Simon Pouil

Michel Warnau

François Oberhänsli

Jean-Louis Teyssié

Dr Marc Metian
email: m.metian@iaea.org

Trophic transfer of 110m Ag in the turbot Scophthalmus maximus through natural prey and compounded feed items

Keywords: Silver, Accumulation, Aquaculture, Feeds, Radionuclides

come

Trophic transfer of 110m Ag in the turbot Scophthalmus maximus through natural prey and compounded feed items

Introduction

The impact of artificial radionuclides on aquatic ecosystems, in which they act as micro-pollutants, is a topic of particular interest, both in terms of radiation protection and of environmental assessment. Under normal operational conditions of nuclear facilities, 110m Ag is one of the radionuclides that can be emitted in small quantities [START_REF] Adam | Kinetics of 110m Ag, 60 Co, 137 Cs and 54 Mn bioaccumulation from water and depuration by the crustacean Daphnia magna[END_REF]. Under particular circumstances, non-negligible amounts of 110m Ag have been measured in the marine environment, such as after the Chernobyl and Fukushima Dai-ichi Nuclear Power Plant accidents [START_REF] Buesseler | Fukushima-derived radionuclides in the ocean and biota off Japan[END_REF][START_REF] Aono | Observation of radionuclides in marine biota off the coast of Fukushima prefecture after TEPCO's Fukushima Daiichi nuclear power station accident[END_REF].

Information on 110m Ag bioaccumulation in fish is until now essentially confined to freshwater organisms (e.g. [START_REF] Baudin | Accumulation, release, and tissue distribution of 110m Ag from natural food (Gammarus pulex) by the common carp, Cyprinus carpio L. Arch[END_REF][START_REF] Ausseil | Influence of metal (Cd and Zn) waterborne exposure on radionuclide ( 134 Cs, 110m Ag, and 57 Co) bioaccumulation by rainbow trout (Oncorhynchus mykiss): A field and laboratory study[END_REF][START_REF] Bertram | Effects of feeding on waterborne silver uptake and depuration in rainbow trout (Oncorhynchus mykiss)[END_REF].

Although no biological role has been identified, these studies showed that the trophic transfer is an important pathway of accumulation in freshwater fish. In the few studies carried out on marine fish, the trophic transfer seemed to be a less important accumulation pathway of 110m Ag [START_REF] Pentreath | The accumulation of 110m Ag by the plaice, Pleuronectes platessa L. and the thornback ray, Raja clavata L[END_REF][START_REF] Long | Assimilation and bioconcentration of Ag and Cd by the marine black bream after waterborne and dietary metal exposure[END_REF]. However, only one type of food per species was investigated in these studies. There is thus currently a lack of knowledge on the possible influence of food quality on 110m Ag assimilation by marine fish.

Dissolved or particulate contaminations occur and contribute simultaneously in natural environment to the global metal bio-accumulation in organisms The delineation of the relative contribution of the different exposure pathways has to be assessed using models (see Thomann, 1981[START_REF] Landrum | Toxicokinetics in aquatic systems: model comparisons and use in hazard assessment[END_REF]or Metian et al., 2008a), to really identify a main uptake route of contamination. These models that are using the kinetic parameters experimentally obtained are more and more considered in the literature but this is not systematic, and some papers argue the importance of one exposure pathway on another without mathematical validation. In the same time, it is key to determine within these models the variability of parameters that can affect the final determination of the main uptake pathway, such as for example the effect of food quality on metal transfer.

In this study, the turbot Scophthalmus maximus was chosen to examine the transfer of 110m Ag from four different types of feeds: three different natural prey (seabream Sparus aurata, shrimp Palaemon serratus, and ragworm Hediste diversicolor) and manufactured pellets; this latter feed was selected since the turbot is a common species produced in aquaculture. In order to better understand assimilation processes, two levels of biological organization were considered: (1) the whole individual (determination of the depuration kinetics and of assimilation efficiencies) and ( 2) the organs and tissues (dissections and determination of body distribution).

Materials and Methods

Origin and acclimation of organisms

In January 2014, one hundred juvenile turbot S. maximus were purchased from a fish farm (France Turbot, France) and shipped to the International Atomic Energy Agency premises in the Principality of Monaco. Fish were acclimated to laboratory conditions for 21 days (open circuit, 500-L aquarium; water renewal: 100 L h -1 ; 0.45 µm filtered seawater; salinity: 38 p.s.u.; temperature: 15 ± 0.5°C; pH: 8.0 ± 0.1; light/dark: 12h/12h). During the acclimation period, the fish were fed to a daily ration of 2% of their estimated biomass with 1.1-mm pellets (Le Gouessant, France).

In order to investigate the influence of the diet on 110m Ag assimilation by S. maximus,1.1-mm manufactured pellets (Le Gouessant, France) and three different natural prey of turbot were used (e.g. Sparrevohn et al., 2008;[START_REF] Florin | Feeding habits of juvenile flatfish in relation to habitat characteristics in the Baltic Sea[END_REF]: juvenile fish (S. aurata; 60-day-old hatchlings, weight: 0.060 ± 0.013g), shrimp (P. serratus, weight : 0.58 ± 0.11g) and worms (H. diversicolor, weight: 0.82 ± 0.14g). Juvenile fish were obtained from a hatchery (Poissons du Soleil, France), shrimp were purchased from a fisherman (Poissons Vivants, France), and worms were purchased from fishing bait seller (Normandie Appats, France). All living feeds were acclimated to the same laboratory conditions than predator fish for a minimum of two weeks prior to experiments. Shrimp and worms were fed a mix of fish feed and crushed mussels whereas juvenile fish were fed 300-µm pellets (Biomar, France). Since body size is known to affect metal bioaccumulation in marine organisms [START_REF] Boyden | Trace element content and body size in molluscs[END_REF][START_REF] Warnau | Allometry of heavy metal bioconcentration in the echinoid Paracentrotus lividus[END_REF], only individuals with homogeneous size of each prey species were used in the experiments.

Radiotracers and Counting

Experiments were carried out using a high-specific 110m Ag radiotracer purchased from POLATOM, Poland ( 110m Ag as AgNO3 in 0.1 M HNO3; [T1/2] = 250 days; specific activity: 520 MBq mg -1 ). The radioactivity of the tracer was measured using a high-resolution γ-spectrometer system composed of 5 Germanium -N or P type -detectors (EGNC 33-195-R, Canberra ® and Eurysis ® ) connected to a multi-channel analyser and a computer equipped with a spectra analysis software (Interwinner 6, Intertechnique ® ). The radioactivity was determined by comparison with standards of known activity and of appropriate geometry (calibration and counting). Measurements were corrected for background and physical radioactive decay. Organisms were placed in counting tubes filled with clean seawater during the counting period. The counting time was adjusted to obtain a propagated counting error less than 5% (e.g. Metian et al., 2008b), typically 25-60 min for whole organism radioanalysis. This counting duration did not affect organism welfare, as shown by their intercounting activity and feeding behavior.

