_REF] Brooks | Pressure effects on lipid membrane structure and dynamics[END_REF][START_REF] Matsuki | Thermotropic and barotropic phase transitions in bilayer membranes of ether-linked phospholipids with varying alkyl chain lengths[END_REF][START_REF] Winter | Effect of pressure on membranes[END_REF][START_REF] Trapp | High hydrostatic pressure effects investigated by neutron scattering on lipid multilamellar vesicles[END_REF][START_REF] Ding | Effects of High Pressure on Phospholipid Bilayers[END_REF].

Nevertheless, not all lipids have the same sensitivity to pressure, for example, lipids with longer hydrophobic chains are more responsive to pressure. This may result in a phase separation in domains on model membranes from a mixture of lipids [START_REF] Mccarthy | Separation of liquid domains in model membranes induced with high hydrostatic pressure[END_REF].

Pressure may have also an impact on more complex macromolecules, as lipoproteins.

Recent studies on human plasma lipoproteins under HHP revealed a reduced flexibility and higher compressibility of its triglyceride rich form, the form associated to pathological health conditions [START_REF] Golub | High hydrostatic pressure specifically affects molecular dynamics and shape of lowdensity lipoprotein particles[END_REF][START_REF] Lehofer | High Hydrostatic Pressure Induces a Lipid Phase Transition and Molecular Rearrangements in Low-Density Lipoprotein Nanoparticles[END_REF].

Cells

Surprisingly, pressure is the unique physical parameter capable of inducing heat-shock and cold-shock proteins' as a cell response to a same stress. Escherichia coli (E. coli) exposed to 53 MPa induces 55 proteins, 11 heat-shock and 4 cold-shock proteins among them. E. coli may try to counteract the damage produced by pressure at different cell levels, such as stability of macromolecules and membrane functionality [START_REF] Welch | Stress response in Escherichia coli induced by elevated hydrostatic pressure[END_REF].

As mentioned, pressures up to 100 MPa affects most of the cellular functions as enzymatic reactions, gene expression, cell motility and morphology, and cell membrane (figure 2). Since pressure is transmitted through a fluid, it will be transmitted uniformly (Pascal's law) over the whole cell and therefore, it makes difficult to identify, if it exists, the main cause of the cell death. Moreover, pressure-induced cell inactivation relies on the type of microorganism and its physiological conditions, such as water content and salt presence.

Overall, eukayotes are more pressure-sensitive than prokaryotes and piezosensitive bacilli and spiral-shaped bacteria are inactivated at lower pressures than cocci [START_REF] Huang | Responses of microorganisms to high hydrostatic pressure processing[END_REF][START_REF] Ludwig | Effects of High Pressure on Bacteria and Funghi[END_REF].

For instance, pressures above 150 MPa usually reduces the viability of mammalian cells and may induce cell death by apoptosis from 200 MPa or through a necrotic-like pathway at 300 MPa [START_REF] Frey | Hydrostatic pressure induced death of mammalian cells engages pathways related to apoptosis or necrosis[END_REF]. On the other hand, bacteria cocci may resist much higher pressure variations, for example, Staphylococcus aureus cell inactivation begins at 350

MPa [START_REF] Ludwig | Effects of High Pressure on Bacteria and Funghi[END_REF].

Figure 2. General effects of pressure on cellular macromolecules (black) and cells (red). Adapted from [66]

In addition, most gram-negative bacteria seem to be less resistant to pressure than grampositive [START_REF] Wuytack | Bacterial inactivation by high-pressure homogenisation and high hydrostatic pressure[END_REF]. Gram-negative bacteria possess a much more complex membrane which makes it a target for pressure damage [START_REF] Ritz | Effects of high hydrostatic pressure on membrane proteins of Salmonella typhimurium[END_REF]. Finally, microorganisms on the exponential growth phase present lower pressure-tolerance than on their stationary growth phase [START_REF] Pagán | Relationship between membrane damage and cell death in pressure-treated Escherichia coli cells: Differences between exponential-and stationary-phase cells and variation among strains[END_REF][START_REF] Patterson | Microbiology of pressure-treated foods[END_REF], for example, exponential-cells may present filamentous shape under pressure which can disrupt membrane functions [START_REF] Abe | Exploration of the Effects of High Hydrostatic Pressure on Microbial Growth, Physiology and Survival: Perspectives from Piezophysiology[END_REF][START_REF] Mañas | Morphological and Physiological Changes Induced by High Hydrostatic Pressure in Exponential-and Stationary-Phase Cells of Escherichia coli: Relationship with Cell Death[END_REF]. Moreover, stationary-cells have the capability to synthesize stress-response proteins to adapt and, therefore, better resist to different harsh conditions [START_REF] Hill | Bacterial stress response in Listeria monocytogenes: Jumping the hurdles imposed by minimal processing[END_REF].

Spores present formidable high resistivity to harsh environments, likely due to their structure with numerous protective layers and their low water content [START_REF] Black | Response of spores to high-pressure processing[END_REF].

Interestingly, relative moderate pressures (50 to 300 MPa) cause the germination of a dormant spore. Though, higher pressures are often less effective to induce germination [START_REF] Michiels | High Pressure Microbiology[END_REF]. Pressure alone is not very effective to inactivate bacterial spores and a treatment with temperature is necessary [START_REF] Wang | Recent Advances in Food Processing Using High Hydrostatic Pressure Technology[END_REF].

Pressure Adaptation in Piezophiles

The biodiversity of piezophiles is huge [START_REF] Orcutt | Microbial Ecology of the Dark Ocean above, at, and below the Seafloor[END_REF][START_REF] Jorgensen | A Starving Majority Deep Beneath the Seafloor[END_REF][START_REF] Howe | Deep-sea hydrothermal vent fauna: evolution, dispersal, succession and biogeography[END_REF][START_REF] Orcutt | Microbial activity in the marine deep biosphere: Progress and prospects[END_REF]. Organisms adapted to pressure include unicellular bacteria, archaea, eukaryotes as invertebrates and fishes and even deep diving marine mammals [START_REF] Grassle | Hydrothermal Vent Animals: Distribution and Biology[END_REF][START_REF] Kelly | High contents of trimethylamine oxide correlating with depth in deep-sea teleost fishes, skates, and decapod crustaceans[END_REF][START_REF] Dover | Evolution and Biogeography of Deep-Sea Vent and Seep Invertebrates[END_REF]. For example, in several hydrothermal vents have been found large invertebrates like mussels, crabs and shrimps and some marine mammals can dive exposed to almost 20 MPa without any negative symptom [START_REF] Castellini | Adaptations to pressure in the RBC metabolism of diving mammals[END_REF][START_REF] Bliznyuk | Marine Mammals' NMDA Receptor Structure: Possible Adaptation to High Pressure Environment[END_REF].

There are differences in the microorganisms adapted to HP, as some are also adapted to low temperatures (psychrophiles) and others to high temperatures (hyperthermophiles), which increases the piezophilic diversity. Hyperthermophilic and piezophilic organisms are found near vent sites, where temperatures can go from 350 °C to 2°C in only few cm distances. Altogether, this represents a source of microorganisms with great potential for technological and pharmaceutical applications, such as new enzymes, antibiotics or cancer cell line active derivatives. Most piezophiles identified in the deep sea are bacteria, they are psychrophiles and piezophiles, but in the niche of hydrothermal vents, most microorganisms are archaea, they are thermophiles and piezophiles [START_REF] Fang | Deep-sea piezosphere and piezophiles : geomicrobiology and biogeochemistry[END_REF] (figure 3). This indicates that piezophiles may follow a different adaptation to temperature and makes it difficult to separate the adaptation to pressure from other stress adaptation since microorganisms may adopted a common strategy to cope with various environmental stresses [START_REF] Zhang | Current developments in marine microbiology: High-pressure biotechnology and the genetic engineering of piezophiles[END_REF]. High pressure has a close relation with other stresses as temperature, pH and salt. Moreover, there are relatively few studies on adaptation mechanisms to pressure and not all adaptations are equally used by the different piezophiles studied. Additionally, not all necessary pressure adaptation mechanisms are deleted at ambient pressure, and therefore the homologue enzymes may use the same mechanism [START_REF] Oger | The many ways of coping with pressure[END_REF]. Finally, piezophilic biospheres are closely related to nutrient limitation and most piezophiles must also be oligotrophes [START_REF] Daniel | Origins of life and biochemistry under high-pressure conditions[END_REF].

Extremophiles have developed great capabilities to adapt to harsh and even fluctuating conditions (e.g. temperature, pressure, composition of the host rocks…) thanks to their own unique macromolecules, such as polysaccharides, lipids or enzymes and even specialized organs [START_REF] Bright | The biology of vestimentiferan tubeworms[END_REF]. These macromolecules adapted to HP present a high potential to develop new biotechnological applications.

Genomes

Marine biosphere contains mobile genetic elements, such as plasmids, bacteriophages, transposons, integrons, integrative conjugative elements and genomic islands [START_REF] Sobecky | Horizontal Gene Transfer and Mobile Genetic Elements in[END_REF]. All of them are essential to understand the high marine microbial diversification and thereby their adaptation [START_REF] Van Wolferen | How hyperthermophiles adapt to change their lives: DNA exchange in extreme conditions[END_REF][START_REF] Li | An integrative genomic island affects the adaptations of the piezophilic hyperthermophilic archaeon Pyrococcus yayanosii to high temperature and high hydrostatic pressure[END_REF]. Consequently, horizontal gene transfer of these mobile genetic elements may play an essential role in the microbial adaptation to pressure and other extreme conditions [START_REF] Fourn | An oxygen reduction chain in the hyperthermophilic anaerobe Thermotoga maritima highlights horizontal gene transfer between Thermococcales and Thermotogales[END_REF].

Recently, it has been described that the gene Ypr153w is possibly responsible for the tryptophan permease's Tat2 stability in Saccharomyces cerevisiae under pressure. It is a gene which has also been identified in other related species as Debaryomyces and Candida strains which have been isolated from sediment samples of deep sea floors [START_REF] Kurosaka | The YPR153W gene is essential for the pressure tolerance of tryptophan permease Tat2 in the yeast Saccharomyces cerevisiae[END_REF]. Another possible HP adaptation could be the 16s rRNA longer stems found in strains from Photobacterium, Colwellia and Shewanella [START_REF] Lauro | The unique 16S rRNA genes of piezophiles reflect both phylogeny and adaptation[END_REF].

The high genetic tractability and hyper-responsiveness to pressure of the piezophile Photobacterium profundum strain SS9 has made it a reference for studies on piezophile adaptation. It has been discovered a pressure-sensitive mutant of SS9 that lacks the recD gene, responsible for a DNA-binding protein. Besides, the transfer of the recD gene from SS9 to E.coli enabled the latter one to divide normally under HP. This indicates that RecD may have an important role for piezoadaptation together with a role of DNA metabolism and cell division [START_REF] Bidle | RecD function is required for high-pressure growth of a deep-sea bacterium[END_REF][START_REF] Abe | Pressure-regulated metabolism in microorganisms[END_REF].

To date, it has not been possible to detect any piezospecific gene. Consequently, it is not possible to determine if an organism is piezophile by molecular approaches, it is necessary to do cultivation approaches and to determine the growth rates at different pressures. However, it has been found that the pressure regulated operons ORFs 1-3 are distributed among different piezophilic Shewanella species [START_REF] Li | Distribution of the pressure-regulated operons in deep-sea bacteria[END_REF].

Proteins

Relatively few enzymes from piezophiles have been studied under pressure. Although there are no apparent differences between the crystal structure of an enzyme from a piezophile and its piezosensitive homologue, there is a variation in the stability between both enzymes caused by a difference in flexibility and hydration of the proteins [START_REF] Kitahara | High pressure NMR reveals active-site hinge motion of folate-bound Escherichia coli dihydrofolate reductase[END_REF].

Most molecular motion studies about pressure adaptation have been done in vitro, or investigating, for example, molecular dynamics [START_REF] Huang | Extreme biophysics: Enzymes under pressure[END_REF]. Nevertheless, nowadays in vivo studies have gained importance thanks to, for example, neutron scattering and NMR experiments that can examine timescales from few nanoseconds to hundreds of milliseconds [START_REF] Peters | Deep sea microbes probed by incoherent neutron scattering under high hydrostatic pressure[END_REF][START_REF] Martinez | High protein flexibility and reduced hydration water dynamics are key pressure adaptive strategies in prokaryotes[END_REF][START_REF] Boehr | Millisecond timescale fluctuations in dihydrofolate reductase are exquisitely sensitive to the bound ligands[END_REF].

It has been shown that some proteins are involved in HP adaptation as well as in adaptation to other stresses (Hsp60, Hsp70, OmpH, RecA, F1F0 ATPases, Cct and Tat2) [START_REF] Bartlett | High pressure influences on gene and protein expression[END_REF]. A system highly studied under pressure is the Omp/Tox system. The proteins ToxS and ToxR from P. profundum SS9 are responsible for regulating the genes that encode the membrane proteins OmpH, OmpL and OmpI. Pressure reduces the abundance and the activity of ToxR, which therefore upregulates the protein OmpH among others, the system acts as a piezometer. Regardless, the systems ToxS and ToxR do not confer HP adaptation and their role under pressure is not clear [START_REF] Simonato | Piezophilic adaptation: a genomic point of view[END_REF][START_REF] Bartlett | Pressure effects on in vivo microbial processes[END_REF].

The protein adaptation to extreme conditions is a balance between the imperative stability (higher number of bounds) to be functional and the flexibility (lower number of bounds) to be capable to adapt to different conditions [START_REF] Feller | Psychrophilic Enzymes: From Folding to Function and Biotechnology[END_REF]. One of the most studied enzymes is dihydrofolate reductase (DHFR). Studies comparing DHFR from the piezosensitive E. coli and from the facultative psychro-piezophile bacteria Moritella profunda reveals that applying pressure decreases EcDHFR activity and increases MpDHFR activity up to 50 MPa before diminishing its activity at higher HP. MpDHFR seems to have higher sensitivity to pressure due to its higher flexibility [START_REF] Ohmae | Pressure dependence of activity and stability of dihydrofolate reductases of the deep-sea bacterium Moritella profunda and Escherichia coli[END_REF]. A more flexible protein may explain the higher absolute activity of piezophile proteins [START_REF] Ichiye | Enzymes from piezophiles[END_REF].

However, most of studies are done in protein-isolated solutions, which differs from their native state. An innovative quasi-elastic neutron scattering study examined the dynamics from whole cells of the piezophile Thermococcus barophilus and the piezosensitive Thermococcus kodakarensis microorganism under atmospheric pressure and 40 MPa. This study revealed that the HP adaptation on whole cells is based on an overall higher proteins' flexibility and, in addition, on the modification of their hydration layers [START_REF] Martinez | High protein flexibility and reduced hydration water dynamics are key pressure adaptive strategies in prokaryotes[END_REF].

Proteins from piezophiles may have a larger total volume of small internal cavities, which makes the protein more compressible and less sensitive to distortion caused by pressure [START_REF] Ichiye | Enzymes from piezophiles[END_REF]. Moreover, the presence of small cavities allows water penetration at HP and consequently increases the hydration but, as seen in MpDHFR, cavities are not big enough to cause the protein denaturation but allow the protein to be more flexible. The presence of more small cavities could decrease the amount of water molecules contained in each cavity (a volume of 15A 3 is necessary for a single water molecule and an increase of approximately 45A 3 is required for each extra molecule [START_REF] Sonavane | Cavities and atomic packing in protein structures and interfaces[END_REF]). It is important to consider that cavities are not mere "packing defects" but that they play a role in conformational changes and in controlling binding and catalysis of the proteins [START_REF] Sonavane | Cavities and atomic packing in protein structures and interfaces[END_REF][START_REF] Collins | Cooperative water filling of a nonpolar protein cavity observed by high-pressure crystallography and simulation[END_REF].

Generally, monomeric proteins are more resistant to pressure than oligomeric proteins.

However, it has been shown that multimeric proteins may be adapted to resist to pressure. For example, studies on the hyperthermophile and piezophile TET3 peptidase from Pyrococcus horikoshii indicate that the protein multimerizes into a dodecamer structure instead of conserving its classical barrel-shape multimer conformation.

Dodecamer multimerization protects the hydrogen bonding between the different subunits and increases its stability against temperature and pressure up to 300 MPa [START_REF] Rosenbaum | Effects of hydrostatic pressure on the quaternary structure and enzymatic activity of a large peptidase complex from Pyrococcus horikoshii[END_REF].

A general extrinsic cell response to pressure-stress is the presence of piezolytes and other low weight organic compounds called osmolytes (e.g., sugars and amino acids) to protect the cell macromolecules, such as proteins, from pressure modification [START_REF] Jebbar | Microbial diversity and adaptation to high hydrostatic pressure in deep-sea hydrothermal vents prokaryotes[END_REF] and therefore adapt its dynamics. Some piezophiles accumulate these low-weight molecules in response to an increase on pressure and others to a decrease, indicating in the latter case that the growth at lower pressure than optimal is perceived as a stress for these piezophiles. For example, trimethylamine oxide (TMAO) is a pressure co-solute that helps proteins to remain active under HP in certain fishes and crustaceans [START_REF] Kelly | High contents of trimethylamine oxide correlating with depth in deep-sea teleost fishes, skates, and decapod crustaceans[END_REF][START_REF] Yancey | Unusual organic osmolytes in deep-sea animals: Adaptations to hydrostatic pressure and other perturbants[END_REF]. On the other hand, the hyperthermophilic and piezophilic Thermococcus barophilus accumulates mannosyl-glycerate when it is grown in non-optimal conditions (ambient pressure) [START_REF] Cario | Molecular chaperone accumulation as a function of stress evidences adaptation to high hydrostatic pressure in the piezophilic archaeon Thermococcus barophilus[END_REF]. This indicates that this archaeon perceives the lack of pressure as a stressful condition.