Experimental Procedure

Radiolabelling of feeds

Twelve grams of pellets were dipped for 1 h in 16.5 mL of seawater spiked with 1.8 kBq 110m Ag mL -1 . Pellets were then dried for 48 hours at 50°C and kept in a dry environment in order to prevent mold growth. Preliminary tests were performed to determine the possible leakage into the water of radioisotopes from the pellets during the feeding. During the feeding, acclimated fish were consuming pellets in less than 1 min. Therefore, preliminary tests consisted in pouring radiolabelled dry pellets (100 mg per treatment) for 1 and 10 min in 50 mL seawater and to measure any radioactivity in the seawater. The leakage of pellet-radioactivity in water did not exceed 0.8% and 2% of the initial activity after 1 min and 10 min, respectively. Although these tests confirmed the single-pathway contamination (viz. food) of the fish, one turbot was used in each treatment, as a control to take into account the possibility of 110m Ag recycling through water (see Section 2.3.2).

Radiolabeling of the natural prey was carried out by exposing them for 7 to 21 days in seawater spiked with the radiotracer (nominal activity of 0.5 kBq 110m Ag L -1 , corresponding to 27 nmol L -1 equivalent stable Ag). No change in pH was detectable in the aquarium (close circuit) after tracer addition. The spiked seawater was renewed daily in order to keep the radiotracer activity as constant as possible. Activity of the metal tracer in seawater was checked daily, before and after each seawater renewal, to determine time-integrated activities (Rodriguez y Baena et al., 2006).

Prey were fed after each seawater renewal and just prior to radioisotope addition. Exposures of prey were made in aerated, 20-L aquaria. For shrimp exposure, each organism was kept individually in a cylindrical PVC container (drilled to allow for free water circulation) during the whole duration of the experiment in order to avoid cannibalism during moulting and to facilitate individual identification. For the worms, the walls of the aquarium were obscured and plastic tubes were added as artificial burrows.

Exposure of turbots via radiolabelled feeds

Four sets of experiments were realized for each type of feed considered. Each time, 8 to 12 juvenile turbots (11.9 ± 5.5 g, wet weight) were randomly picked and transferred in an aerated, open circuit, 70-L aquarium. One week before the exposure to radiolabelled feed, fish were daily fed the selected type of feed and individually identified by slits cut in fins.

Each experiment consisted in a single exposure to radiolabelled feed (single feeding method also called pulse chase feeding; see e.g. [START_REF] Hédouin | Influence of food on the assimilation of selected metals in tropical bivalves from the New Caledonia lagoon: qualitative and quantitative aspects[END_REF][START_REF] Metian | Metal and metalloid bioaccumulation in the Pacific blue shrimp Litopenaeus stylirostris (Stimpson) from New Caledonia: Laboratory and field studies[END_REF]. For each experiment, turbot were fed ad libitum for 30 min; the uneaten food was then removed. Ragworms and shrimp were cut into pieces prior to the feeding in order to facilitate ingestion. Two hours after the initiation of the feeding, each fish was whole-body γ-counted alive and then replaced in clean, flowing seawater conditions (parameters as previously described). No regurgitation of the radiolabelled feeds was observed. Fish were then fed daily non-labeled pellets (2% of their biomass). In parallel, after the feeding period, an additional turbot was placed in each aquarium within in a net to control any possible radiotracer recycling from seawater due to radiotracer leaching from the contaminated food or from fish depuration. Fish were then whole-body γ-counted alive daily over 21 days (including control individuals). They were moved to another, clean 70-L aquarium after each counting to avoid contamination from 110m Ag contained in faeces.

After the depuration period, 4 individuals were dissected in 7 compartments: (1) the 4 fillets (skinned muscles), (2) the kidney, (3) the liver, (4) the gall bladder, (5) the digestive tract, (6) the head (including gills) and ( 7) the remaining parts (including skin, skeleton, fins, heart and muscle residues) and were separated, weighed (wet wt) and radioanalysed to determine the radiotracer body distribution. During experiments, no mortality was recorded.

110m Ag compartmentalization in feeds

For the natural prey, the distribution of 110m Ag between the soluble and insoluble fractions was determined according to a method adapted from [START_REF] Bustamante | Subcellular and body distributions of 17 trace elements in the variegated scallop Chlamys varia from the French coast of the Bay of Biscay[END_REF]. Briefly, an aliquot of contaminated feed stored at -80°C was crushed and tissues homogenized using a T25 Ultra-Turrax Basic (IKA ® ) in approximately 10 volumes (w:v) of TRIS-HCl buffer 0.02 M sucrose 0.25 M with 1 mM protease inhibitor (PMSF, phenylmethylsulfonylfluoride) and 5 mM reducing agent (DTT, dithiothreitol), at pH 8.06. The homogenates were centrifuged at 45 000 G for 2 h at 4°C (Sorvall Evolution RC Superspeed Centrifuge, Sorvall instruments ® ) to separate particle-free supernatant from the debris. Aliquots of the homogenates and trophically available metal (TAM) obtained were radioanalyzed in order to determine the radiotracers compartmentalization. The same procedure was repeated over time for each prey.

Data treatment and statistical analyses

Depuration of radiotracers was expressed as the percentage of remaining radioactivity (radioactivity at time t divided by the initial radioactivity measured in the organism at the beginning of the depuration period; [START_REF] Warnau | Biokinetics of selected heavy metals and radionuclides in the common Mediterranean echinoid Paracentrotus lividus: Sea water and food exposures[END_REF]. Depuration kinetics were fitted using non-linear model. They were best fitted using the following two-component model that includes an exponential component and a constant (Eq. 1) (decision based on F test and ANOVA tables):

𝐴 𝑡 = 𝐴 0𝑠 . 𝑒 -𝑘 𝑒𝑠 𝑡 + 𝐴E (Eq. 1)
where At and A0s are the remaining activities (%) at time t (d) and 0, respectively; kes is the depuration rate constant (d -1 ) and AE is the assimilation efficiency (%). The first component (𝐴 0𝑠 . 𝑒 -𝑘 𝑒𝑠 𝑡 ) represents the depuration kinetics of the radiotracer fraction that is weakly associated to the organisms and rapidly eliminated, whereas the second component (𝐴E) describes the radiotracer fraction that is retained in the body [START_REF] Warnau | Biokinetics of selected heavy metals and radionuclides in the common Mediterranean echinoid Paracentrotus lividus: Sea water and food exposures[END_REF]. A short-term biological halflife can be calculated (Tb1/2) from the depuration rate constant according to the relation Tb1/2s = ln2/kes. By definition, biological half-life of the "AE" component is infinite. Model constants were estimated by iterative adjustment of the model using the nonlinear curve-fitting routines in the Statistica ® software 7.0. Distribution patterns of 110m Ag among organs and tissues were compared by a G-test using R software 3.0.1 (R Development Core Team, 2014). The level of significance for statistical analyses was always set at α = 0.05.

Results

In order to evaluate the influence of food quality on 110m Ag assimilation in the turbot S. maximus, 4 food items were first radiolabelled through seawater. The seabream S. aurata, the shrimp P.

serratus, the ragworm H. diversicolor and the manufactured compounded feed concentrated 110m Ag differently: the average activities measured in these feeds just before the feeding of the turbots were respectively 7 Bq g -1 , 37 Bq g -1 , 117 Bq g -1 and 1072 Bq g -1 (Table 1).

When turbots were fed seabream or shrimp, activities could be measured accurately in turbots after feeding. However, after 48 h of depuration, uncertainties of counts became very high (> 20%).

After two weeks, the activities were below detection limits. Therefore, depuration kinetics could not be modeled for these two types of food.

When turbots were fed with pellets, whole-body depuration kinetics were best fitted by a twocomponent model (an exponential phase and a constant; Figure 1; R 2 : 0.99). 110m Ag was poorly assimilated by the fish (AE = 0.32 ± 0.75%, p>0.05). Virtually all ingested 110m Ag was very rapidly lost (Tb½s= 0.77 ± 0.02 d; mean ± SD; Fig. 1). When ragworms were used for feeding turbots, depuration kinetics were best fitted by the same two-component model as with pellets (Fig. 1; R 2 : 0.99). The short-term depuration of 110m Ag was hereto fast (Tb½s= 0.66 ± 0.03 d; mean ± SD; Fig. 1) but, conversely to the other feeds, 110m Ag assimilation was high enough to be accurately estimated (AE: 3 ± 0.7%; mean ± SD; Table 1, p<0.05).

Post-feeding distribution of 110m Ag in turbots at the end of the 21-d depuration period is shown in Table 2, for the different types of food. Similar patterns of 110m Ag distribution among compartments were observed in turbots fed with the different feeds (p>0.05): liver and digestive tract accounting for until 65% of the total 110m Ag load (Table 2). These body compartments represented, respectively, less than 10% and 1% of the body weight. Most of the remaining activity was distributed in the head and in remaining tissues.

Regarding subcellular partitioning of 110m Ag in natural prey, the majority was found in the soluble fraction: the average proportion ranged from 50 to 72%. Significant difference was recorded for the two extreme values (p <0.01): the highest proportion of 110m Ag in the soluble fraction (72 ± 5%) was measured in shrimp whereas this proportion was 50 ± 4% in ragworms (Fig. 2).

Discussion

110m Ag can contribute up to 20% of the low level gamma radioactivity in liquid effluents from some pressurized reactors under normal operating conditions [START_REF] Baudin | Accumulation, release, and tissue distribution of 110m Ag from natural food (Gammarus pulex) by the common carp, Cyprinus carpio L. Arch[END_REF]. For several years, radioecological field studies have detected 110m Ag in the main components of aquatic ecosystems downstream from nuclear power stations (e.g. [START_REF] Eyrolle | Radioactivity levels in major French rivers: summary of monitoring chronicles acquired over the past thirty years and current status[END_REF][START_REF] Aono | Observation of radionuclides in marine biota off the coast of Fukushima prefecture after TEPCO's Fukushima Daiichi nuclear power station accident[END_REF].

Moreover, the recent Fukushima Dai-ichi Nuclear Power Plant accident has led to the release of a massive amount of this radionuclide in the environment [START_REF] Buesseler | Fukushima-derived radionuclides in the ocean and biota off Japan[END_REF][START_REF] Aono | Observation of radionuclides in marine biota off the coast of Fukushima prefecture after TEPCO's Fukushima Daiichi nuclear power station accident[END_REF].

In this context, it is essential to understand the transfer of this radionuclide in aquatic organisms to better assess the risk for the ecosystem (food chain) and for Human.

Despite the potential importance of trophic transfer, the contribution of the food pathway for 110m Ag global bioaccumulation in marine fish has been poorly investigated (e.g. [START_REF] Pentreath | The accumulation of 110m Ag by the plaice, Pleuronectes platessa L. and the thornback ray, Raja clavata L[END_REF]Rouleau et al., 2000;[START_REF] Wood | Silver[END_REF]. Experimental approaches appear to be the most appropriate to better understand, estimate and model radionuclide transfer from prey to predators [START_REF] Adam | Bioaccumulation of 110m Ag, 60 Co, 137 Cs and 54 Mn by the freshwater crustacean Daphnia magna from dietary sources (Scenedes musobliquus and Cyclotella meneghiana)[END_REF].

To undertake this study, different natural prey and compound pellet were exposed to waterborne 110m Ag. Concentration Factors (CFs) were calculated at the end of the exposure period; they were 76 ± 9 in shrimp, 119 ± 77 in seabream and 203 ±101 in ragworm. Although the exposure time varied among prey (7-21 days), results suggested that ragworms, exposed for 7 days in this study, have a higher 110m Ag accumulation capacity than the other feeds.

To get a better understanding of 110m Ag biological storage mechanisms at intracellular level in natural prey, ultracentrifugation was used for determining the subcellular fractioning of this element. The method used allows separating insoluble fraction (i.e., intracellular metal-rich granules and membrane fragments) from soluble fraction (i.e., cytosol and soluble metallothioneins and heat-sensitive proteins; [START_REF] Vijver | Internal metal sequestration and its ecotoxicological relevance: A review[END_REF]. Most studies using such a technique were focusing on target organs and showed that 110m Ag was mainly stored in an insoluble form, for example in the gills of the trout Oncorhynchus mykiss (80-95%; [START_REF] Galvez | The distribution kinetics of waterborne silver-110m in juvenile rainbow trout[END_REF][START_REF] Wood | Kinetics of radiolabelled silver uptake and depuration in the gills of rainbow trout (Oncorhynchus mykiss) and European eel (Anguilla anguilla): the influence of silver speciation[END_REF] or in the digestive gland of the scallop Pecten maximus (93 ± 2%; Metian et al., 2008a). Subcellular storage of 110m Ag (and thus partitioning) may vary among organs for a given species, as shown in the bivalve species, Gafrarium tumidum; where 65% of 110m Ag was found in the insoluble fraction of the visceral mass whereas 90% was found in this fraction of the gills [START_REF] Metian | Use of radiotracer techniques to study subcellular distribution of metals and radionuclides in bivalves from the Noumea Lagoon, New Caledonia[END_REF].

Therefore, in the present study, we decided to centrifuge the whole organisms (prey) to get a better idea of the integrated subcellular partitioning. Results revealed that a large proportion of the 110m Ag was included in the soluble fraction of prey (50 -72%).