Few studies have been done on pressure adaptation of higher complex pluricellular organisms. For example, it is thought that the regulation of N-methyl-D-aspartate receptor (NMDR), a cell membrane protein found in nerve cells, is responsible for the absence of the high-pressure nervous syndrome (HPNS) on deep dive mammalians [START_REF] Bliznyuk | Marine Mammals' NMDA Receptor Structure: Possible Adaptation to High Pressure Environment[END_REF][START_REF] Brown | NMDA receptor regulation is involved in the limitation of physiological tolerance to both low temperature and high hydrostatic pressure[END_REF]. The regulation of this protein may be done by modulating its interaction with lipids, for example by the presence of cholesterol, and thanks to protein's particular tertiary structure in piezo-tolerant organisms.

Membrane lipids

Cells have the capability to modify their cell membrane lipid composition metabolically to maintain it in a functional liquid crystalline phase with specific functional physicochemical properties, such as fluidity, permeability and membrane curvature in spite of environmental stresses. This process is known as homeoviscous adaptation [START_REF] Sinensky | Homeoviscous Adaptation-A Homeostatic Process that Regulates the Viscosity of Membrane Lipids in Escherichia coli[END_REF].

Eukarya and bacteria possess lipids different from those in archaea but their homeoviscous adaptations have similarities. Eukaryal and bacterial lipids are composed by straight hydrocarbon chains linked by ester bonds on 1,2 -sn-glycerol and a phosphodiester-linked polar group or sugar. On the other hand, archaeal lipids have isoprenoid hydrocarbon chains bounded by ether bonds on 2,3-sn-glycerol. Partly, the adaptation of archaea to extreme conditions may thus rise from their particular lipid structure [START_REF] Van De Vossenberg | The essence of being extremophilic: The role of the unique archaeal membrane lipids[END_REF].

The common routes of lipid adaptation of bacterial and archaeal membranes are the change of the acyl chain length, the addition or removal of mono-unsaturated lipids and the change in the polar headgroups [START_REF] Oger | Adaptation of the membrane in Archaea[END_REF][START_REF] Siliakus | Adaptations of archaeal and bacterial membranes to variations in temperature, pH and pressure[END_REF]. Longer acyl chains are translated into more rigid membranes, in contrary adding just one unsaturation to lipid chains makes the membrane more permeable and larger headgroups increase the disruption of the membrane packing by increasing the membrane fluidity. Furthermore, archaea possess tetraether lipids, which may form a monolayer instead of the common bilayer. In addition, some archaeal species comprise lipids with cyclopentane rings and isoprenoid chains that are crosslinked. The change in the different ratios from di-and tetra-ether lipids and the presence of cyclopentanes and crosslinked chains modifies as well the properties of the cell membrane. Finally, psychrophilic bacteria present polyunsaturated fatty acids (PUFAs), such as eicosapentaenoic acid (EPA) and docosahexaenoic acid (DHA), which, just as lipids with one unsaturation, increase the permeability of the membrane under low temperatures [START_REF] Usui | Eicosapentaenoic acid plays a role in stabilizing dynamic membrane structure in the deep-sea piezophile Shewanella violacea: A study employing high-pressure time-resolved fluorescence anisotropy measurement[END_REF]. The function of PUFAs is not clear, one of the hypotheses is that requires less carbon and energy to obtain the same effect than the mono-unsaturated lipids [START_REF] Simonato | Piezophilic adaptation: a genomic point of view[END_REF][START_REF] Usui | Eicosapentaenoic acid plays a role in stabilizing dynamic membrane structure in the deep-sea piezophile Shewanella violacea: A study employing high-pressure time-resolved fluorescence anisotropy measurement[END_REF][START_REF] Myka | Adaptations of Cold-and Pressure-Loving Bacteria to the Deep-Sea Environment: Cell Envelope and Flagella[END_REF]. Another hypothesis is that they may play a role in cell division under HP, as demonstrated for bacteria Shewanella violacea [START_REF] Kawamoto | Favourable effects of eicosapentaenoic acid on the late step of the cell division in a piezophilic bacterium, Shewanella violacea DSS12, at high-hydrostatic pressures[END_REF].

Only two studies have been done to examine the lipid composition under pressure: one on Methanocaldococcus jannaschii and another on T. barophilus [START_REF] Kaneshiro | Pressure Effects on the Composition and Thermal-Behavior of Lipids From the Deep-Sea Thermophile Methanococcus-Jannaschii[END_REF][START_REF] Cario | Membrane homeoviscous adaptation in the piezo-hyperthermophilic archaeon Thermococcus barophilus[END_REF]. Both present an increase in the diether : tetraether lipid ratio to counteract the increase in rigidity provoked by pressure on the cell membrane.

Pressure Biotechnological Applications

HP application is mostly used in food processing since it does not affect non-covalent interactions (at least up to 2 GPa) and it can inactivate bacteria and viruses without changing markedly nutrients and flavours of food. Furthermore, pressure can change the reaction rates, which may favour the extraction of the required product [START_REF] Balasubramaniam | Principles and Application of High Pressure-Based Technologies in the Food Industry[END_REF].

Besides, pressure may be used for diverse biotechnological and biopharmaceutical applications, for example to explore new therapies [START_REF] Hradilova | Generation of dendritic cell-based vaccine using high hydrostatic pressure for non-small cell lung cancer immunotherapy[END_REF][START_REF] Adkins | High hydrostatic pressure in cancer immunotherapy and biomedicine[END_REF] and conserve vaccines [START_REF] Adkins | High hydrostatic pressure in cancer immunotherapy and biomedicine[END_REF], improve cryopreservation [START_REF] Gu | Improved cryotolerance and developmental competence of human oocytes matured in vitro by transient hydrostatic pressure treatment prior to vitrification[END_REF] or for orthopaedics' surgery [START_REF] Brouillet | Inactivation of Staphylococcus aureus in calcium phosphate biomaterials via isostatic compression[END_REF].

Food Industry

HP (400 -600 MPa) inactivates microorganisms, like yeast, molds and viruses. It affects the cell at different levels, such as nutrient transport and cell reproduction, reaching to the cell death [START_REF] Wang | Recent Advances in Food Processing Using High Hydrostatic Pressure Technology[END_REF][START_REF] Huang | Current status and future trends of high-pressure processing in food industry[END_REF]. Moreover, HP hardly affects low-molecular weight compounds (ex: vitamins, amino acids, flavour molecules) so, organoleptic and nutritional properties are only slightly modified [START_REF] Farkas | High pressure processing[END_REF]. On the other hand, HP alone cannot inactivate bacterial spores and thus a combination with other variations of pH, chemicals or thermal processes maybe needed. Nevertheless, pressure reduces considerably the working temperature, as 70 °C instead of 180 °C is enough to inactivate spores if it is combined with 600 MPa [START_REF] Wang | Recent Advances in Food Processing Using High Hydrostatic Pressure Technology[END_REF]. Such decrease in temperature can help to preserve quality and minimise off-flavour generations. Therefore, HP techniques are useful as a complement on thermal process but also to inactivate microorganisms on products where temperature cannot be applied. As an illustration, high pressure pasteurization of cold-pressed juices eliminate pathogens of juices without impairment of its fresh-like qualities and increasing the shelf life of the product [START_REF] Polydera | Comparative shelf life study and vitamin C loss kinetics in pasteurised and high pressure processed reconstituted orange juice[END_REF][START_REF] Bull | The effect of high pressure processing on the microbial, physical and chemical properties of Valencia and Navel orange juice[END_REF].

HP extends shelf life of a high variety of food products. For example, fresh shrimp treated at 435 MPa has a shelf life of 15 days, three times longer than the shelf life of the untreated shrimp [START_REF] Kaur | Chilled storage of high pressure processed black tiger shrimp (Penaeus monodon)[END_REF]. Fresh cheese treated at 300-400 MPa has a shelf life at 4 °C of 14-21 days, which is greatly higher than the 7 days for the untreated cheese [START_REF] Evert-Arriagada | Effect of high pressure on fresh cheese shelf-life[END_REF].

Food is a complex matrix and inactivation efficiency depends on different factors as treatment conditions, microorganisms to inactivate and its food matrix characteristics.