Differences of 110m Ag accumulation in food items (in terms of concentration or subcellular distribution) did not have any influence on assimilation of the radionuclide by turbots. Indeed, our results indicated clearly that 110m Ag was poorly assimilated and retained by the turbots whatever the feed considered. The assimilation efficiency of the radioelement ingested with ragworms and pellets was <3%. AEs could not even be calculated in the case where turbots were fed with seabream and shrimp (activities measured in turbots were rapidly below the detection limits; Table 1). These results are consistent with the literature, although a wide range of food items was never looked at. Estimated AEs in individuals fed ragworms and pellets (3 ± 0.7% and 0.3 ± 0.8%, respectively; mean ± SD) are similar to the AE of 4.2 ± 2.8% estimated in plaice Pleuronectes platessa, fed with radiolabelled ragworms (46 days of depuration; [START_REF] Pentreath | The accumulation of 110m Ag by the plaice, Pleuronectes platessa L. and the thornback ray, Raja clavata L[END_REF]. To the best of our knowledge, no study has been using pellets as food for studies on 110m Ag AE. However, Rouleau et al. (2000) followed Ag depuration in individuals of the American plaice, Hippoglossoides platessoides, fed with a 110m Ag-labelled wet paste (composition close to manufactured pellets; [START_REF] Provencher | Essais d'alimentation de la morue franche (Gadus morhua) (in french)[END_REF]; authors found a very low retention of 110m Ag with this food, with an estimated AE of 8.6 ± 4.8%. For this latter study, authors have used gravid females (adults) and thus direct comparison is not possible as we used juvenile fish and sexual maturity may affect the retention of trace elements due to the energy allocation for the oocyte production [START_REF] Bang | The relation between concentrations of ovarian trace elements and the body size of Atlantic cod Gadus morhua[END_REF].

Some authors are linking AE of elements in predators with the subcellular partitioning of these elements in their prey. The concept of Trophically Available Metal (TAM), as defined by [START_REF] Wallace | Subcellular compartmentalization of Cd and Zn in two bivalves. II. Significance of trophically available metal (TAM)[END_REF], states that the fraction of elements found in the soluble fraction of prey should reflect the element fraction that is bioavailable to predators ingesting that prey. In our study, results revealed that although 110mAg was preferentially found in the soluble fraction of prey (50e72%), AEs were always low. Therefore, the positive relationship between TAM and AE stated by [START_REF] Wallace | Subcellular compartmentalization of Cd and Zn in two bivalves. II. Significance of trophically available metal (TAM)[END_REF] could be true for simple organisms (as shown for bivalves fed phytoplankton; Wang and Fisher, 1996;Reinfelder et al., 1997) (1) the occurrence of neutralization processes such as speciation or complexation of the element due to acidic conditions within the stomach of the fish, that would limit the element accessibility in the digestive tract or (2) the occurrence of detoxification mechanisms in the wall of the digestive tract, preventing 110mAg to penetrate into the internal compartments of the fish. The first hypothesis is backed up with our pellet experiment for which 110mAg was not biological bound but chemically adsorbed. Indeed, although the radionuclide was virtually 100% under "soluble" form in the latter experiment, 110mAg was not assimilated either by the fish. Regarding the second hypothesis, our body distribution results show that most of the assimilated 110m Ag is located in the liver and digestive tract and not eliminated (AE component), suggesting storage of the element in these organs and tissues of the turbots. This can be seen in relation to previous observations made, for example, in several bivalves (freshwater and seawater species; [START_REF] Berthet | Bioaccumulation toxicity and physico-chemical speciation of silver in bivalve molluscs: ecotoxicological and health consequences[END_REF], the sea urchin Paracentrotus lividus [START_REF] Warnau | Biokinetics of selected heavy metals and radionuclides in the common Mediterranean echinoid Paracentrotus lividus: Sea water and food exposures[END_REF] and the fish Oryzias latipes [START_REF] Chae | Evaluation of the toxic impact of silver nanoparticles on Japanese medaka (Oryzias latipes)[END_REF], where Ag is stored within the intestinal wall as amorphous Ag (Ag2S) generally found precipitated in the basement membranes and underlying connective tissues: [START_REF] Berthet | Bioaccumulation toxicity and physico-chemical speciation of silver in bivalve molluscs: ecotoxicological and health consequences[END_REF].

The occurrence of mechanism(s) impeding the transfer of the element across the fish digestive barrier was also mentioned in several studies, which suggested that intestinal tissues help buffer diet-borne metal uptake (Cu and Zn), by acting as a barrier [START_REF] Maage | Assessment of zinc status in juvenile Atlantic salmon (Salmo salar) by measurement of whole body and tissue levels of zinc[END_REF]Clearwater et al., 2000;[START_REF] Handy | Mechanisms of gastrointestinal copper absorption in the African walking catfish: copper dose-effects and a novel aniondependent pathway in the intestine[END_REF].

Results from the transfer of 110m Ag to the turbot via manufactured pellets highlighted a low risk of contamination of fish produced in aquaculture in the case of an environmental contamination by 110m

Ag. The potential for pellet to accumulate 110m Ag by adsorption exists in the environment but the related AE in fish would be extremely low (Table 1). However, the bioaccumulation of 110m Ag in fish through seawater was not investigated in our study and should be considered to establish a comprehensive risk to fish and consumers.

As radioactive isotopes have nearly identical physical and chemical properties to equivalent stable isotopes, the findings of the present study can also be extended to the stable element (viz. Ag) and its bioaccumulation mechanisms in fish (e.g. Metian et al., 2008a;[START_REF] Metian | Metal and metalloid bioaccumulation in the Pacific blue shrimp Litopenaeus stylirostris (Stimpson) from New Caledonia: Laboratory and field studies[END_REF]. Here, we demonstrated the low silver retention regardless the 4 type of feeds used (natural prey and manufactured pellet) in a marine fish. This information is also of interest in the context of the occurrence of silver pollution in the marine environment, e.g. in relation to the rapid increase in the use of silver nanoparticles (Savery et al., 2013) or to the urban poor sewage treatment (Sañudo-Wilhelmy and Flegal, 1992). Table 1. Activities (Bq g -1 wwt, mean ± SD) in the different feeds (n=4) and in the turbots (n = 8-12) after the single-feeding and at the end of depuration period. Assimilation efficiencies (%) related to the different feeds are also indicated.

Conclusion

Feed Activity in feed Activity in turbot Remaining (Bq g -1 ) (2h after the single-feeding, Bq g -1 ) (end of depuration, Bq g -1 ) activity (%)

Figure 1 .Figure 2 .

 12 Figure 1. Influence of type of food on whole-body depuration of 110m Ag in turbots (% remaining

  but was not verified in the case of our more complex prey (teleost, crustacean and polychaete). Even considering the lower value of soluble fraction of 110mAg (23.7e26.8%) previously found in the whole body H. diversicolor by Rainbow et al. (2006), the AE of 110mAg is still too low compared to what would be expected by TAM concept. Possible explanations for this low AE in turbot despite a theoretically important source of bioavailable 110mAg are:

  Rainbow, P.S.,Poirier, L., Smith, B.D., Brix, K.V., Luoma, S.N., 2006. Trophic transfer of trace metals: subcellular compartmentalization in a polychaete and assimilation by a decapods crustacean.Mar. Ecol. Prog. Ser. 308,[91][92][93][94][95][96][97][98][99][100]