For example, meat treated at 300 MPa has a cooked like appearance but if it is processed at 100-200 MPa and 60 °C, it is more tender than the untreated meat [START_REF] Sazonova | Application of high-pressure processing for safety and shelf-life quality of meat-a review. 11th[END_REF].

Consequently, inactivation conditions must be defined for every food product. HP is not efficient for low water content food (such as flour) or food with high content of air bubbles and if the food needs to be wrapped before treatment only plastic packaging is acceptable, as packaging material needs a compressibility of at least 15% [START_REF] Huang | Current status and future trends of high-pressure processing in food industry[END_REF].

Finally, HP may also be applied as a pre-treatment. On the one hand, it has been demonstrated that the application of sublethal HP on cells gives them cross-resistance to other stresses. For example, the most studied probiotic, Lactobacillus rhamnosus, is more resistant to heat after an application of 100 MPa for 10 min [START_REF] Ananta | Evidence on the role of protein biosynthesis in the induction of heat tolerance of Lactobacillus rhamnosus GG by pressure pre-treatment[END_REF]. This opens the possibility to inactivate the pathogens on probiotic products, conferring a health benefit.

On the other hand, pre-treatment can be useful to facilitate the extraction of internal nutritional components. HP makes the cells more permeable, increasing the mass transfer rate and as a result incresing the release of extracts. It has been shown that the time extraction of caffeine from green tea leaves is reduced from 20h to 1 min if a pressure of 500 MPa is applied; extraction of anthocyanin red grape skin is increased by 23% by applying 600 MPa of pressure and the extraction yield of gingenosides from Panax quinquefolium root increases linearly between 100 MPa and 500 MPa [START_REF] Zhao | Pretreatments for the Efficient Extraction of Bioactive Compounds from Plant-Based Biomaterials[END_REF].

3.2.

Allergenicity and digestibility

Several proteins can provoke allergic reactions caused by an immune disorder on the IgE binding. Because HP tend to denature proteins and as a result, it has been shown to induce a modification of their allergenicity [START_REF] Vanga | Review of conventional and novel food processing methods on food allergens[END_REF], both on protein solutions and on food systems. For example, pressures of 300-700 MPa reduce the allergenicity of a ginkgo seed protein and of soybean allergens [START_REF] Zhou | Improvement of allergenicity and functional properties of proteins from ginkgo seeds by high hydrostatic pressure treatment[END_REF]. Another interesting example is the use of HP together with proteases to obtain hypoallergenic rice [START_REF] Kim | Effects of high hydrostatic pressure treatment on the enhancement of functional components of germinated rough rice (Oryza sativa L.)[END_REF][START_REF] Peñas | High hydrostatic pressure effects on immunoreactivity and nutritional quality of soybean products[END_REF]. However, the effect of HP on allergenicity is not universal. There is no allergenicity change caused by pressure on almonds, or on the protein Mald1 from apples [START_REF] Li | Effect of Pulsed Ultraviolet Light and High Hydrostatic Pressure on the Antigenicity of Almond Protein Extracts[END_REF][START_REF] Houska | Is high-pressure treatment able to modify the allergenicity of the main apple juice allergen, Mal d1?[END_REF]. Mald1 native state possess a high internal cavity occupied by water [START_REF] Ahammer | Structure of the Major Apple Allergen Mal d 1[END_REF] and therefore, pressure may not be able to conform significant hydration changes since the protein is already highly hydrated.

HP may have not only an effect on the allergenicity of the food products but can help to increase its digestibility by exposing inaccessible sites of proteins and, thereby, enhancing the efficiency of protein hydrolysis. For example, the time required for proteolysis of β-lactoglobulin, the major allergen in cow's milk, is reduced from 48h to 20 min at 200 MPa [START_REF] Chicón | Hydrolysis under High Hydrostatic Pressure as a Means To Reduce the Binding of β-Lactoglobulin to Immunoglobulin E from Human Sera[END_REF].

Antiviral vaccines

Several viruses are inactivated or dissociated by pressure. Under pressure, the atomic contacts between subunits are replaced by interactions with the solvent and therefore once pressure is released, viruses cannot come back to their native form. For example, pressure inactivates picornaviruses by causing the lack of VP4 from the intern capsid [START_REF] Oliveira | Low temperature and pressure stability of picornaviruses: Implications for virus uncoating[END_REF]. Both viruses with polyhedral and helicoidal symmetry are sensitive to pressure.

Even so, not all viruses are equally reactive to pressure, for example, the foot-andmouth-disease picanovirus is highly sensitive and poliovirus is much higher resistant against HP [START_REF] Oliveira | Low temperature and pressure stability of picornaviruses: Implications for virus uncoating[END_REF].

Interestingly, virus re-associate under their fusogenic state under pressure, a less infectious and highly immunogenic form [START_REF] Oliveira | Low temperature and pressure stability of picornaviruses: Implications for virus uncoating[END_REF][START_REF] Silva | The use of hydrostatic pressure as a tool to study viruses and other macromolecular assemblages[END_REF][START_REF] Dumard | Stability of different influenza subtypes: How can high hydrostatic pressure be a useful tool for vaccine development?[END_REF]. This is why high pressure has been suggested for antiviral vaccine development. It has been demonstrated that immunization against HP-inactivated virus is equally effective as against intact virus and have higher immunity response than isolated viral subunits [START_REF] Silva | Pressure induces folding intermediates that are crucial for protein-DNA recognition and virus assembly[END_REF][START_REF] Silva | Effects of hydrostatic pressure on a membrane-enveloped virus: high immunogenicity of the pressureinactivated virus[END_REF].

Bio-purification

An antigen may be purified from its medium by affinity chromatography due to a steric recognition with an antibody linked to a matrix. The recognition causes an increase in molecular volume and, as pressure causes a volume decrease, it could be useful to apply pressures to dissociate the product of interest without using drastic elution process which reduce the lifetime of matrices [START_REF] Lemay | The use of high pressure for separation and production of bioactive molecules[END_REF]. This has been demonstrated for the recovery of β-galactosidase: four 15 min cycles of 150 MPa at 4°C recovers 32% of E.

coli β-galactosidase compared with the 46% recovered by adding a solution of pH=11 [START_REF] Estevez-Burugorri | High-pressure induced recovery of β-galactosidases from immunoadsorbents: Stability of antigens and antibodies. Comparison with usual elution procedures[END_REF]. Although the product yield is lower when using HP, the method is simpler and have a lower impact on matrices than the current elution process.

HP ability to disrupt immune complexes has been proved on anti-prostate specific antibody from its antigen [START_REF] Estevez-Burugorri | High-pressure induced recovery of β-galactosidases from immunoadsorbents: Stability of antigens and antibodies. Comparison with usual elution procedures[END_REF][START_REF] Cheung | High-pressure-mediated dissociation of immune complexes demonstrated in model systems[END_REF], its dissociation was increased by 22-37% when pressures from 140 to 550 MPa were applied. Pressure may also optimize the dissociation of amphiphilic biomolecules from a fixed adsorbent: 80% of Triton-X can be recovered form a bed absorption if a pressure of 250 MPa is applied on the system [START_REF] Niemeyer | An innovative approach for sorptive separation of amphiphilic biomolecules applying high hydrostatic pressure[END_REF].

Finally, as pressure can dissociate aggregates, it may be used for the recovery of proteins from inclusion bodies, i.e. aggregates of incomplete folded proteins.

Traditionally, to separate proteins from inclusion bodies is necessary to use high concentrations of agents that destroy the spatial structure of proteins with a necessary subsequent difficult refolding. However, a pressure of 240 MPa is effective to dissociate the inclusion bodies of endostatins and a subsequent application of 40 MPa induces the refolding of 78% of the protein [START_REF] Chura-Chambi | An analysis of the factors that affect the dissociation of inclusion bodies and the refolding of endostatin under high pressure[END_REF].

Modulation of Cell Activity

Already relative low applied pressures can enhance the cell activity to our profit, as for example observed at 10 MPa for ethanol production by Saccharomyces cerevisiae [START_REF] Picard | In situ monitoring by quantitative Raman spectroscopy of alcoholic fermentation by Saccharomyces cerevisiae under high pressure[END_REF] which occurs 3 times faster than at atmospheric pressure. Another example is on the fermentation by Clostridium thermocellum [START_REF] Bothun | Metabolic selectivity and growth of Clostridium thermocellum in continuous culture under elevated hydrostatic pressure[END_REF], this Clostridium converts cellobiose to biofuels and other chemicals but also synthesizes other non-desired products (acetate, H 2 , CO 2 ). When the fermentation happens under pressure of 7 or 17 MPa, the microorganism modifies the metabolic pathways and shifts the production to desired metabolites, reaching an increase of 60-fold.

However, as HP is considered a stress for most cells, it will translate into the expense of additional energy for cell maintenance and growth, reducing the product yield. For example, HP reduces the fermentation rate of lactic acid fermentation due to the inhibitory effect on the growth of Streptococcus thermophilus, Lactobacillus bulgaricus and Bifidobacterium lactis [START_REF] Mota | Probiotic yogurt production under high pressure and the possible use of pressure as an on/off switch to stop/start fermentation[END_REF]. To avoid the loss of efficiency rates under HP, efforts are made to enhance the resistance to HP of mesophilic microorganisms, leading to HP resistant organisms with higher performance under HP [START_REF] Mota | Microorganisms under high pressure -Adaptation, growth and biotechnological potential[END_REF].

Biotechnological Applications of Piezophiles

There are more than 3000 enzymes identified to date and most of them are used for biotechnological applications. Nevertheless, these enzymes are not enough to respond to the new technological challenges that appear each day [START_REF] Dumorné | Extremozymes: A potential source for industrial applications[END_REF]. One of the problems is the stability of the enzymes under industrial conditions, so it is necessary to find enzymes which are highly resistant to harsh conditions and here deep-sea enzymes may play a major role. Pressure-stable enzymes are capable of sustaining biocatalysis under HP, modifying therefore specific enzymatic reactions, and have even higher thermostability.

For example, Biolabs ® has already commercialized a DNA polymerase from a hyperthermophile and piezophile Pyrococcus, which presents a half-life of 23 hours at 95 °C. Moreover, piezophilic enzymes may possess different properties than their surface homologues, which may open new possibilities for industry [START_REF] Schroeder | Bioactive Marine Molecules and Derivatives with Biopharmaceutical Potential[END_REF]. The market for industrial enzymes is growing every year and the exploitation of extremozymes is a huge and mostly unexplored resource [START_REF] Dalmaso | Marine extremophiles a source of hydrolases for biotechnological applications[END_REF].

As we have seen, lipids from extremophiles are unique. Archaea in particular contain lipids which confer to the cell a highly stable and impermeable membrane. The unique stability may be used in biotechnological or pharmaceutical applications, for example to protect therapeutic peptides from the harsh environment of the gastrointestinal tract [START_REF] Benvegnu | New Generation of Liposomes Called Archaeosomes Based on Natural or Synthetic Archaeal Lipids as Innovative Formulations for Drug Delivery[END_REF][START_REF] Jacobsen | Archaeal lipids in oral delivery of therapeutic peptides[END_REF]. Additionally, many piezophilic bacteria contain omega 3-PUFAs associated with their cell membrane, which are precursors to hormones and hormonelike molecules in many animals. Consequently, it could be used for hypertriglyceridemia diseases and clinical studies for this purpose have already been approved [START_REF] Schroeder | Bioactive Marine Molecules and Derivatives with Biopharmaceutical Potential[END_REF].

The high marine biodiversity has woken up the interest to search new compounds with biopharmaceutical potential [START_REF] Tortorella | Antibiotics from Deep-Sea Microorganisms: Current Discoveries and[END_REF]. It has been discovered marine derived molecules with, for example, antitumor potential, for the treatment of pain or antimicrobial activities [START_REF] Schroeder | Bioactive Marine Molecules and Derivatives with Biopharmaceutical Potential[END_REF][START_REF] Jacobsen | Archaeal lipids in oral delivery of therapeutic peptides[END_REF][START_REF] Tortorella | Antibiotics from Deep-Sea Microorganisms: Current Discoveries and[END_REF]. For example, studies have identified some bioactive compounds from marine echinoderms (such as the piezotolerant Cucumaria frondosa)

with antiproliferative, antimetastatic and immunomodulatory activities [163].

All pressure-specific impacts allow to modify macromolecules and cells in unique ways.

Food industry was pioneer in using pressure to inactivate microorganisms and as a pretreatment, but pressure capability does not stop here. Promising applications, such as antiviral vaccines, the use of pressure for bio-purification or to vary cell activities has led to a greater interest on this physical parameter. In addition, piezophile organisms opens a range of possibilities to use pressure-adapted molecules, and to increase, for example, the reaction rates, or to find new active macromolecules.

EXTRA Biopharmaceutical Bacterial ghosts as delivery systems

Bacterial ghosts are usually obtained by the expression of a lyses gene that leads to the formation of a transmembrane tunnel with its consequent cell material leakage. These products retains their immunogenic properties as their cell surface is not affected.

Bacterial ghosts obtained by the expression of lysis gene does

Vaccines preservation

HP can be also considered as a stabilizing method for vaccine preservations. It has been demonstrated that high pressure stabilizes attenuated poliovirus against temperatures of 37°C, giving a higher thermal resistance to this virus. Actual trivalent oral polio vaccine (OPV), as most live vaccines, is heat labile, particularly OPV is between the most heat labile vaccines and it needs to be stored frozen and used immediately after thawing. HP treatment may solve the problems of refrigeration guarantying a higher stabilised and effective poliovirus vaccine. This technique is still under development; nevertheless, it is under clinical testings for several cancers indications, such as lung, prostate and ovarian cancers.

Oncology

Orthopaedics surgery

The actual treatments for exvivo devitalization of infected or tumour affected bone segments are based on irradiation, autoclaving, thermal treatment or use of chemicals.

Even if the devitalization is total using these techniques, they reduce the biomechanical and biological integrity of the bone. The procedure is based on the resection, the extracorporeal disinfection and finally the devitalization before its reimplantation.

High pressure can also be used to exvivo devitalization of bone segments without altering the main proteins present in bone tissue (fibronectine, vitronectine and type I collagen). HP inactivates several bacteria, fungi and virus, however it is necessary to take intoaccount that bacteria in bones are less sensitive to HP than in solution.

For example, HP treatment of 300 MPa for 10 min of a cancerous bone segment is sufficient to inactivate all bone cells, including the bone tumour cells. This possible treatment mainly concerns bone segments and tendons. It has been demonstrated that HP treated Achilles tendons and trabecular bones does not loss their properties after a pressure treatment at 600 MPa. Finally, the revitalization of high pressure treated bone segments has been demonstrated indicating that a successful implementation of HP treated bone segments may be possible.

Moreover, HP may be useful to disinfect biomaterials as prosthesis or bone plates and surgical equipment as screws. It has been demonstrated that a pressure of 300 MPa and 40°C for 30 min disinfect screws contaminated with Staphylococcus aureus.

Cartilage stimulus

With the development of tissue engineering, engineered meniscus have been a new option to combat knee cartilage diseases. However, engineered meniscus constructs, without the correct stimulation, possess a lower organization and mechanical properties than the native meniscus.

Cartilage is an avascular and aneural tissue that detects HP, with an adequate stimulation, as a mechanical sign to increase its regeneration. In a normal activity, knee cartilage and synovial joint are hydrostatically pressurized reaching pressures up to 10 MPa. For example, it has been demonstrated that static pressure of 5 MPa for 1.5h increases the glycosaminoglycan incorporation in explants cultures. Interestingly, the effect of pressure on cartilage differs if the study is done on a chondrocytes monolayer or on a three dimensional matrix, which indicates that the interaction cellextracellular matrix may play a role in HP regulation.

Figure 3 .

 3 Figure 3. Op/mal growth temperature and pressure for some piezophilic bacteria (blue) and archaea (red).

Dendritic cell based vaccines

  Shinitzky and colleagues have been explored the use of HP killed tumor cells as a wholecell vaccine, they have demonstrated that a pressure of 100-200 MPa increases the cell immunogenicity. However, since their phase I study in 2000, there are no more clinical studies using this approach.Recently, dendritic cell (DC, antigen presenting cells from the mammalian immune system) based vaccines have been studied to obtain a vaccine for tumours. This approach uses pressures of 200 -300 MPa to kill cancer cell lines, pressure will reveal immunogenic sites on the killed cancer cell surface. Then, these molecules interacts with the specific receptors presents on the surface of DC and activates the phagocytosis of killed cells by DC. Finally, mature DC will activate tumour cell specific immune responses and therefore it could modify the clinical response of the patient with cancer. Moreover, HP killed cells have easy cryopreservation without affecting their immunogenicity.


domain: info.info-ni math info

text:
Gayane Vardoyan

Saikat Guha

Philippe Nain

Don Towsley

On the Capacity Region of Bipartite and Tripartite Entanglement Switching

We study a quantum switch serving a set of users in a star topology. The function of the switch is to create bipartite or tripartite entangled state among users at the highest possible rates at a fixed ratio. We model a set of randomized switching policies. Discovering that some are better than others, we present analytical results for the case where the switch stores one qubit per user, and find that the best policies outperform a time division multiplexing (TDM) policy for sharing the switch between bipartite and tripartite state generation. This performance improvement decreases as the number of users grows. The model is easily augmented to study the capacity region in the presence of qubit decoherence, obtaining similar results. Moreover, decoherence appears to have little effect on capacity. We also study a smaller class of policies when the switch stores two qubits per user.