	R Development Core Team, 2014. R: a language and environment for statistical computing. R
	Foundation for Statistical Computing, Vienna, Austria.
	Rodriguez y Baena, A.M., Metian, M, Teyssié, J.-L., De Broyer, C., Warnau, M., 2006.
	Experimental evidence for 234 Th bioaccumulation in three Antarctic crustaceans: Potential
	implications for particle flux studies, Mar. Chem. 100, 354-365.
	Toxicol. 03.
	Sparrevohn, C.R., Støttrup, J., 2008. Diet, abundance and distribution as indices of turbot (Psetta
	maxima L.) release habitat suitability. Rev. Fish. Sci. 16, 338-347.
	Thomann, R.V., 1981. Equilibrium model of fate of microcontaminants in diverse aquatic food

Rouleau, C., Gobeil, C., Tjälve, H., 2000. Accumulation of silver from the diet in two marine benthic predators: The snow crab (Chionoecetes opilio) and American plaice (Hippoglossoides platessoides). Environ. Toxicol. Chem. 19, 631-637. Sañudo-Wilhelmy, S.A., Flegal, R., 1992. Anthropogenic silver in the southern California 552 bight: A new tracer of sewage in coastal water. Environ. Sci. Technol. 26, 2147-2151. Savery, L.C., Wise, S.S., Falank, C., Wise, J., Gianios, C.J., Thompson, W.D., Perkins, C., Mason, M.D., Payne, R., Kerr, I., John Pierce Sr, W., 2013. Global assessment of silver pollution using sperm whales (Physeter macrocephalus) as an indicator species. J. Environ. Anal.

chains. Can. J. Fish.

Aquat. Sci. 38,[280][281][282][283][284][285][286][287][288][289][290][291][292][293][294][295][296]

Table 2 .

 2 Body distribution (%, mean ± SD, n = 4) of 110m Ag in the turbots fed with radiolabelled feeds and then maintained for 21d in uncontaminated seawater. Values with high uncertainty (>5%) are shown in italic. Weights were expressed as a percentage of total mass. For this experiment, counting errors exceed 5% due to the low activities measured.

	Seabream	7	0.67 ± 0.39	<0.03 ± 0.009 a	<6.5 ± 5.2 b
	Shrimp	37	0.98 ± 0.45	<0.02 ± 0.007 a	<2.1 ± 0.6 b
	Pellet	1072 c	15.44 ± 7.17	0.19 ± 0.09	1.3 ± 0.65
	Worm	117	11.65 ± 5.61	0.29 ± 0.10	2.3 ± 1.2

a Values indicated are the detection limits. b Values calculated using detection limits. c Concentration (Bq g -1 ) dry weight *Different from 0 (p< 0.0001) a For these experiments, activities are closed to the detection limits.

b

Acknowledgments

The IAEA is grateful to the Government of the Principality of Monaco for the support provided to its Environment Laboratories. MW is an Honorary Senior Research Associate of the National Fund for Scientific Research (NFSR, Belgium).

This present study confirmed the low assimilation efficiency of 110m Ag from food in a marine fish, the turbot S. maximus, with the use of a range of food items that are the main components of the diet of the fish in natural conditions or in aquaculture production. Results showed a poor assimilation and retention of 110m Ag by the predator fish regardless of the type of food.


Yoshihiro Yamanishi

Jean-Philippe Vert
email: jean-philippe.vert@mines.org

Kernel Matrix Regression

Keywords:

come

Introduction

Kernel methods such as support vector machines (SVM) enable the use of powerful statistical analysis for various datasets as soon as kernel matrices for the dataset are available [START_REF] Schoelkopf | Learning with Kernels: Support Vector Machines, Regularization, Optimization, and Beyond[END_REF]. When a dataset contains N objects, the N objects are represented as an N × N positive semidefinite matrix whose elements can be thought of as objectsobject similarities. An advantage of kernel methods is that they can be applied not only to real-valued data but also to complex structured objects such as strings, trees and graphs [START_REF] Schoelkopf | Kernel Methods in Computational Biology[END_REF]. The kernel matrix not only plays an important role as the input to kernel methods, but also provides important information regarding the similarity between objects.

In this paper we consider the problem of estimating missing entries in a kernel matrix. More precisely we assume that two datasets describing the same objects are available; however, although all data are available for the first dataset, only part of the second dataset is available. In this case, a full kernel Gram matrix can be obtained for the first dataset, while only a partial kernel Gram matrix is obtained for the second dataset, that is, a matrix with missing entries. If we are more interested in the second dataset rather than in the first dataset, it is natural to think of estimating the missing part of the second kernel matrix, e.g., by looking for correlations between the two kernels.

This problem arises commonly in applications such as bioinformatics, where informative data for a given classification task is expensive to produce while less informative data are easily available. As an example, the DNA sequences of all proteins of a given organism are easily obtained from the sequenced genome, while the 3D structures of most proteins are still unknown and difficult to obtain. In this case, we want to know the 3D structure information for the proteins whose structure have not been determined. As another example, high-throughput genome-wide data (e.g., gene expression data) are available for the full genes of an organism, while the metabolic network information is known only for a limited number of genes. In this case, we want to predict the unknown part of gene network using the fully available genome-wide data. A variety of methods have been proposed in order to tackle such problems related to kernel matrix completion. Examples include the use of kernel canonical correlation analysis (kernel CCA) for gene classification [START_REF] Vert | Graph-driven features extraction from microarray data using diffusion kernels and kernel CCA[END_REF] and gene network inference [START_REF] Yamanishi | Protein network inference from multiple genomic data: a supervised approach[END_REF], and the use of the em-algorithm based on information geometry for protein 3D structure prediction [START_REF] Kim | Protein classification via Kernel Matrix Completion[END_REF].

In this paper we attack the kernel matrix completion problem from the viewpoint of regression, considering the two kernel matrices represent inner products between explanatory and response variables, respectively. We propose a variant of the regression model based on the underlying features in the reproducing kernel Hilbert space by modifying the idea of kernel CCA, and we estimate the missing entries by fitting this model to the existing samples. In the experiment, we show promising results on the prediction of missing edges in a gene network from genome-wide data and on the prediction of protein 3D structure information from their sequences. We also discuss the mathematical and numerical relationships between the proposed methods and the em-algorithm based on information geometry.

Formalism of the question

Suppose that we have an explanatory random variable x ∈ R d and a response random variable y ∈ R l . Let us now consider the situation where the data is available for all the N objects for the explanatory variable x, while the data is available for the first n objects and unavailable for the remaining (N -n) objects for the response variable y. We refer to the first n objects as training set, and we refer to the remaining N -n objects as test set below.

Let k and g be symmetric positive definite kernels defined on R d and R l , respectively. When we compute the kernel matrix for the explanatory variable x, we obtain an N × N kernel matrix K, where (K) ij = k(x i , x j ) (1 ≤ i, j ≤ N ), x i belongs to a set X and N is the number of all objects. On the other hand, when we compute the kernel matrix for the response variable y, we obtain an N × N kernel matrix G, where (G) ij = g(y i , y j ) (1 ≤ i, j ≤ n), y i belongs to a set Y, and n is the number of available objects (n < N ). Note that G contains in fact missing values for all entries (G) ij with max(i, j) > n. We want to estimate the missing part of G using full Gram matrix K, taking into account a form of correlation between the two kernels.