INTRODUCTION

Multi-qubit entangled states are fundamental ingredients of several quantum computation, sensing, and security applications. Consequently there is a need for a quantum network that can generate such entanglement on demand between pairs and groups of users [START_REF] Pant | Routing Entanglement in the Quantum Internet[END_REF][START_REF] Schoute | Shortcuts to quantum network routing[END_REF][START_REF] Van Meter | Quantum networking[END_REF]. In this paper, we study the performance of the simplest multi-user network, a star-topology quantum switch connecting k users, where each user is connected to the switch via a separate link. Bipartite, two-qubit maximally-entangled states, i.e., Bell pairs (or EPR states) are generated at a constant rate across each link, with the qubits stored at local quantum memories at each end of the links. As these link entanglements start appearing, the switch uses two-qubit Bell-state measurement (BSM) between pairs of locally-held qubits and three-qubit Greenberger-Horne-Zeilinger (GHZ) basis measurements between triples of locally-held qubits to provide two-qubit and three-qubit entanglements to pairs and triples of users, respectively [START_REF] Nielsen | Quantum Computation and Quantum Information[END_REF]. The capacity of such a switch to provide these two types of entanglements to the users depends on the switching mechanism, the number of quantum memories and their decoherence rates, and the number of links.

In this paper, we study the capacity region when the switch can store either B = 1 or B = 2 qubits for each link at any given time. The number of quantum memories available to a link is referred to as its buffer size. We consider a

Copyright is held by author/owner(s).

simple time division multiplexing (TDM) policy between the two types of entanglements, along with a class of randomized policies. When properly configured, the latter provide higher capacities than TDM. However the relative difference between the two policies goes to zero as k → ∞. We also observe that increasing the number of memories from one to two increases capacity but that the increase diminishes as k increases. We also explore the effect that decoherence-the locally stored qubits at each end of the link being subject to a noise process that reduces the entanglement between the two qubits-has on capacity. In the cases of B = 1 with and without decoherence, we have simple closed form expressions for capacity whereas for the case of B = 2, our results are numerical.

The remainder of this paper is organized as follows: in Section 2, we provide relevant background and related work. In Section 3, we formulate the problem and propose a method for solving it. In Section 4, we present the case where the system has a per-link buffer of size one, and provide analytical and numerical results. In Section 5, we present numerical results for the case where the system has a per-link buffer of size two and observe similar behavior to the buffer size one case. In Section 6, we introduce a simple technique for modeling quantum state decoherence and use it to examine the effect of decoherence on the bipartite-tripartite capacity region for systems with per-link buffer sizes one and two. For the former, we also have analytical results. We make concluding remarks in Section 7.

BACKGROUND AND RELATED WORK

Bell states are an integral part of a diverse set of distributed quantum applications, including Quantum Key Distribution (QKD) [START_REF] Bennett | Quantum cryptography without Bell's theorem[END_REF][START_REF] Ekert | Quantum cryptography based on Bell's theorem[END_REF], superdense coding [START_REF] Bennett | Communication via one-and two-particle operators on Einstein-Podolsky-Rosen states[END_REF], teleportation [START_REF] Bennett | Teleporting an unknown quantum state via dual classical and Einstein-Podolsky-Rosen channels[END_REF], and distributed quantum computation [START_REF] Jiang | Distributed quantum computation based on small quantum registers[END_REF]. Similarly, GHZ states can be used to implement a variety of quantum protocols, such as cryptographic conferencing [START_REF] Grasselli | Finite-key effects in multipartite quantum key distribution protocols[END_REF], quantum sensing [START_REF] Eldredge | Optimal and secure measurement protocols for quantum sensor networks[END_REF], and multipartite generalization of superdense coding [START_REF] Hao | Controlled dense coding using the Greenberger-Horne-Zeilinger state[END_REF]. The advantage of these applications is that they offer functionality that cannot be achieved classically, e.g., information-theoretic security. However, quantum distributed tasks typically require reliable transport of quantum states; this can be a significant challenge due to the exponential rate-versus-distance decay [START_REF] Pirandola | Fundamental limits of repeaterless quantum communications[END_REF]. Quantum repeaters positioned between communicating parties alleviate this issue [START_REF] Guha | Rate-loss Analysis of an Efficient Quantum Repeater Architecture[END_REF]. In this work, we use the term "quantum switch" instead of "repeater" to indicate that the former is equipped with entanglement switching logic.

A mathematical model for a quantum switch was origi-nally introduced in [START_REF] Vardoyan | On the stochastic analysis of a quantum entanglement switch[END_REF]. There, the authors study a switch that serves only BSMs, but state decoherence, link heterogeneity, and arbitrary buffer sizes (including infinite) are considered. In [START_REF] Nain | On the Analysis of a Multipartite Entanglement Distribution Switch[END_REF], the authors study a multipartite entanglement distribution switch that serves n-partite GHZ states to users, for n ≥ 3. In this work, links are assumed to be identical and the effects of state decoherence negligible. In contrast to this prior work, we no longer assume that the switch serves only one type of entangled state, i.e., we allow n to be either two or three, and our goal is to design and evaluate a suitable switching policy. Another contrast to [START_REF] Nain | On the Analysis of a Multipartite Entanglement Distribution Switch[END_REF] is that there, the quantum switch is modeled to have an infinite number of quantum memories, while in this work we consider finite buffer sizes that scale with the number of links.

SYSTEM DESCRIPTION

We consider a switch that connects k users over k separate links. The creation of an end-to-end entanglement requires two steps. First two-qubit Bell states are generated pairwise between a qubit stored locally at the switch and a qubit owned by a user. Once such link-level two-qubit entangled states have been created, the switch performs joint (entangling) measurements (over j ≥ 2 locally-held qubits that are entangled with qubits held by j distinct users), which, if successful, produces a j-qubit maximally-entangled state between the corresponding j users. Link-level entanglement generation, as well as entangling measurements, when realized with practical systems, are inherently probabilistic [START_REF] Guha | Rate-loss Analysis of an Efficient Quantum Repeater Architecture[END_REF]. We assume that only two-user (two-qubit) and three-user (three-qubit) entanglements are created, i.e., BSMs and 3qubit GHZ basis measurements are done at the switch. For simplicity, we will assume that these j = 2 or 3 qubit measurements at the switch take negligible time and always succeed.

Each link attempts two-qubit entanglements in each time slot of length τ seconds, and with probability p, establishes one entangled pair successfully. For simplicity, we model the time to successfully create a link entanglement as an exponential random variable with mean 1/µ = τ /p. We assume that each link can store B = 1, 2, . . . qubits. We also assume that qubits at the switch can decohere and model decoherence time as an exponential r.v. with mean 1/α. We assume a step-function decoherence model where the twoqubit entanglement goes from a maximally-entangled qubit pair (one ebit) to zero entanglement. In this paper, we only consider B = 1, 2. Last, when a qubit is stored at the switch, with its entangled pair stored at a user, we refer to this as a stored link entanglement.

We assume that all possible bipartite and tripartite user entanglements are of interest and consider two classes of probabilistic policies, one for B = 1 and the second for B = 2, that provide the flexibility to generate both types of entanglements with arbitrary rates. Policies in both classes incorporate the oldest link entanglement first (OLEF) rule whereby when a link entanglement is created it is always matched up with stored link entanglements when possible rather than be stored. This has the nice consequence, when coupled with the assumption that links are homogeneous but statistically independent, that the system can be modeled by a continuous time Markov chain (CTMC) where the state simply tracks the number of stored entanglements for two users. The next section describes the class of policies Figure 1: CTMC for a system with at least three links and buffer size one for each link. k is the number of links, µ is the rate of entanglement generation, and r1, r2, and r3 are parameters that specify the scheduling policy.

for B = 1 and Section 5 for the class of B = 2 policies.

PER-LINK BUFFER SIZE ONE

In this section, we assume that each link can store one qubit in the buffer, so that the per-link buffer size B = 1. We model this system using a CTMC, and by obtaining its stationary distribution, we are able to compute the capacity region of the switch. We discover that it is always possible to configure a randomized policy that outperforms TDM, although as the number of links grows, the advantage of using such a policy diminishes.

Description

In a system where the switch can make tripartite measurements, we must keep track of two variables for each state of the CTMC: each representing a link with a stored qubit. Hence, (1, 1) represents the state where two of the k links have a qubit stored, one each. Note that we do not need to keep track of all links individually due to the OLEF rule and link homogeneity assumption. States (1, 0) and (0, 0) represent cases where only one link has a stored qubit or no link has a qubit, respectively.