In this study we express each kernel matrix by splitting the matrix into four parts. We denote by K tt (resp. G tt ) the n × n kernel matrix for the training set versus itself, K pt (resp. G pt ) the (N -n) × n kernel matrix for the test set versus the training set, and K pp (resp. G pp ) the (N -n) × (N -n) kernel matrix for the test set versus itself:

K = K tt K ⊤ pt K pt K pp , G = G tt G ⊤ pt G pt G pp (1)
Note that K pt and K pp are known, while G pt and G pp are unknown. The goal is to predict G pt and G pp from K and G tt .

Methods

In this section we describe four approaches that can be used for the problem of kernel matrix completion: the direct approach (Section 3.1) is a baseline straightforward approach, the kernel CCA approach (Section 3.2) has been proposed in previous work [START_REF] Yamanishi | Protein network inference from multiple genomic data: a supervised approach[END_REF], while the kernel matrix regression (Section 3.3) and penalized kernel matrix regression (Section 3.4) are new.

Direct approach

A straightforward approach is to directly plug the entries of the kernel matrix K for the explanatory variable x into the missing entries of the kernel matrix G for the response variable y, that is, to choose G pt = K pt and G pp = K pp . We refer to this approach as the direct approach.

Kernel CCA (kCCA)

The use of kernel canonical correlation analysis (kCCA) has been proposed to estimate the unknown part of the metabolic network form genomic data [START_REF] Yamanishi | Protein network inference from multiple genomic data: a supervised approach[END_REF]. We make a brief review of this approach in this subsection. This approach amounts to searching low-dimensional feature spaces derived from both kernels that are maximally correlated during the training phase. The reconstruction of missing entries in G is then obtained by projecting the corresponding points onto the feature space for the kernel K, and computing their inner product in this feature space as an approximation of the kernel G. More formally, let us write a feature u : R d → R for the explanatory variable x, and a feature v : R l → R for the response variable y in the reproducing kernel Hilbert spaces as follows:

u(x) = n j=1 k(x, x j )α j , v(y) = n j=1 g(y, y j )β j , (2)
and use the notation

α = (α 1 , α 2 , • • • , α n ) ⊤ and β = (β 1 , β 2 , • • • , β n ) ⊤ .
The objective is to find features u and v that are as correlated as possible, that is, which maximize the following correlation

ρ = Corr(u, v) = Cov(u, v) V ar(u) 1/2 V ar(v) 1/2 , (3)
where

Cov(u, v) = E(uv) -E(u)E(v) and V ar(u) = E(u 2 ) -E(u) 2 .
For theoretical and practical reasons [START_REF] Bach | Kernel Independent Component Analysis[END_REF] it is better to compute features u and v which maximize the following penalized canonical correlation:

ρ = α ⊤ K tt G tt β α ⊤ (K tt + λ x I) 2 α β ⊤ (G tt + λ y I) 2 β , (4)
where I is an identity matrix, and λ x and λ y are positive regularization parameters, and the matrices K tt and G tt are assumed to be centered. When m different features u (j) and v (j) associated the m largest canonical correlations ρ (j) , j = 1, 2, • • • , m are obtained, they can be merged into feature vectors u(x) and v(y) as u

(x) = (u (1) (x), • • • , u (m) (x)) ⊤ and v(y) = (v (1) (y), • • • , v (m) (y)) ⊤ .
The missing entries in G are then estimated as ĝ(y, y ′ ) = u(x) ⊤ u(x ′ ).

Kernel matrix regression (KMR)

An apparent drawback of the kCCA approach is that the objective function of kCCA is different from that of correctly predicting the values of the kernel G. In particular, by computing features v for the response variable y, the notion of similarity between response variable y is changed. In the problem of kernel matrix completion, we do not want to change the similarity space for the response variable y. We want instead to change the object-object similarity space only for the explanatory variable x to make it fit the the object-object similarity space for the response variable y. In this section we propose a variant of the regression model based on the underlying features in the reproducing kernel Hilbert space by modifying the idea of kernel CCA. The ordinary regression model between an explanatory variable x ∈ R d and a response variable y ∈ R can be formulated as follows:

y = f (x) + ǫ, (5)
where f : R d → R and ǫ is a noise term. By analogy we propose to regard (x, x ′ ) ∈ R d × R d as an explanatory variable and g(y, y ′ ) ∈ R as a response variable in our context. Assuming the underlying feature u(x) ∈ R m in the reproducing kernel Hilbert space, we formulate a variant of the regression model as follows:

g(y, y ′ ) = f (x, x ′ ) + ǫ = u(x) ⊤ u(x ′ ) + ǫ, (6)
where f : R d × R d → R. We refer to this model as kernel matrix regression model. We note that imposing f to be of the form f (x, x ′ ) = u(x) ⊤ u(x ′ ) for some feature u : R d → R m ensures that the regression function is positive definite and the number of dimension m of the feature u is allowed to be infinite.

Following a classical approach in kernel methods, we consider features in the reproducing kernel Hilbert space of the kernel K that possess an expansion of the form:

u(x) = n j=1 k(x, x j )w j , (7)
where w = (w 1 , w 2 , • • • , w n ) ⊤ is a weight vector and n is the number of objects in the training set. When m different features are considered, we express them by a feature vector u as u(x) = (u (1)

(x), u (2) (x), • • • , u (m) (x)) ⊤ .
In order to represent the set of features for all the objects, we define feature score matrices

U t (x) = [u(x 1 ), u(x 2 ), • • • , u(x n ) for the training set and U p (x) = [u(x n+1 ), u(x n+2 ), • • • , u(x N )] ⊤ for the test set.
In the matrix form, we can actually compute the feature score matrices as U t = K tt W for the training set and U p = K pt W for the test set, where W = [w (1) , w (2)

, • • • , w (m) ].
The inner products of the feature vectors between two objects are q(x,

x ′ ) = u(x) ⊤ u(x ′ ).
To represent all the object-object similarities in the feature space, we define the similarity matrix Q as

(Q) ij = q(x i , x j ) = u(x i ) ⊤ u(x j ), 1 ≤ i, j ≤ N.
Splitting the matrix Q into several parts according to the training set, test set and their interaction, we can compute them as follows:

Training set versus Training set:

Q tt = U t U ⊤ t = K tt W W ⊤ K ⊤ tt , (8)
Test set versus Training set:

Q pt = U p U ⊤ t = K pt W W ⊤ K ⊤ tt , (9)
Test set versus Test set:

Q pp = U p U ⊤ p = K pt W W ⊤ K ⊤ pt . (10)
Here we want to find the n × m weight matrix W such that Q tt fits G tt as much as possible. If we set A = W W ⊤ , this problem can be replaced by finding A which minimizes the difference between G tt and Q tt . It means that, this enables us to avoid considerable computational burden for computing W itself, even if m is infinite. Therefore, we attempt to find A(= W W ⊤ ) which minimizes

L = G tt -K tt AK ⊤ tt 2 F , (11)
where • F indicates the Frobenius norm. We can rewrite the above equation in the trace form as

L = tr (G tt -K tt AK ⊤ tt )(G tt -K tt AK ⊤ tt ) ⊤ . ( 12
)
The derivative of L with respect to A is obtained as

1 2 ∂L ∂A = -K tt G tt K tt + K 2 tt AK 2 tt ,
From setting ∂L ∂A = 0, the solution is analytically obtained by

A = W W ⊤ = K -1 tt G tt K -1 tt .
Then, the feature score matrix U can be computed for training set and test set, respectively, as follows:

U t = K tt K -1 tt G 1/2 tt , U p = K pt K -1 tt G 1/2 tt . (13)
Therefore, we can compute the feature-based similarity matrix Q involving the test set as follows: Test set versus Training set:

Q pt = U p U ⊤ t = K pt K -1 tt G tt , (14)
Test set versus Test set:

Q pp = U p U ⊤ p = K pt K -1 tt G tt K -1 tt K ⊤ pt . (15)
By using the Q pt and Q pp , we can predict the missing entries in the kernel matrix G, which correspond to G pt and G pp .

Penalized kernel matrix regression (PKMR)

Here we consider introducing the idea of regularization in the KMR proposed in the previous section. To do so, we attempt to find A(= W W ⊤ ) which minimizes the following penalized loss function:

L = G tt -K tt AK tt 2 F +λP EN (A), (16)
where λ is a regularization parameter and P EN (A) is a penalty term for A defined as follows. Each positive semidefinite matrix A can be expanded as A = n i=1 w i w ⊤ i , where the (w i ) i=1,...,n form an orthogonal basis of eigenvectors. To each w i is associated a feature u i : R d → R by [START_REF] Tsuda | The em Algorithm for Kernel Matrix Completion with Auxiliary Data[END_REF], whose norm in the RKHS of K is given by:

u i 2 RKHS = n j,k=1 w i,j w i,k k(x j , x k ) = tr(w i w ⊤ i K) .
To enforce regularity of the global mapping u, we therefore define the following penalty for A:

P EN (A) = 2 n i=1 u i 2 RKHS = 2 n i=1 tr(w i w ⊤ i K tt ) = 2tr(AK tt ) .
In this case, the optimization problem is reduced to finding A which minimizes

L = tr (G tt -K tt AK ⊤ tt )(G tt -K tt AK ⊤ tt ) ⊤ + 2λtr {AK tt } . ( 17
)
The derivative of L with respect to A is

1 2 ∂L ∂A = -K tt G tt K tt + K 2 tt AK 2 tt + λK tt .
Therefore, the solution of the above penalized optimization problem is obtained by

A = K -1 tt (G tt -λK -1 tt )K -1 tt .
We note that the justification for the penalty used is only valid for positive semidefinite matrices, which will be obtained at least for small enough λ. Therefore, we can compute the feature-based similarity matrix Q involving the test set as follows: Test set versus Training set:

Q pt = U p U ⊤ t = K pt K -1 tt (G tt -λK -1 tt ), (18)
Test set versus Test set:

Q pp = U p U ⊤ p = K pt K -1 tt (G tt -λK -1 tt )K -1 tt K ⊤ pt . (19)
By using the Q pt and Q pp , we can predict the missing entries in the kernel matrix G, which correspond to G pt and G pp .

Relationship with the em-algorithm

For the kernel matrix completion problem, the use of the em algorithm based on information geometry has been proposed [START_REF] Tsuda | The em Algorithm for Kernel Matrix Completion with Auxiliary Data[END_REF]. There the kernel matrix completion problem is defined as finding missing entries that minimize the Kullback-Leibler divergence between the resulting completed matrix and a spectral variant of the full matrix.

It is interesting to observe that the final algorithms between em and KMR are very similar. The em algorithm results in the following equations for estimating the incomplete parts G pt and G pp in G:

Test set versus Training set:

Q pt = K pt K -1 tt G tt , (20)
Test set versus Test set:

Q pp = K pp + K pt K -1 tt K ⊤ pt + K pt K -1 tt G tt K -1 tt K ⊤ pt . ( 21
)
We note that the Q pt of the em-algorithm is equivalent to that of the kernel matrix regression. On the other hand, the Q pp of the em-algorithm is not equivalent to that of the kernel matrix regression. It differs by

K pp + K pt K -1 tt K ⊤ pt .
This stems from the difference of the geometry space between the two methods. The em-algorithm is based on the information geometry, while the proposed KMR is based on the Euclidean geometry.

Experiments

In this section we report an empirical comparison of different methods: 1) direct method, 2) kCCA method, 3) em-method, 4) KMR method, 5) PKMR method applied to the problem of gene network inference and protein 3D structure prediction.

Estimation of missing edges in the metabolic gene network

The metabolic gene network is an important biological network. However, most parts of the metabolic gene network remain unknown, and many enzyme genes are still missing in our current knowledge. Determining new enzyme genes and their position in the metabolic network is an expensive and painstaking process that requires many wet experiments. On the contrary, we can easily obtain various genome-wide genomic datasets representing gene/protein information, such as gene expression data, protein localization data, and phylogenetic profiles. We therefore attempt to predict missing edges in the metabolic gene network by using such genomic data.

We gathered a kernel matrix of the genomic data (consisting of 769 genes) by combining three kernel matrices obtained from three datasets: gene expression data, protein localization data, and phylogenetic profiles, and we regard this matrix as an explanatory kernel matrix K. We used the same datasets and corresponding kernels as those used by [START_REF] Yamanishi | Protein network inference from multiple genomic data: a supervised approach[END_REF]. We obtained a kernel matrix for the gene network from the graph information of the gene network by using the diffusion kernel [START_REF] Kondor | Diffusion Kernels on Graphs and Other Discrete Input[END_REF] with parameter σ = 1, and we regard this matrix as a response kernel G. The kernel matrix K is invertible in this case. The kernel similarity values in G (transformed by the diffusion kernel from the graph of the gene network) are expected to represent the intensity of graphical association between genes, which can be considered as a possibility of the existence of the edge. Therefore, if the gene pairs sharing similarities higher than a threshold, they are predicted to interact with each other.

To compare the performance between different methods, we applied the direct approach, the kCCA, the emalgorithm, the KMR, and the penalized KMR (PKMR) to the gene network prediction. We tested their performance by cross-validation. In each cross-validation iteration, we randomly split the genes in the gold standard data into training set and test set. We learned the model based on the training dataset only and we applied the model to the test set in order to predict the missing edges involving the test set on the metabolic network. We are also interested in the effect of the rate of the test samples against the training samples, so we carried out the same experiment with different percentages of the test samples in the data splitting process in each cross-validation iteration.