The system is fully described in Figure 1. For a variable x ∈ [0, 1], we use the notation x ≡ 1 -x. When the system is in state (0, 0), new entanglements are generated with rate kµ; this is the rate of transitioning from (0, 0) to (1, 0). When the system is in state (1, 0), any new entanglements generated on the link that already has one stored qubit causes the switch to drop one of the qubits. New entanglements on other links are generated with rate (k -1)µ, and the switch must decide whether to immediately use the two qubits for a BSM or keep both and wait for a new link entanglement. To generalize the policy as much as possible, we add a policy parameter, r1 ∈ [0, 1], that specifies the fraction of time the switch performs a BSM. Note that r1 = 1 corresponds to the policy of always using qubits for BSMs. While this maximizes C2, it also means that C3 = 0. Now, suppose that the system is in state (1, 1) and a third link generates an entanglement. This event occurs with rate (k -2)µ. The switch has two choices: either use all three qubits for a tripartite measurement, or choose two of them for a BSM. We add another policy parameter, r2 ∈ [0, 1], that specifies fractions of times the switch performs a BSM and tripartite measurements in the event of three qubits on Figure 2: Capacity region for a system of buffer size one and three links. The red line represents the set of TDM policies. three different links. Another event that can occur in the (1, 1) state is the generation of an entanglement on either of the two links that already have stored entanglements. This event occurs with rate 2µ. Since B = 1, the switch cannot store the new entanglement. A decision must be made: to either discard one of the link entanglements (and remain in state (1, 1)) or perform a BSM on two of them and keep the third (and transition to state (1, 0)). Since it is not clear which policy is most advantageous, we add another parameter, r3 ∈ [0, 1], which specifies the fraction of time that the switch performs a BSM when it resides in this state.

Numerical Results

We plot the capacity region for the switch with B = 1 for all values of r1, r2, r3 ∈ [0, 1] and compare it against TDM. The entanglement generation rate µ simply scales the capacities, so we set it equal to one. In Figure 2, the number of links is three, and the TDM line (comprised by the set of TDM switching policies) is shown in red. Clearly, it is possible to design a policy that yields better performance than TDM: the triangular blue region above TDM represents the maximum capacities of the set of such policies.

Note that the TDM line connects the points (0, C * 2 ) and (C * 3 , 0), where C * 2 and C * 3 are the maximum achievable capacities for bi-and tripartite measurements, respectively. The point farthest from and above TDM (the vertex of the triangular region above the line, shown in green in Figure 2) is achieved by setting r1 = 0 and r2 = r3 = 1. In other words, the most "efficient" policy in terms of being the farthest from the TDM line is the following: (i) never perform BSMs in state (1, 0); and (ii) when in state (1, 1) and a third entanglement is generated on a different link, always use it in a tripartite measurement, but when a third entanglement is generated on one of the links that already has a stored qubit, always perform a BSM. Note that the latter rule directs the switch to not waste an entanglement whenever it is possible to use it in a measurement.

The capacity regions for k = 10 and 50 are shown in Figure 3. Note that as the number of links increases, the differences between TDM and the more efficient random policies diminish. In the next section, we provide an analytical proof of this phenomenon.

Analysis

Let π(0, 0), π(1, 0), and π(1, 1) represent the stationary distribution of the CTMC in Figure 1. The balance equations (excluding µ, as it cancels out due to every transition rate being its multiple), are:

π(0, 0)k = π(1, 0)(k -1)r1 + π(1, 1)(k -2)r2, π(1, 1)((k -2)r2 + (k -2)r2 + 2r3) = π(1, 0)(k -1)r1, π(0, 0) + π(1, 0) + π(1, 1) = 1.
Solving these equations yields

π(1, 1) = k(k -1)r 1 D , π(1, 0) = k(k -2 + 2r 3 ) D
, where

D = (k -2 + 2r 3 )((k -1)r 1 + k) + (k -1)r 1 ((k -2)r 2 + k).
Then the bipartite and tripartite capacities for this system, C2 ≡ C2(r1, r2, r3) and C3 ≡ C3(r1, r2, r3), are

C 2 = π(1, 0)(k -1)µr 1 + π(1, 1)((k -2)µr 2 + 2µr 3 ) = k(k -1)µ(k -2 + 2r 3 -(k -2)r 2 r1 ) (k -2 + 2r 3 )((k -1)r 1 + k) + (k -1)r 1 ((k -2)r 2 + k) , C 3 = π(1, 1)(k -2)µr 2 = k(k -1)(k -2)µr 2 r1 (k -2 + 2r 3 )((k -1)r 1 + k) + (k -1)r 1 ((k -2)r 2 + k) .
Claim 1. The maximum value of C2 is given by C * 2 = C2(r1, 0, 1) = C2(1, 0, r3), where r1 and r3 are arbitrary values in [0, 1]. The maximum value of C3 is given by C * 3 = C3(0, 1, 0).

Proof. We start by proving this for C2. First, note that to maximize C2's numerator and minimize its denominator, r2 must be set to 0. This yields

C2(r1, 0, r3) = k(k -1)µ(k -2 + 2r3) (k -2 + 2r3)((k -1)r1 + k) + k(k -1)r1 = k(k -1)µ (k -1)r1 + k + k(k-1)r 1 k-2+2r 3 . Now, r3 = 1 maximizes C2(r1, 0, r3), which yields C2(r1, 0, 1) = k(k -1)µ (k -1)r1 + k + (k -1)r1 = k(k -1)µ 2k -1 .
Note that C2(1, 0, r3) yields the same expression as C2(r1, 0, 1). Next, consider the expression for C3. To minimize the denominator, we should set r3 = 0. This yields

C 3 (r 1 , r 2 , 0) = k(k -1)(k -2)µr 2 r1 (k -2)((k -1)r 1 + k) + (k -1)r 1 ((k -2)r 2 + k) = k(k -1)(k -2)µr 2 r1 (k -1)r 1 ((k -2)r 2 -k) + k(k -2) + (k -1)((k -2)r 2 + k) .
It is clear that r1 must be 0, which yields

C3(0, r2, 0) = k(k -1)(k -2)µr2 k(k -2) + (k -1)((k -2)r2 + k) = k(k -1)(k -2)µr2 k(2k -3) + (k -1)(k -2)r2 = k(k -1)(k -2)µ k(2k-3) r 2 + (k -1)(k -2)
.

From above, we can see that r2 must be 1, so the maximum is at C * 3 = C3(0, 1, 0).

For brevity, let (C3(0, 1, 1), C2(0, 1, 1)) ≡ ( Ĉ3, Ĉ2); this is the point farthest above the TDM line within the achievable capacity region (e.g., the green point in Figure 2). We prove this as part of the proof of the claim below.

Claim 2. Any point (C3, C2) in the achievable capacity region satisfies the following constraints:

C2 ≤ - 3k -2 2k -1 C3 + µk(k -1) 2k -1 and (1)
C2 ≤ - k(k -2) + 2(k -1) 2 k(k -2) C3 + µ(k -1), (2) C2, C3 ≥ 0. (3)
Moreover ( 1) and ( 2) define a tight upper bound on the achievable capacity region.

Proof. First, we must show that the point ( Ĉ3, Ĉ2) is indeed the farthest from the TDM line. We do this by first assuming that there exists an achievable capacity region above the TDM line that is shaped like a triangle, as in Figures 2 and3, and later prove this to be true. With this assumption, the point farthest from the TDM line will be located at the vertex of this triangle. Thus, let us find a point (C3, C2) on the plane such that the (potentially negative) slope of the line that passes through it and (0, C * 2 ) is maximized. This is equivalent to minimizing the quantity

C * 2 -C2 C3 = (3k -2)(k -2)r2 + 2(k -1)(1 -r3) r2(2k -1)(k -2) .
To do so, we must set r3 = 1. Next, note that the TDM line is given by the equation f (x, y) = y -C * 2 (1 -x/C * 3 ), and the distance between it and any point (C3, C2) is given by |f

(C3, C2)|/ 1 + (C * 2 /C * 3 ) 2 . Hence, it is sufficient to maximize |f (C3(r1, r2, 1), C2(r1, r2, 1))|, given by 2µk(k -1) (2k -1) k -2 + 2k 2 -k (k-1)r 2 (1-r 1 )
.

It is clear that we must set r2 = 1 and r1 = 0, yielding ( Ĉ3, Ĉ2) as the point farthest from the TDM line, as expected.

Next, consider the line passing through (0, C * 2 ) and ( Ĉ3, Ĉ2):

y1 = - 3k -2 2k -1 x1 + µk(k -1) 2k -1 , (4)
and the line passing through ( Ĉ3, Ĉ2) and (C * 3 , 0):

y2 = - k(k -2) + 2(k -1) 2 k(k -2) x2 + µ(k -1). ( 5
)
It is not hard to show that for any point (C3, C2), (1) and ( 2) are satisfied. In other words, all points in the achievable capacity region fall on or below these two lines. To prove that this upper bound is tight, it remains to show that all points on lines (4) and ( 5) are achievable. To see this, let r1 = 0 and r3 = 1:

C2(0, r2, 1) = (k -(k -2)r2)k(k -1)µ k 2 + (k -1)(k + (k -2)r2) , C3(0, r2, 1) = (k -2)r2k(k -1)µ k 2 + (k -1)(k + (k -2)r2)
.