As a measure of the performance, we used the AUC score (area under the ROC curve), because the performance depends on the threshold given in advance. The ROC curve is defined as a function of the true positive rates against the false positive rates based on several threshold values. "True positive" means that the predicted gene-pairs are actually present in the gold standard network, while "false positive" means that the predicted gene-pairs are absent in the gold standard network. In the case of the kCCA, we set the regularization parameters λ x and λ y as 0.1 and 0.1, and we used 30 features, as suggested by [START_REF] Yamanishi | Protein network inference from multiple genomic data: a supervised approach[END_REF]. In the case of the PKMR, the regularization parameter λ is optimized by applying the internal cross-validation within the training set with the AUC score as a target criterion, which provides us with λ = 0.1. All the result of the experiments are summarized in Tables 1 and2. Table 1 shows the result in estimating Q pt , while Table 2 shows the result in estimating Q pp . In each table, the rows correspond to the percentage of the training samples against the test samples and the columns correspond to the methods. It appears that the direct approach performs significantly worse than the other supervised learning based methods. It seems that, the kCCA performs worse than the em-algorithm, the KMR and PKMR in estimating Q pt , while the kCCA performs better than the em-algorithm and the KMR, and at competitive level with the PKMR in estimating Q pp . Focusing on the comparison of the performance between the em-algorithm and the KMR, both the em-algorithm and the KMR show the same performance in estimating Q pt , as expected from the mathematical relationship between the em-algorithm and the KMR. On the other hand, the em-algorithm and the KMR behave differently in estimating Q pp , but their performances are at competitive level. The penalized KMR (PKMR) slightly outperforms the other methods in estimating both Q pt and Q pp , suggesting that the introduction of regularization can be meaningful in this context.

Prediction of protein 3D structures from their sequences

Protein 3D structures are strongly associated with evolutionary history and biological functions, compared with protein sequences. Here we attempt to classify proteins into superfamilies based on the structure information. However, the number of proteins whose structures are determined is limited even nowadays and the structure information of most proteins is almost unknown. Therefore, we performed protein classification by predicting missing similarity elements of protein 3D structures.

The sequence similarities are obtained by marginalized kernel [START_REF] Tsuda | Marginalized kernels for biological sequences[END_REF] and the 3D structural similarities are obtained by the result of MATRAS [START_REF] Kawabata | Protein tertiary structure comparison using the Markov transition model of evolution[END_REF]. We used the same datasets and corresponding kernel matrices used in previous work [START_REF] Kim | Protein classification via Kernel Matrix Completion[END_REF], where K corresponds to the similarity matrix for protein sequences and G corresponds to the similarity matrix for protein 3D structures. The kernel matrix K is invertible in this case. We applied the support vector machine (SVM) to the dataset of TIM beta/alpha-barrel protein fold (18 classes, 90 proteins), and conducted one-versus-other supervised classifications for class 1 and class 3, respectively. The reason why we selected the above two classes is that they have more than 10 members in each class. The leave-one-out cross-validation is conducted and the performance is evaluated by using sensitivity and specificity, where sensitivity is defined as #T P/(#F N + #T P ) and specificity is defined as #T P/(#F P + #T P ), respectively. In the case of PKMR, the regularization parameter λ is optimized by applying the internal cross-validation within the training set, and set to be 0.1. 3 and Table 4 show the results of computing sensitivities and specificities depending on the rate of training set for class 1 and class 3, respectively, in the leave-one-out cross-validation experiments. The direct method with 0% means that we use sequence information only, while the direct method with 100% means that we use structure information only. Looking at the tables, the PKMR seems to outperform the other methods in this context especially from the viewpoint of sensitivity. In contrast, the performance seems to be at competitive level across different methods from the viewpoint of specificity.

Discussions and conclusions

In this paper we addressed the problem of filling missing entries in a kernel Gram matrix, given a related full kernel Gram matrix. We attacked this kernel matrix completion problem from the viewpoint of regression. We proposed the kernel matrix regression (KMR) based on the underlying features in the reproducing kernel Hilbert space by modifying the idea of kCCA. Through the development of the KMR, we also clarified the mathematical relationship between the kCCA, the em-algorithm, and the proposed methods in the context of the kernel matrix completion problem. In the experiment on gene network inference and protein 3D prediction, we confirmed that the performance of the KMR is competitive with that of other methods, and we showed that the penalized version of the KMR works the best when an appropriate regularization parameter is chosen.

Table 1 :

 1 Comparison of AUC scores with varying rates of training set: Q tp (training versus test).

	RATE DIRECT KCCA	EM	KMR PKMR
	90 %	0.598	0.840 0.889 0.889	0.892
	80 %	0.570	0.824 0.844 0.844	0.848
	70 %	0.580	0.783 0.805 0.805	0.814
	60 %	0.575	0.780 0.786 0.786	0.801
	50 %	0.579	0.772 0.783 0.783	0.772
	40 %	0.569	0.714 0.760 0.760	0.758
	30 %	0.571	0.682 0.732 0.732	0.738
	20 %	0.565	0.633 0.674 0.674	0.676
	10 %	0.593	0.669 0.672 0.672	0.676

Table 2 :

 2 Comparison of AUC scores with varying rates of training set: Q pp (test versus test).

	RATE DIRECT KCCA	EM	KMR PKMR
	90 %	0.531	0.785 0.766 0.774	0.787
	80 %	0.593	0.727 0.724 0.723	0.743
	70 %	0.602	0.680 0.686 0.700	0.703
	60 %	0.558	0.673 0.683 0.678 0.675
	50 %	0.581	0.661 0.644 0.651	0.662
	40 %	0.569	0.646 0.635 0.635 0.642
	30 %	0.583	0.610 0.621 0.627	0.637
	20 %	0.579	0.587	0.587 0.567 0.576
	10 %	0.568	0.591 0.585 0.573	0.589

Table 3 :

 3 Comparison of sensitivities (left) and specificities (right): Class 1.

	Rate	direct kCCA em KMR PKMR	Rate	direct kCCA em KMR PKMR
	0%	0.47	-	-	-	-	0%	0.66	-	-	-	-
	20%	-	0.82	0.76 0.76	0.88	20%	-	0.45	0.43 0.41	0.42
	40%	-	0.70	0.70 0.70	0.76	40%	-	0.36	0.4	0.36	0.38
	60%	-	0.70	0.88 0.88	0.94	60%	-	0.5	0.53 0.48	0.51
	80%	-	0.76	0.88 0.88	0.88	80%	-	0.81	0.93 0.93	0.93
	100% 0.94	0.94	0.94 0.94	0.94	100%	1.0	1.0	1.0	1.0	1.0

Table 4 :

 4 Comparison of sensitivities (left) and specificities (right): Class 3.

	Rate	direct kCCA em KMR PKMR	Rate	direct kCCA em KMR PKMR
	0%	0.58	-	-	-	-	0%	0.62	-	-	-	-
	20%	-	0.64	0.76 0.76	0.82	20%	-	0.40	0.44 0.34	0.35
	40%	-	0.64	0.82 0.88	0.88	40%	-	0.39	0.53 0.45	0.42
	60%	-	0.76	0.82 0.82	0.88	60%	-	0.56	0.51 0.51	0.53
	80%	-	0.76	0.94 0.94	1.0	80%	-	0.68	0.94 0.94	0.85
	100%	1.0	1.0	1.0	1.0	1.0	100% 0.94	0.94	0.94 0.94	0.94
	Table