Note that any point (C3(0, r2, 1), C2(0, r2, 1)) is on line (4), and these two functions are continuous in r2 ∈ [0, 1]. Similarly, letting r1 = 0 and r2 = 1, we have

C2(0, 1, r3) = 2r3k(k -1)µ k(k -2 + 2r3) + 2(k -1) 2 , C3(0, 1, r3) = k(k -1)(k -2)µ k(k -2 + 2r3) + 2(k -1) 2 .
Any point (C3(0, 1, r3), C2(0, 1, r3)) is on line (5), and these two functions are continuous in r3 ∈ [0, 1]. Using these facts, we conclude that all points on (4) and ( 5) are achievable. Claim 3. As k → ∞, the benefit of using an alternate policy (one that lies above TDM) diminishes.

Proof. We prove this by showing that as k → ∞, the ratio of the achievable area above the TDM line, which we call A (because this area has the shape of a triangle) to the total area below the capacity region, which we call AT , goes to zero. For A , the length of the base of the triangle is simply the distance between the points (0,

C * 2 ) and (C * 3 , 0), or (C * 2 ) 2 + (C * 3 ) 2 . The height is given by |f ( Ĉ3, Ĉ2)|/ 1 + (C * 2 /C * 3 ) 2 . Then A = |f ( Ĉ3, Ĉ2)|C * 3 2 .
Then, the area below the TDM line is given by

AT DM = (C * 2 ) 2 + (C * 3 ) 2 4
, so the total area is

AT = A + AT DM = 2|f ( Ĉ3, Ĉ2)|C * 3 + (C * 2 ) 2 + (C * 3 ) 2 4
.

Then the ratio of the area above the TDM to the total area is

A A T = 2|f ( Ĉ3 , Ĉ2 )|C * 3 2|f ( Ĉ3 , Ĉ2 )|C * 3 + (C * 2 ) 2 + (C * 3 ) 2 = 1 1 + (C * 2 ) 2 +(C * 3 ) 2 2|f ( Ĉ3 , Ĉ2 )|C * 3 .
To prove that this ratio goes to zero with k, it suffices to show that the second term in the denominator goes to ∞. It can be shown that ((

C * 2 ) 2 +(C * 3 ) 2 )/2|f ( Ĉ3, Ĉ2)|C * 3 simplifies to 39k 6 -220k 5 + 493k 4 -568k 3 + 362k 2 -120k + 16 4(6k 5 -33k 4 + 67k 3 -62k 2 + 26k -4) k→∞ -→ ∞.

PER-LINK BUFFER SIZE TWO

In a system with per-link buffer size two, there are three additional states, as shown in Figure 4. The goal of this part of the study is to show the existence of better policies than TDM, rather than to find the optimal policy. Hence, the design in Figure 4 does not encapsulate all possible policies: for instance, there is no r1 parameter here, since our exhaustive search over the entire parameter space for the system with B = 1 revealed that r1 is best set to zero. In addition, note that if the system is in state (1, 1) and another entanglement is generated on one of the links that already has a stored qubit, the system is not allowed to use two of the qubits for a BSM. The reasoning is that since B = 2, there is enough space to keep the new qubit. Similarly, when the system is in state (2, 1) a BSM is only allowed if (i) another entanglement is generated on one of the k -2 links that does not have a qubit, or (ii) another entanglement is generated on the link that already has two qubits stored.

0,0 1,0 2,0 kμ μ 2μ (k -1)μ μ (k -1)μ 1,1 (k -2)μr2 2,2 2,1 (k -2)μr2 (k -2)μr2 (k -2)μr ¯2 (k -2)μ + μ r ¯2 r3 (k -2)μ + 2μ r ¯2 r3
In the latter scenario, not performing a BSM would cause a qubit to be discarded. While this design does not grant the switch access to the full range of policies, it does enable us to find a class of policies that are more efficient than TDM. Figure 5 shows capacity regions for B = 2 with number of links k = 3 and 10. We observe that policies more efficient than TDM can be found, but as the number of links grows, the advantage of such policies relative to TDM diminishes. This phenomenon mimics that of the B = 1 switch. Figure 6 shows a comparison of B = 1 and B = 2 switches for three and ten links. We observe that while there is a clear benefit to extra buffer space for a small number of users, the advantage becomes less apparent as the number of users grows. In addition, it appears that C3 benefits more from the extra buffer space than C2.

MODELING DECOHERENCE

In this section, we present a simple way to augment the model from Section 4 to account for the decoherence of quantum states. For switches with B = 1, we present both analytic and numerical results. We also augment the model from Section 5 to incorporate decoherence, but for switches with B = 2, we present only numerical results. Our decoherence model is described in Section 3. For B = 1, the resulting CTMC is illustrated in Figure 7.

The analysis of this model is almost identical to that of the B = 1 system without decoherence. As with the latter, the capacity region is bounded above by two lines:

y 1 = - µ(3k -2)(α + (k -2)µ) + 2α 2 µ(k -2)((2k -1)µ + α) x 1 + k(k -1)µ 2 (2k -1)µ + α , y 2 = - 2(k -1) 2 µ 2 + (kµ + α)((k -2)µ + 2α) µ(k -2)(kµ + α) x 2 + k(k -1)µ 2 kµ + α .
To avoid ambiguity, let C 2 and C 3 denote the bi-and tripartite capacities of a system with decoherence. As with the previous model, C 2 is maximized at r1 = 1, r2 = r3 = 0; C 3 is maximized at r1 = r3 = 0, r2 = 1, and the point farthest from TDM is obtained by setting r1 = 0, r2 = r3 = 1. The first bounding line passes through the points (0, C 2 (1, 0, 0)) and (C 3 (0, 1, 1), C 2 (0, 1, 1)); and the second line passes through (C 3 (0, 1, 1), C 2 (0, 1, 1)) and (C 3 (0, 1, 0), 0). Moreover, all points on the bounding lines are achievable, indicating that the bound is tight. Note that the denominator is quadratic in α. This causes both C 2 and C 3 to tend to zero as α → ∞.

Figure 8 shows a comparison of the capacity regions for systems with B = 1, for three and ten links and different decoherence rates. For all cases, µ is set to one: for qualitative results, we only need to concern ourselves with the value of α relative to µ. In real scenarios, we expect α to be at least one order of magnitude less than µ. From numerical results, we observe that the effect of decoherence on the capacity region is not significant, especially as the number of links grows. Analysis supports this observation, since we can show that Figure 9 shows a comparison for systems with B = 2 and varying number of links and decoherence rates. Results are consistent with that of the case B = 1: the effects of decoherence on capacity are less apparent for larger k values.

CONCLUSION

In this work, we explore a set of policies for a quantum switch that can store up to two qubits per link and whose objective is to perform bi-and tripartite joint measurements to distribute two and three qubit entanglement to pairs and triples of users. We present analytical results for the case where the per-link buffer has size one. By comparing against TDM policies, we discover that better policies in terms of achievable bi-and tripartite capacities exist, but that as the number of links grows, the advantage of using such policies diminishes. We also compare the capacity regions for systems with different per-link buffer sizes and observe that systems with fewer links benefit more from the extra storage space than systems with a larger number of links. Finally, we model decoherence for both types of systems and present analytical results for the case with per-link buffer size one. Observations and analysis show that as the number of links increases, the effects of decoherence become less apparent.

Figure 3 :

 3 Figure 3: Capacity region for a system of buffer size one and varying number of links. The red line represents the set of TDM policies.

Figure 4 : 10 Figure 5 :

 4105 Figure4: CTMC for a system with at least three links and buffer size two for each link. k is the number of links, µ is the rate of entanglement generation, and r2 and r3 are parameters that specify the scheduling policy.

= 10 Figure 6 :

 106 Figure 6: Comparison of capacity regions for systems of buffer sizes one and two with varying number of links k, and entanglement generation rate µ = 1.

Figure 7 :

 7 Figure7: CTMC for a system with at least three links and buffer size one. k is the number of links, µ is the rate of entanglement generation, α is the decoherence rate, and r1, r2, r3 are parameters that specify the scheduling policy.

(a) k = 3 (b) k = 10 Figure 8 : 5 (b) k = 10 Figure 9 :

 31085109 Figure 8: Capacity region for a system of buffer size one and varying number of links k, decoherence rates α, and entanglement generation rate µ = 1. The solid lines are the upper boundaries of the capacity region, and the dashed are TDM lines.